Open Problems in Nonsmooth Dynamics

Conference: Open Problems in
Nonsmooth Dynamics
Centre de Recerca Matemàtica
February 1 to 5, 2016
Acknowledgements: The Conference: Open Problem in Nonsmooth Dynamics
is made possible by the generous support from the following agencies: Simons
Foundation, Italian Society for Chaos and Complexity (SICC), Engineering and
Physical Sciences Research Council, wiris, Generalitat de Catalunya, Universitat
Politècnica de Cataluna, Societat Catalana de Matemàtiques and BREUDS.
3
Organizing Committee
Mike Jeffrey, University of Bristol
Alessandro Colombo, Politecnico di Milano
J. Tomás Lázaro, Universitat Politècnica de Catalunya
Josep M. Olm, Universitat Politècnica de Catalunya
Scientific Committee
Bernard Brogliato, INRIA Grenoble
Jaume Llibre Saló, Universitat Autonòma de Barcelona
Tere Martı́nez-Seara, Universitat Politècnica de Catalunya
Gerard Olivar Tost, Universidad Nacional de Colombia
Petri Piiroinen, National University of Ireland Galway
Enrique Ponce Núñez, Universidad de Sevilla
Marco Antonio Teixeira, Universidade Estadual de Campinas
Nathan van de Wouw, Technische Universiteit Eindhoven
Speakers
Mireille Broucke, University of Toronto
Rachel Kuske, The University of British Columbia
Sarah Spurgeon, University of Kent
Gábor Stépán, Budapest University of Technology and Economics
Peter Varkonyi, Budapest University of Technology and
Economics
5
Contents
1. Timetable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2. Abstracts of the Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Mireille E. Broucke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Reach control problem
Rakel Kuske . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Escapes and exits in noisy non-smooth models
17
Sarah K. Spurgeon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The case for discontinuous control theory
18
Gábor Stépán . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Machined surface quality and the non-smooth dynamics of metal
cutting
19
Peter Varkonyi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Painleve’s paradox in multi-contact mechanical systems
19
3. Abstracts of the Contributed Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Mate Antali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Sliding dynamics on codimension-2 discontinuity surfaces
Viktor Avrutin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
On border collisions in m-th iterate map with a large m
Elena Blokhina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
The nonsmooth dynamics of a microelectromechanical energy
harvester
Carles Bonet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Tautologies in the regularization of piecewise-smooth dynamical
systems
Alan Champneys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Nonsmooth dynamics of industrial pressure relief valves
David Chillingworth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Single-impact orbits for an impact oscillator near periodic orbits with
degenerate graze
Zoltan Dombóvári. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Nonsmooth dynamics of milling processes
6
Manuel Dominguez-Pumar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sliding mode control of heterogeneous systems
28
Abdelali El Aroudi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
What ruined the bandcount doubling in an AC-DC boost PFC circuit?
Vasfi Eldem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Mechanism of autonomous switching and the challenges in the
stability of bimodal systems
Mike Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Asynchronous networks and event driven dynamics
39
Laura Gardini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Bifurcation structure in a family of discontinuous linear-power maps
Paul Glendinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dimension, dynamics and bifurcations of attractors in piecewise
linear maps
40
Albert Granados . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The period adding bifurcation in one and n-dimensional piecewisesmooth maps
41
John Hogan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2007 and all that
41
Luigi Iannelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
On dissipative time varying monotone evolution equations
Jun Jiang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Switching sensitive and insensitive responses in a piecewise smooth
nonlinear planar motion systems
43
K. Uldall Kristiansen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
On the interpretation of the piecwise smooth visible-invisible two-fold
singularity in R3 using regularization and blowup
Claude Lacoursière . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Variational time stepping for nonsmooth analytical system dynamics
Julie Leifeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Homoclinic bifurcation in a climate application
Jaume Llibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Averaging theory for computing periodic solutions of nonsmooth
differential systems with applications
7
Tamás G. Molnár . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Higher-order estimation of the amplitude of regenerative machine tool
vibrations
Karin Mora . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Non-smooth Hopf-type bifurcations arising from impact/friction
contact events
49
Gerard Olivar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Challenges from system dynamics to complexity and piecewisedeterministic Markov processes: market modeling
50
Yizhar Or . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Analysis of foot slippage in simple theoretical models of dynamic
legged locomotion in sagittal plane
Petri T Piiroinen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Discontinuity geometry – an approach to discover the landscape
of impact oscillators
Camille Poignard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Oscillations for non monotonic smooth negative feedback systems
bounded by two hybrid systems
Enrique Ponce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The boundary focus in planar Filippov systems: a wolf in sheep’s
clothin
52
Thibaut Putelat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Nonlinear dynamics of localized frictional slip
Tere M. Seara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Regularization of a planar Filippov vector field near a visible-invisible
fold point
David J.W. Simpson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Border-collision bifurcations: Myths, facts and open problems
54
Pascal Stiefenhofer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of nonsmooth periodic orbits in dynamical systems with
discontinuous right hand side
55
Iryna Sushkoa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2D Border collision normal form and smale horseshoe construction
8
Antonio E. Teruel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Folded nodes, canards and mixed-mode oscillations in 3D piecewiselinear systems
Joan Torregrosa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Limit cycles in piecewise planar systems via ET-systems with
accuracy
Catalina Vich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimation of synaptic conductances in a McKean neuron model
57
Marian Wiercigroch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grazing induced bifurcations: Innocent or dangerous?
58
4. Abstracts of the Posters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Elena Bossolini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Non-robustness of non-smooth systems to regularization
Juan Castillo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Global connections in a class of discontinuous piecewise linear
systems
Christian Erazo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamic cell-to-cell mapping for computing basins of attraction in
bimodal Filippov systems
61
Davide Fiore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Incremental stability of bimodal Filippov systems: analysis and
control
62
Wilker Thiago Resende Fernandes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Investigation of isochronous center conditions for a family of vector
fields
64
Ricardo Miranda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chaos in piecewise smooth vector fields on compact surfaces
64
Chara Pantazi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Quadratic systems with a singular curve of degree 3
Zsolt Verasztó . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Hardware-in-the-loop test of stick-slip phenomena: model, analysis,
experiment
9
Si Mohamed Sah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reconstructing non-smooth forces in atomic force micoscopy with
frequency combs
66
5. List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
11
1. Timetable
Monday, February 1
13:30 – 14:00
Registration
14:00 – 14:30
Opening
14:30 – 15:20 Sarah K. Spurgeon, University of Kent
The case for discontinuous control theory
15:20-15:45
Mike Field, Imperial College
Asynchronous network and event driven dynamics
15:45 – 16:15
Coffee break
16:15 – 16:40 Enrique Ponce, Universidad de Sevilla
The boundary focus in planar Filippov systems: a wolf in sheep’s
clothing
16:40 – 17:05 David J.W. Simpson, Massey University
Border-Collision Bifurcations: Myths, Facts and Open Problems
17:05 – 17:30 Julie Leifeld, University of Minnesota
Homoclinic bifurcations in a climate application
18:30 – 20:00
Wine and cheese reception at Hotel Campus
(Sala Martı́ i Franqués)
12
Tuesday, February 2
09:25 – 10:15 Gábor Stépán, Budapest University of Technology and
Economics
Machined surface quality and the non-smooth dynamics of metal
cutting
10:15 – 10:40 Manuel Dominguez-Pumar, Universitat Politècnica de
Catalunya
Sliding mode control of heterogeneous systems
10:40 – 11:10
Coffee break & Poster presentation
11:10 – 11:35 Thibaut Putelat, University of Bristol
Nonlinear dynamics of localized frictional slip
11:35 – 12:00 Karin Mora, University of Paderborn
Non-smooth Hopf-type bifurcations arising from impact/friction
contact events
12:00 – 12:25 Alan Champneys, University of Bristol
Nonsmooth dynamics of industrial pressure relief valves
12:25 – 12:50 Marian Wiercigroch, University of Aberdeen
Grazing induced bifurcations: Innocent or dangerous?
12:50 – 14:30
Lunch break
14:30 – 14:55 Yizhar Or, Technion Israel Institute of Technology
Analysis of foot slippage in simple theoretical models of dynamic
legged locomotion in sagittal plane
14:55 – 15:20 Jun Jiang, Xi’an Jiaotong University
Switching sensitive and insensitive responses in a piecewise
smooth nonlinear planar motion systems
15:20 – 15:45 Zoltan Dombóvári, Budapest University of Technology and
Economics
Nonsmooth dynamics of milling processes
15:45 – 16:15
Coffee break & Poster presentation
16:15 – 16:40 Mate Antali, Budapest University of Technology and Economics
Sliding dynamics on codimension-2 discontinuity surfaces
16:40 – 17:05 Luigi Ianelli, University of Sannio in Benevento
On dissipative time varying monotone evolution equations
17:05 – 17:30 Claude Lacoursière, University of Umea
Variatioinal time stepping for nonsmooth analytical systems
dynamics
13
Wednesday, February 3
09:25 – 10:15 Rachel Kuske, The University of Britisch Columbia
Escapes and exits in noisy non-smooth models
10:15 – 10:40 Gerard Olivar, Universidad Nacional de Colombia
Challenges from system dynamics to complexity and piecewisedeterministic Markov processes: Market modeling
10:40 – 11:10
Coffee break
11:10 – 11:35 Viktor Avrutin, Università degli studi di Urbino
On border collisions in m-th iterate map with a large m
11:35 – 12:00 Laura Gardini, University of Urbino
Bifurcation structure in a family of discontinuous linear-power
maps
12:00 – 12:25 Albert Granados, Technical University of Denmark
The period adding bifurcation in one and n-dimensional
piecewise-smooth maps
12:25 – 12:50 Iryna Sushko, National Academy of Sciences of Ukraine
2D border collision normal form and smale horseshoe construction
12:50 – 14:30
Lunch break
14:30 –15:30
Discussion session
17:00 – 19:00
Guided visit
20:00 – 23:00
Conference dinner
(more information to be given during registration)
14
Thursday, February 4
09:25– 10:15 Mireille E. Broucke, University of Toronto
Reach control problem
10:15 – 10:40 Jaume Llibre, Universitat Autònoma de Barcelona
Averaging theory for computing periodic solutions of nonsmooth
differential systems with applications
10:40 – 11:10
Coffee break
11:10 – 11:35 Paul Glendinning, University of Manchester
Dimension, dynamics and bifurcations of attractors in piecewise
linear maps
11:35 – 12:00 Carles Bonet, Universitat Politècnica de Catalunya
Tautologies in the regularization of piecewise-smooth dynamical
systems
12:00 – 12:25 Tere M. Seara, Universitat Politècnica de Catalunya
Regularization of a planar Filippov vector field near a visibleinvisible fold point
12:25 – 12:50 Kristian U. Kristiansen, Technical University of Denmark
On the interpretation of the piecwise smooth visible-invisible
two-fold singularity in R3 using regularization and blowup
12:50 – 14:30
Lunch break
14:30 – 14:55 Antonio E. Teruel, Universitat de les Illes Balears
Folded nodes, canards and mixed-mode oscillations in 3D
piecewise-linear systems
14:55 – 15:20 Joan Torregrosa, Universitat Autònoma de Barcelona
Limit cycles in piecewise planar systems via et-systems with
accuracy
15:20 – 15:45 Catalina Vich, Universitat de les Illes Balears
Estimation of synaptic conductances in McKean neuron model
15:45 – 16:15
Coffee break
16:15 – 16:40 Abdelali El Aroudi, Universitat Rovira i Virgili
What ruined the bandcount doubling in an AC-DC boost PFC
circuit?
16:40 – 17:05 Camille Poignard, Inria Biocore
Oscillations for non monotonic smooth negative feedback
systems bounded by two hybrid systems
17:05 – 17:30 Vasfi Eldem, Okan University
Mechanism of autonomous switching and the challenges in the
stability of bimodal systems
15
Friday, February 5
09:25 – 10:15 Peter Varkonyi, Budapest University of Technology and
Economics
Painleve’s paradox in multi-contact mechanical systems
10:15 – 10:40 Tamás G. Molnár, Budapest University of Technology and
Economics
Higher-order estimation of the amplitude of regenerative
machine tool vibrations
10:40 – 11:10
Coffee break
11:10 – 11:35 Petri T. Piiroinen, National University of Ireland
Discontinuity geometry – an approach to discover the landscape
of impact oscillators
11:35 – 12:00 David Chillingworth, University of Southampton
Single-impact orbits for an impact oscillator near periodic orbits
with degenerate graze
12:00 – 12:25 Elena Blokhina, University College of Dublin
The nonsmooth dynamics of a microelectromechanical energy
Harvester
12:25 – 12:50 John Hogan, University of Bristol
2007 and all that
12:50 – 13:00 Mike Jeffrey, Program Organizer
Closing Remarks & Opening of the Intensive Research Program
13:00
Lunch
17
2. Abstracts of the Speakers
Mireille E. Broucke
Reach control problem.
Abstract: We discuss a class of control problems for continuous time dynamical systems featuring synthesis of controllers to meet certain logic specifications.
Such problems fall in the area of hybrid systems. Hybrid systems have been studied for some time; unfortunately the area has not delivered all that it promised:
a theory of control synthesis has remained elusive. Some work has been done
at the high level on synthesis of controllers for logic specifications inspired by
discrete event system theory. These approaches do not confront where the true
challenge lies: a (hopefully structural) characterization of the intrinsic limits of
a continuous time control system to achieve a non-equilibrium specification.
We study affine systems and logic specifications encoded as inequality constraints. Mathematically, the model is an affine system defined on a polytopic
state space, and control synthesis typically yields piecewise affine controllers. By
studying this special model, synthesis tools have been recoverable. The core synthesis problem has been distilled in the so-called Reach Control Problem (RCP).
Roughly speaking, the problem is for an affine system xdot = Ax+Bu+a defined
on a simplex to reach a pre-specified facet (boundary) of the simplex in finite time
without first exiting the simplex. The significance of the problem stems from its
capturing the essential requirements of logic specifications: state constraints and
the notion of trajectories reaching a goal set of states in finite-time.
In the talk I will give highlights of nearly 10 years of research on RCP: solvability by affine feedback, continuous state feedback, time-varying affine feedback,
and piecewise affine feedback; an associated Lyapunov theory; a geometric structure theory; and emerging applications.
Contact address: [email protected]
Rakel Kuske
Escapes and exits in noisy non-smooth models.
Abstract: The concept of escaping or exiting is an important one in stochastic
modeling. Exit times, exit distributions, probability of escape routes appear frequently as quantities of interest and are valuable characterizations of stochastic
dynamics, robustness, and sensitivity to noisy fluctuations. This talk reviews
a variety of contexts where it is important to understand the stochastic effects
on escape, comparing and contrasting some well-known situations in smooth dynamics with some basic questions in non-smooth models. Models in biology,
engineering, and the environment are discussed.
Contact address: [email protected]
18
Sarah K. Spurgeon
The case for discontinuous control theory .
Abstract: Discontinuous or Variable Structure Control Systems (VSCS) are
characterised by a suite of feedback control laws and a decision rule. The decision rule, termed the switching function, has as its input some measure of the
current system behaviour and produces as an output the particular feedback
controller which should be used at that instant in time. Within the discipline
of control engineering, the need to ensure the resulting system performance is
robust to changes in the system dynamics as well as to the effects of external
perturbations is key and the discontinuous control paradigm has received a great
deal of attention in the literature largely due to its inherent robustness properties.
This robustness is achieved at the price of a discontinuous control signal which
for some applications may be undesirable. For example, in mechanical systems,
a discontinuous control signal can induce wear of components. For this reason,
in the domain of control applications, considerable efforts have been expended
on achieving similar performance levels via continuous approximation.
In implementing any control system, the issue of how much information can
be directly measured becomes pertinent. Frequently measurement of all the system states may be impractical or even impossible and it becomes of interest to
design dynamical systems which can be used to estimate any unmeasurable system states. Such dynamical systems are called observers and the development
of observers driven by discontinuous injection signals is a very popular current
area of research. The issue of discontinuity in the injection signal is typically not
a problem for any observer and the ability to provide robust state estimates as
well as estimates of unknown input signals or fault signals is extremely useful in
practice.
This presentation will give a general overview of the positive features and
opportunities of discontinuous control and observer theory and will outline some
specific application areas where results have been particularly promising. Specific
results on finite time stabilisation of a perturbed double integrator with jumps in
velocity will then be presented. Rigid body inelastic impacts are considered and
a non-smooth state transformation is employed to transform the original system
into a jump-free system. The transformed system is shown to be a switched
homogeneous system with negative homogeneity degree whose solutions are well
defined. Secondly, a non-smooth Lyapunov function is identified to establish
uniform asymptotic stability of the transformedsystem. The global finite time
stability then follows from thehomogeneity principle of switched systems. A
finite upper bound on the settling time is also computed. Simulation results are
presented to illustrate the presented theory.
Contact address: [email protected]
19
Gábor Stépán
Machined surface quality and the non-smooth dynamics of metal
cutting .
Abstract: Machine tool vibrations are one of the most intricate vibration problems that are often compared to the problem of turbulence in fluid mecanics.
These vibrations cause uncomfortable noise, may damage the edges of cutting
tools or certain parts of themachine tool, but most importantly, they always
have negative effect on the quality of the machined surface of workpieces.
The lecture summarizes the basic types of machine tool vibrations that include
free, forced, self-excited, and even parametrically forced vibrations together with
their different combinations with delayed oscillators. A common feature of these
vibrations is the presence of the non-smooth cutting force characteristics. The
basic concepts are presented on turning processes where the idea of regenerative
effect is introduced. The relation to machined surface quality is demonstrated for
thread cutting. The modelling and the corresponding cutting stability of milling
and especially high-speed milling processes are explained and the development
of the related surface quality parameters are presented. As an inverse application, vibration based experimental methods are also introduced to identify the
nonlinear characteristics of cutting forces. Industrial case studies are used to
demonstrate the results.
Contact address: [email protected]
Peter Varkonyi
Painleve’s paradox in multi-contact mechanical systems.
Abstract: The non-uniqueness and non-existence of a classical instantaneous
solution in mechanical systems with frictional contacts (often referred to as
Painleve’s paradox) have been known for more than a century. However most of
our knowledge is based on the analysis of a few simple examples. Most notably,
hardly anything is known about the generic behavior of systems with multiple
contacts. In my talk, I will outline qualitatively new phenomena associated with
multiple contacts, and refute some common beliefs that are based on the analysis
of the widely known simple examples.
Contact address: [email protected]
21
3. Abstracts of the Contributed Talks
Mate Antali
Sliding dynamics on codimension-2 discontinuity surfaces.
Abstract: In Filippov systems, switching surfaces are analysed thoroughly in
the literature (see e.g. [1]). Mathematically, switching surfaces are codimension1 manifolds in the phase space, where the vector field is discontinuous. Higher
codimensional discontinuity surfaces can also exist, for example at the intersection
of two switching surfaces. This scenario results a codimension-2 manifold in the
phase space, where so-called multiple sliding dynamics can occur (see [2] and [3]).
However, isolated codimension-2 discontinuity sets can also exist if the vector
field is continuous everywhere except on a codimension-2 manifold. A possible
source of this type of discontinuity is the Coulomb friction between the surfaces
of 3D bodies. Several concepts from the Filippov systems can be generalised
to these types of systems, most importantly, the existence of sliding dynamics
can be defined to these codimension-2 discontinuity manifolds. Based on this
idea, the authors recently obtained results for problems connected to the rolling
and slipping of bodies (see [4] and [5]). Bifurcations can also be found in these
systems, which are the codimension-2 analogy of the nonsmooth fold and the
persistence bifurcation of Filippov systems.
References
[1] M. di Bernardo et al. (2008) Piecewise-smooth Dynamical Systems. Springer, London
[2] Mike R. Jeffrey (2014) Dynamics at a Switching Intersection: Hierarchy, Isonomy, and
Multiple Sliding, SIAM J. Appl. Dyn. Syst. 13(3), p. 1082-1105
[3] Luca Dieci, Fabio Difonzo (2015) The Moments Sliding Vector Field on the Intersection
of Two Manifolds, Journal of Dynamics and Differential Equations, 2015, online first
[4] Mate Antali, Gábor Stépán, (2015) Discontinuity-induced bifurcations of a dual-point
contact ball Nonlinear Dynamics 2015, online first
[5] Mate Antali, Gábor Stépán, (2015) Loss of stability in a nonsmooth model of dual-point
rolling, Proceedings of the 24st IAVSD Symposium, Vienna, 17-21 August 2015
Acknowledgements. The research leading to these results has received funding
from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Advanced Grant Agreement n. 340889.
This is a joint work with Gábor Stépán.
Contact address: [email protected]
Viktor Avrutin
On border collisions in m-th iterate map with a large m.
Abstract: Power converters represent one of the standard application fields for
piecewise smooth dynamical systems. Over the years, many results were obtained
22
for models derived from DC/DC converters, while DC/AC converters (inverters)
received less attention. However, in the last few years the interest on inverters
has been significantly increased because of their use in solar (photovoltaic) panel
and wind turbine systems, as well as in the power supply systems and motor
drives of electric and hybrid cars.
From the mathematical point of view, to deal with a generic model of an
inverter is a challenging task. If the behaviour of the inverter during one switching
cycle is described by some map f (which is necessarily piecewise smooth), then to
describe the overall dynamics we need to face the m-th iterate of f, i.e., the map
f m with a high value of m (for practical purposes, values of m between 50 and
5000 are of interest). Due to such values of m, the map f m has by construction
an extremely high number of border points and belongs therefore to a class of
models for which neither analytic results nor suitable numerical techniques are
available in a sufficient extent.
In the present talk some recent results regarding border collisions in the considered class of models is discussed. In particular:
1) We discuss numerical problems of border collision detection in such models
and present a simple but numerically robust technique based on symbolic
dynamics which solves these problems.
2) We report an unusual phenomenon which appears to be characteristic in
the considered class of models: transition to chaos via irregular cascades
of border collisions.
3) Initially, the the parameter domain in which the map f m has a globally
attracting fixed point has been assumed to be uniform and suitable for
practical work. We show that this is not completely true and this domain has a complicated interior structure formed by boundaries defined
by persistence border collisions.
From the applied point of view, the presented results make possible to detect
regions in the parameter space leading to qualitatively different quality of the
output signals of the inverter, ranging from the optimal (in the sense low values of the total harmonic distortion) to completely unacceptable (although still
corresponding to a globally attracting fixed point of f m ). From the theoretical
point of view, we show that border collisions associated with a persistence of a
fixed point are much less trivial than typically assumed and may form complex
structures in the parameter space.
This is a joint work with Zhanybai T. Zhusubaliev, Erik Mosekilde.
Contact address: [email protected]
23
Elena Blokhina
The nonsmooth dynamics of a microelectromechanical energy
harvester .
Abstract: Energy harvesting is the conversion of ambient energy in an environment to electrical energy and then using this electrical energy to power a device
or to extend the life-cycle of a battery [1]. There are numerous ambient energy
sources in an environment that can be used to harvest energy e.g., ambient radio
frequency electromagnetic waves, light, temperature gradient, airflow, mechanical
vibrations or heel strike amongst others [2].
Ambient mechanical vibrations can be found in many environments e.g., car
engines, the beating of a heart, the opening of a door. Vibration energy harvesters
(VEHs), sometimes referred to as kinetic energy harvesters (KEHs) are devices
that convert mechanical energy into electrical energy. A VEH is generally defined
by its transduction method. There are three common types of transducer used
in VEHs: electromagnetic, piezoelectric, and electrostatic/capacitive.
Electrostatic vibration energy harvesters (eVEHs) are very beneficial for lowpowered, wireless applications and are suited to miniaturisation through MEMS
implementation [1]. EVEHs consist of an electrostatic transducer and some form
of conditioning electronics, which implements a desired electromechanical conversion process. The transducer of an eVEH comprises a high-Q resonator (either
linear or nonlinear) coupled with a variable capacitor. A high-level schematic of
an eVEH can be seen in Fig. 1.
Figure 1. Generic schematic for an eVEH. The mechanical resonator can be described by the mass-springdamper equation. The
external driving force, from which the mechanical energy is harvested, is assumed to be sinusoidal. The transducer force Ftran
depends on the architecture of the conditioning circuitry. The directions of the forces Ftran and kx depend on x, where x is the
resonator displacement.
Although an eVEH appears to be relatively a simple system, it can display
some very complex, and very interesting, nonlinear behaviour. This is due to
24
the switched nature of its conditioning electronics. The conditioning electronics
control whether or not a charge is placed on the capacitive transducer upon the
detection of a local extremum of resonator displacement. If there is a charge
on the transducer, then a transducer force affects the dynamics of the system.
If there is no charge on the transducer then there is no transducer force. The
switched nature of the system leads to a discontinuity in the vector fields governing the system dynamics. Such a discontinuity makes this hybrid system not
only nonsmooth but also of Filippov’s type. As a result, discontinuity-induced
behaviour can be seen in the dynamics of eVEHs along with classic nonlinear
behaviour e.g., period-doubling and chaos. This makes eVEHs a novel type of
nonsmooth oscillators.
The discontinuity-induced behaviour that is observed in eVEH dynamics manifests itself in the form of sliding bifurcations and sliding motion. Sliding motion
is a parasitic effect in eVEHs and thus, is unwanted in these systems [3]. The
reason for this is because the sliding motion consists of multiple high-frequency
crossings of the discontinuity switching surface. Every one of these crossings leads
to a detection of a local extrema, which in turn causes the switches in the system
to operate. Thus, sliding motion costs a lot of energy in switching, which is very
detrimental to eVEH performance.
Sliding motion appears and disappears on a trajectory through discontinuityinduced bifurcations or sliding bifurcations [4]. There are four types of sliding
bifurcation reported in [4]. Two of these sliding bifurcations can be observed in
eVEHs: crossing-sliding bifurcations and adding-sliding-bifurcations.
Sliding bifurcations, and therefore sliding motion, appear when a system’s
trajectory interacts with specific regions on a switching surface called sliding
regions [4]. When a trajectory reaches the switching surface, and if the particular
area it contacts on the switching surface is a sliding region, the trajectory then
slides until it hits the boundary of the sliding region, at which point it leaves the
switching surface and evolves in a more regular fashion. As can be seen in Fig.
2, the sliding region boundaries of an eVEH evolve as parameter values of the
system vary. This can result in the appearance of very large sliding regions on the
switching manifold. These very large regions mean that sliding segments appear
on trajectories that also display classic nonlinear phenomena e.g. period-doubling
and chaos. This mixed-scenario interaction of “classic” and discontinuity-induced
nonlinear behaviour is a very interesting aspect of eVEH dynamics [5].
A novel type of sliding behaviour, which has been termed “virtual” sliding
appears in eVEHs that implement the constant-charge energy conversion process
[6]. This is as a result of the transducer force’s dependence on the most recently
detected local maximum of resonator displacement. Virtual sliding explains some
very peculiar behaviour witnessed in eVEH dynamics.
An eVEH is a hybrid system, with a practical application, that displays not
only classic nonlinear behaviour but also discontinuity-induced phenomena e.g.,
sliding motion and the novel virtual sliding. Although this sliding motion is
25
Figure 2. Evolution of eVEH sliding regions on the switching
surface as the acceleration amplitude of external vibrations, Aext,
is varied. As Aext increases, the sliding region boundaries join
together and the sliding region grows. These large sliding regions
allow for the existence of novel, mixed-scenario, nonlinear dynamics
[5].
detrimental to the performance of eVEHs it is still very interesting to study.
No other system we have come across displays such behaviour. Is this novel
phenomenon observable in any other system and can it be used to benefit the
performance of other systems?
This is a joint work with Peter Harte, Dimitri Galayko, Manuel DominguezPumar, Orla Feely.
References
[1]
P. Mitcheson, E. Yeatman, G. Rao, A. Holmes, and T. Green, “Energy harvesting from
human and machine motion for wireless electronic devices”, Proceedings of the IEEE, vol.
96, no. 9, pp. 1457–1486, 2008.
[2] P. Basset, D. Galayko, A. Paracha, F. M. A. Dudka, and T. Bourouina, “A batch-fabricated
and electret-free silicon electrostatic vibration energy harvester”, Journal of Micromechanics and Microengineering, vol. 19, no. 11, p. 115025, november 2009.
26
[3]
P. Harte, E. Blokhina, D. Galayko, and O. Feely, “The nonlinear dynamics of a micro-scale
electrostatic vibration energy harvester”, in Proc. of the 16th Design, Test, Integration and
Packaging of MEMS/MOEMS - DTIP 2014, 1–4 April 2014, Cannes, France, 2014.
[4] M. Di Bernardo, P. Kowalczyk, and A. Nordmark, “Bifurcations of dynamical systems with
sliding: derivation of normal-forms mappings”, Physica D, vol. 170, no. 12, pp. 175–205,
2002.
[5] P. Harte, E. Blokhina, O. Feely, D. Fournier-Prunaret, and D. Galayko, “Electrostatic
vibration energy harvesters with linear and nonlinear resonators”, International Journal
of Bifurcation and Chaos, vol. 24, no. 11, p. 1430030, 2014.
[6] E. Blokhina, D. Galayko, D. Fournier-Prunaret, and O. Feely, “Sliding in a piecewisesmooth dynamical system with a hold-on effect”, Physics Letters A, vol. 378, no. 42, pp.
3085–3092, 2014.
Contact address:
Carles Bonet
Tautologies in the regularization of piecewise-smooth dynamical
systems.
Abstract: The two most commonly used formalisms to define solutions on a
switching surface of non smooth dynamical system are due to Filippov and Utkin.
But it is well known that the Filippov and Utkin methods give different solutions
in some situations,and the precise conditions defining those situations are less
well understood. To decide between the two solutions we must improve the
discontinuous model, but we must be aware of tautologies: both solutions can be
rigorously proven to be valid under different assumptions. We try to clarify the
situation by showing that a class of smooth systems may tend to either Filippov’s
or Utkin’s solutions in different well-defined limits. Essentially we replace an
ideal switch with a boundary layer, and find that a ‘positive’ boundary gives a
continuous flow that tends to Utkin’s solution, while a ‘negative’ boundary layer
induces hysteresis which tends to Filippov’s solution.
This is a joint work with Tere M. Seara, Enric Fossas, Mike R. Jeffrey.
Contact address: [email protected]
Alan Champneys
Nonsmooth dynamics of industrial pressure relief valves.
Abstract: In this talk I shall present a simple 3D autonomous system which
contains impacts, that models the behaviour of a direct spring operated pressure
relief valve. The system undergoes two different kinds of grazing bifurcation, a
chattering sequence and a novel global bifurcation that appears to be the interaction between a Shilnikov-type orbit and a complete chattering sequence. In
addition the model has industrial relevance when modelled in conjunction with
acoustic waves in the inlet piping. Reduced order modelling reveals a 5D model
27
with a rich variety of smooth and nonsmooth bifurcations including robust examples of torus grazing. There remain many open questions about the dynamics
of the model, as well as active research with industrial valve manufacturers in
understanding how to alleviate the effects of the dynamics in practice. This is
joint work with Csaba Hos of the Technical University of Budapest and with
Pentair Valves and Controls, Texas.
Contact address: [email protected]
David Chillingworth
Single-impact orbits for an impact oscillator near periodic orbits with
degenerate graze.
Abstract: An impact oscillator here means a second order ODE with one degree
of freedom x and with T −periodic forcing, and such that the velocity x0 is replaced
by −rx0 with 0 < r < 1 whenever the value x = c (clearance) is attained. A
grazing orbit has x0 = 0 when x = c. A natural question is whether under change
of parameter in the equations or of the clearance c a ‘free’ T −periodic orbit
persists as a single-impact T −periodic orbit after colliding with the clearance.
For impact with nonzero acceleration (codimension 1 bifurcation) a companion
single-impact T −periodic orbit typically exists either before or after the collision
(Nordmark 2001) in a configuration sometimes called a non-smooth fold. In this
talk we consider degenerate cases where the grazing impact acceleration is zero,
and show how simple geometric templates characterise the bifurcation scenarios.
Contact address: [email protected]
Zoltan Dombóvári
Nonsmooth dynamics of milling processes.
Abstract: A non-smooth model of milling processes is presented. A simple
geometric representation of the infinite dimensional phase space is shown where
the time periodic switching surfaces can be traced and the dynamics can be
described in three different states: full cutting, partial fly-over and complete flyover. By means of the corresponding switching surfaces, the fly-over effect is
captured and the related phenomenon of the bistable zone is identified by the
continuation of quasi-periodic tori. These unstable topological structures emerge
from the secondary Hopf bifurcation of the time-periodic orbits corresponding to
the desired stationary milling process. In this model, the limits where these tori
graze the switching surface are traced by using two-parameter continuation. As
the parameters of the linear structural behaviour of the tool/machine tool system
can be obtained by means of standard modal testing, the developed numerical
algorithm provides efficient support for the design of milling processes with quick
28
estimates of those parameter domains where chatter can still appear in spite of
setting the parameters into linearly stable domains.
This is a joint work with Gábor Stépán.
Contact address: [email protected]
Manuel Dominguez-Pumar
Sliding mode control of heterogeneous systems.
Abstract: In heterogeneous systems at least some of the state variables are
not part of an electronic circuit or a mechanical system. In some cases they
can be described by multiexponential models, which are ubiquitous to many
unrelated phenomena such as the thermal response of devices, charge trapping
and detrapping in dielectrics, relaxation times in Magnetic Resonance Imaging
or even chemical reactions in sensors or biology. All these are multicomponent
processes in which different coexisting physical mechanisms generate different
time responses. The best performance is often obtained when working in closedloop configuration, be it because the time response of the system is improved, or
because unwanted drifts in the characteristic of the devices are avoided.
Figure 1. Sigma-delta controls of dielectric charging: a) First
order, b) Second order.
In this paper, we will focus on the analysis of sigma-delta modulators used
to implement closed control loops in heterogeneous systems described by multiexponential models. As practical study-case examples we will present thermal
sigma-delta modulators working with anemometers, [1], and a class of dielectric
charge controllers recently proposed by the authors, [2], [3]. All these control
systems can be connected both to sigma-delta modulators and to sliding mode
controllers. Using the first connection it is possible to study the properties of
the control bitstreams, such as noise shaping. Using the second connection it is
possible to infer the closed-loop time dynamics of the filtered output bitstreams.
It is also possible to analyze the effect of external disturbances such as that introduced by ionizing radiation (X-rays or gamma) in dielectric charge control, [4].
Analytical and experimental results will be provided showing that this double
point of view is possible: first and second order noise shaping characteristic of
29
sigma-delta modulators (Fig. 1), while filtered output bitstreams will be analyzed
and compared with sliding mode analysis.
In the dielectric charge control loop case, two different voltage waveforms,
called BIT0 and BIT1 (see Fig. 2), are applied to a MEMS device with a double
purpose: indirectly sensing the total charge in the dielectric, while also providing
an adequate actuation. The application of a BIT0 waveform serves to increase the
net charge while BIT1 decreases it. A first order control can then be implemented
in which at each sampling period the net dielectric charge is compared with a
given target value and, depending on the result, either a BIT0 or a BIT1 is applied
during the next cycle to reach and keep such target level of net charge. The
interesting point is that the provided actuation is bipolar, and the switching takes
place at the surface of the state variables determined by ‘total charge constant’
and equal to the desired target value. This means that under some conditions a
sliding movement takes place on this surface.
Figure 2. Bipolar voltage symbols used to actuate the MEMS,
BIT0 (a) applies a constant voltage V − < 0 for a times (1 − δ)TS ,
followed by V + > 0 for a short time δTS .IN BIT1 (b), V + is applied
during (1 − δ)TS , then V − for δTS . TS is the sampling period.
The first step in the analysis is to generate ‘equivalent average systems’ in
order to take into account the predetermined affine switching due to the voltage
switching within each BIT0 and BIT1 waveform. In a second step the system is
analyzed using these average systems in order to predict the conditions to have
a sliding motion and the obtained behaviour of the bitstreams.
Fig. 3 shows a dielectric charging experiment in which three different target
charges (related to three C-V voltage shift values) have been programmed as
a function of time. Fig. 4 shows the comparison between the experimental
bitstreams obtained, using both first and second order charging controllers, and
the sliding model analysis proposed in this work. This figure shows an excellent
agreement between the discrete time simulations, taking into account all the
switching during each control symbol, and the sliding mode analysis.
30
Figure 3. Voltage shift as a function of time from a charge control
experiment. The target voltage shifts are +0.5V , −0.75V , 0V . The
experiments parameters are V + = −V − = 4V , δ = 0.2, TS = 2−5s.
Each target value is applied for 48 hours.
Figure 4. Comparison between the sliding mode analysis with the
actual experimental results obtained with first and second order
sigma-delta modulators.
With this approach it is therefore possible to analyze the behaviour of different
heterogeneous systems. The wealth of different dynamics provided by thermal or
chemical systems, among others, will pose a challenge from a theoretical point of
view in the coming future.
This is a joint work with Sergi Gorreta, Teresa Atienza, Elena Blokhina, Joan
Pons-Nin.
31
References
[1] M. Domı́nguez, V. Jiménez, J. Ricart, L. Kowalski, J. Torres, S. Navarro, J. Romeral, and
L. Castañer, A hot film anemometer for the Martian atmosphere. Planet. Space Sci., vol.
56, no. 8, pp. 1169–1179, 2008.
[2] E. Blokhina, S. Gorreta, D. Lopez, D. Molinero, O. Feely, J. Pons-Nin, and M. DominguezPumar, Dielectric charge control in electrostatic MEMS positioners / varactors. IEEE
JMEMS, vol. 21, pp. 559–573, 2012.
[3] S. Gorreta, J. Pons-Nin, E. Blokhina, O. Feely, and M. Dominguez-Pumar, Delta-Sigma
Control of Dielectric Charge for Contactless Capacitive MEMS. Microelectromechanical
Syst. J., vol. 23, no. 4, pp. 829–841, aug 2014.
[4] M. Domı́nguez-Pumar, S. Gorreta, J. Pons-Nin, F. Gómez-Rodrı́guez, and D. M. GonzálezCastaño, Charge induced by ionizing radiation understood as a disturbance in a sliding mode
control of dielectric charge. Microelectron. Reliab., vol. 55, no. 9–10, pp. 6–11, 2015. [Online].
Available: http://linkinghub.elsevier.com/retrieve/pii/S0026271415001766
Contact address: [email protected]
Abdelali El Aroudi
What ruined the bandcount doubling in an AC-DC boost PFC circuit?
Abstract: Power Factor Correction (PFC) AC-DC converters are switched
mode power supplies for regulat- ing an output voltage while providing a near
unity power factor (PF) in the sense that the average input current is proportional to the input voltage. Due to nonlinear effects, PFC AC-DC converters
working in continuous conduction mode can exhibit nonlinear phenomena such
as subharmonic and chaotic oscillations. In this paper, a discrete time model is
used to study the behavior of a boost AC-DC converter. The equation describing
the dynamics of the system is:
dıl
= m1 (t)δ(t) − m2 (t)(1 − δ(t))
dt
where m1 (t) = vg (t)/L and m2 (t) = (V0 − vg (t))/L; δ(t) is the driving signal
generated by T -periodic PWM process; V0 is the output voltage supposed for
simplicity to be constant; and vg (t) = Vg | sin(wt)| is the rectified sinusoidal source
voltage with a constant amplitude Vg and th eline angular frequency w = 2π/Tl .
By integrating (1) over one switching interval the following discrete-time model
is obtained:
(2a)
Vg
kn
Vg
kn + 1
V0
xn+1 = f (xn , kn ) = xn +
cos 2π
−
cos 2π
− dn T
Lw
m
Lw
m
L
(2b)
kn+1 = h(kn ) = (kn + 1) mod m/2, k0 = 0
(1)
32
where
(2c)
dn =


0
rn −xn
 V0
1
if xn < rn
if rn ≤ xn ≤ rn + I0
if xn > rn + I0
kn
rn = gVg sin 2π
(2d)
m
The variable x represents the inductor current, while the variable k corresponds
to the discrete time n restricted to the interval K = [0, m/2 − 1] and therefore
taking integer values kn = n mod m/2.
Figure 5. Bifurcation diagrams obtained numerically from f m/2
at different values of k.
Because of the possible saturation of the duty cycle and its expression given in
Eq. (2d), the phase space R+ ×K of map (2) consists of three partitions separated
by the following two borders
(3a)
s− = {(x, k) | xk = pk }
(3b)
s+ = {(x, k) | xk = pk + I0 }
corrresponding to dn = 0 and dn = 1, respectively.
After validating the correctness of the proposed model by numerical simulations from the circuit- based switched model, bifurcation diagrams for the system
are obtained using a stroboscopic map defined by the (m/2)th iterate of f calculated at a fixed falue of k. The bifurcation diagrams calculated for different
values of k are shown in Fig. 1. We observe a sequence of bifurcations forming a cascade which is similar to a bandcount doubling cascade well known for
piecewise smooth 1D maps. Recall that in continuous maps such a cascade is
formed by merging bifurcations and shows a sequence of chaotic attractors with
K = p · 2i bands, i = 0, 1, 2, . . ., accumulating to a parameter value at which a
p-cycle changes its stability. In our 2D map, a similar scenario can be observed
33
but the bifurcations diagrams calculated at different fixed values of k appear incomplete and exhibit remarkable gaps (see Fig. 1). The location of these gaps
depends on k and at the present, the observed phenomenon is far from being completely understood. It can already be said that the appearance of these gaps is
not related to inaccuracies of numerical simulations but results from interactions
between two oscillations, namely a low frequency input signal and high frequency
switching process. However, the mechanism which causes the bifurcations leading
the number of bands to change is still to be investigated.
Contact address: [email protected]
Vasfi Eldem
Mechanism of autonomous switching and the challenges in the
stability of bimodal systems.
Abstract: Piecewise Linear Systems (PLS) constitute a subclass of switched systems where subsystems are linear, time invariant and the switching is autonomous
(state dependent). Bimodal Piecewise Linear Systems (BPLS) are simplest type
of PLS which consist of two subsystems coupled on a separating hyperplane. In
spite of its simplicity, BPLS exhibit very rich dynamic behavior. For instance,
subsystems may be stable, but BPLS may be unstable. Conversely, subsystems
may be unstable but BPLS may be stable. If the vector field is discontinuous on
the switching plane, then the eigenvectors of subsystems can be changed without
changing the eigenvalues and this change can make BPLS stable or unstable.
Well-posedness i.e., the existence and uniqueness of solutions, is a central issue in BPLS. This issue is Örst resolved by Imura and van der Schaft [1]. An
interesting dynamic behavior encountered in PLS is the existence of trajectories
which change mode infinite number of times in a finite time interval. This paradoxical behavior is called Zeno behavior and investigated in a series of papers by
Çamlibel [4], Çamlibel et. al. [5], and Thuan and Çamlibel [6]. It was shown
that well-posed PLS do not exhibit Zeno behavior. Moreover, well-posedness is
also an essential issue in problems such as stability, stabilizability and feedback
control. Along this line, the reader may refer to Example 13 in Iwatani and Hara
[7], where well-posedness conditions guarantee global asymptotic stability of a
BPLS in R2 . Recently, Şahan and Eldem [8] provided the necessary and sufficient conditions for well-posedness of bimodal piecewise affine systems (BPAS).
It is shown that these conditions induce a joint structure for subsystem matrices
of BPAS in Rn . It is also shown that, in the absence of affine terms, these conditions are equivalent to well-posedness conditions given in Imura and van der
Schaft [1].
Global Asymptotic stability (GAS) of BPLS is studied extensively in the literature. The necessary and sufficient conditions for GAS of BPLS (with continuous
vector fields) in R2 is given by Çamlibel [9, 10]. The same problem (with discontinuous vector fields) is investigated by Iwatani and Hara [7].
34
Stability of BPLS in R3 have also attracted considerable attention in literature
lately. Carmona, Freire, Ponce and Torres [2, 3] have considered the stability of
BPLS in R3 with continuous vector fields. Iwatani and Hara [7] have provided
separate necessary and sufficient conditions for GAS of BPLS in Rn where n > 2.
Recently, Eldem and Şahan [11, 12] and Eldem and Öner [13] introduced a verifiable set of necessary and sufficient conditions for GAS of certain classes of BPLS
in R3 .
In what follows, we outline the approach used in Eldem and Şahan [11, 12] and
Eldem and Öner [13] and point out the challenges which arise in GAS of BPLS
in R3 .
1. Outline of the approach
(1) (Well-posedness) Consider the following BPLS.
(1)
Σ0 : ẋ(t) =
A1 x(t) if cT x(t) ≥ 0
A2 x(t) if cT x(t) ≤ 0
where x, c ∈ R3 , A1 and
in R3×3. Note that the switching plane
A2 are matrices
T
H defined as follows H := {x(t) : c x(t) = 0} . H divides R3 into two open
half-spaces as described below
H+ := {x(t) : cT x(t) > 0} and H− := {x(t) : cT x(t) < 0}.
Since both modes are allowed to be active on the plane H, the issue of wellposedness has to be resolve first. Let us first assume that
Assumption1 (A1): The pairs (cT , A1 ) and (cT A2 ) are observable where only
(cT A2 ) is in observable canonical form. Assumption2 (A2): ker cT ∩ ker (cT A1 ) =
ker cT ∩(cT A2 ) (or equivalenty a31 = 0) and a21 , a32 > 0 (equivalently the system
is well-posed).
In this setup the vector field is allowed to be discontinuous on H and
this fact distinguishes this approach from some of the works in the literature, [2],
[3], [9] and [10].
The assumption A2 induces a geometry in R3 which is discussed in detail
in Eldem and Şahan [8]. Note that since the pairs (cT , A1 ) and (cT , A2 ) are
observable, L : ker cT ∩ ker (cT A1 ) = ker cT ∩ ker (cT A2 ) is a line passing
through the origin which divides H into two open half-planes defined as
P+ : = {x(t) : cT x(t) = 0 and cT Ai x(t) > 0 for i = 1, 2},
P− : = {x(t) : cT x(t) = 0 and cT Ai x(t) < 0 for i = 1, 2}.
35
Similarly, the origin divides the line L into two open half lines defined as
L+ : = {x(t) : cT x(t) = cT Ai x(t) = 0 and cT A2i x(t) > 0 for i = 1, 2},
L+ : = {x(t) : cT x(t) = cT Ai x(t) = 0 and cT A2i x(t) < 0 for i = 1, 2}.
Following the notation used in Imura and van der Schaft [1], let Si denote the
set of initial conditions in R3 where a solution starts and continues in
mode i; for i = 1, 2. Then, it follows that
S1 = H+ ∪ P+ ∪ L+ and S2 = H− ∪ P− ∪ L− .
Furthermore, we have S1 ∪ S2 = R3 and S1 ∩ S2 = {0}, which is equivalent to
well-posedness conditions given in Imura and van der Schaft [1].
(2) (Choice of eigenvectors and solutions) Here, we consider the case where
there is one real eigenvalue and a conjugate pair of imaginary eigenvalues in both
modes. The case where the second mode has only real eigenvalues can be treated
in a similar way as given in Eldem and Öner [13]. Thus, our next assumption is
as follows.
Assumtption 3 (A3): The eigenvalues of Ai ’s are {λi , σi ± jwi } where λi , σ : i
and wi ∈ R and wi > 0.
In this case, system matrices are




2
2
λ2 (σ2 + w2 )
a11 a12 a13 
0 0




 T 

2
2 
(2) A1 = 
a21 a22 a23  , A2 = 1 0 −2σ2 λ2 − (σ2 + w2 ) , c = 0 0 1




0 a32 a33
0 1
2σ2 + λ2
where a32 and a21 > 0.
Let {ri } and {xi ± jyi } (i = 1, 2) denote the real and complex eigenvectors
of Ai . As shown in [11, Lemma 2] the eigenvectors can be chosen uniquely as
follows
cT yi = 0, cT A1 y1 > 0 and cT A2 y2 < 0,
cT xi = cT ri = (−1)‘i + 1,
where
(3)


(a11 −σ1 )2 +w12 +a12 a21
a32 a21




r1 = 


−λ1 −a33
a32
1


(a11 +λ1 )(a11 −σ1 )+a12 a21
a32 a21






 , x1 = 




−σ1 −a33
a32
1
(i = 1, 2),


w1 (a11 +λ1 )
 a32 a21 




 , y1 = 




w1
a32
0





36
and



T
2
2
σ2 λ2
−w2 λ2 
−(σ2 + w2 ) 






−(λ − σ ) −w  .
r2 := 
,
=

2σ
x
y
2
2
2
2
2 

2






−1
0
1
(4)
Then, any initial condition in R3 can be expressed as a linear combination of the
eigenvectors of each mode as δ1 ri +βi xi +γi yi where δi , βi and γi are real numbers.
Thus, one can use two different bases for R3 . The set of trajectories that start
out with initial conditions δi ri where δi > 0, will decay to the origin without
going into the other mode if and only if λi < 0. Therefore, we need to investigate
only the trajectories with nontrivial sinusoidal parts. More specifically, let zi (t)
(i = 1, 2) denote the trajectories which start from Si and continue in Si with
initial conditions such that either βi 6= 0 and/or γi 6= 0. Then, the behavior of
such trajectories in the ith mode can be written as
(5)
zi (t) = Ki {αi exp(λi t)ri + exp(σi t)(sin(θi + wi t)xi + cos(θi + wi t)yi )}, for i = 1, 2
where Ki := (βi2 + γi2 )1/2 > 0, αi :=
γi
. This implies that
(β 2 +γ 2 )1/2
i
δi
(βi2 +γi2 )1/2
and sin θi :=
βi
,
(βi2 +γi2 )1/2
cos θi :=
i
cT zi (t) = Ki {αi exp(λi t) + exp(σi t) sin(θi + wi t)} = Ki exp(λi t){fi (t)}
where fi (t) := αi + exp((σi − λi )t) sin(θi + wi t). Note that if a trajectory starts
form H, then αi = − sin(θi ). Thus, for a trajectory starting from S1 capH, z1 (0)
can be written as z1 (0) = K1 v1 (θ1 ). Here, v1 (θ1 ) is defined as
v1 (θ1 ) := x̂1 sin θ1 + y1 cos θ1
where x̂1 := (x1 − r1 ), −φ1 ≤ θ1 < π − φ1 , and
σ1 − λ1
b1 := cot φ1 :=
.
w1
(3) (First classification of trajectories) The first classification is based on the
following definition.
Definition 1. Let zi (t) be a solution of σ0 with initial condition in Si (i = 1, 2).
If there exists a finite τi > 0 such that cT zi (τi ) = 0 and zi (t) changes mode at
t = τi , then zi (t) is called a transitive trajectory. Otherwise, it is called a
nontransitive trajectory.
If σi − λi > 0, then it turns out that all trajectories in mode i are transitive
except the ones with initial conditions Kri where K > 0 and ri is the real
eigenvector. If σi −λi ≤ 0, then there is a closed cone of nontransitive trajectories
on H and all other trajectories are transitive which constitute an open cone on H.
37
Furthermore, one can define a continuous scalar map Fi (θi ) = θi + wi τi (i = 1, 2)
which represents the map from initial condition to final condition where the
trajectory hits H and changes mode.
(Behavior after mode change) A transitive trajectory starting from the first
mode will hit H and change mode and start to evolve in the second mode. In
order to follow up this evolution it is necessary to change bases. The change of
bases can be represented by another scalar function G1 (θ1 + w1 τ1 ) = θ2 If the
trajectory hits the cone of nontransitive trajectories of the second mode, then
it will always stay in the second mode. Otherwise, it will again hit H, change
mode which can be represented by F2 (θ2 ) = θ2 + w2 τ2 . Furthermore, it is again
necessary to change bases which can be represented by another continuous scalar
function G2 (θ2 + w2 τ2 ) = θ11 . In order to formalize these observations, let
T1 (θ1 ) := G2 (F2 (G1 (F1 (θ1 )))).
Since Fi ’s and Gi ’s are continuous, it follows that T1 is also continuous. Thus, we
have
F1
G1
F2
G2
θ1 −→
θ1 + w1 τ1 −→
θ2 −→
θ2 + w2 τ2 −→
θ11 := T1 (θ1 ).
The function T1 (θ1 ) represents a Poincare map for H. At this stage, a second
classification is in order.
a. the trajectories which change mode finite number of times as
t → ∞,
b. the trajectories which change mode infinite number of times as
t→∞
(a) Note that the trajectories which change mode finite number of times decay
to the origin if and only if λi < 0 for both modes. In view of the this observation,
we need to investigate only the stability of the class of trajectories which change
mode infinite number of times as t → ∞. Since such trajectories change mode at
H, we can restrict our investigation to trajectories which start from H without
loss of any generality.
(5) (Stability) Note that T1 (θ1 ) is defined only for trajectories which change
mode at least two times. For trajectories which changes mode 2k times; we
define T1k (θ1 ) as follows.
T1k (θ1 ) := T1 (T1 (· · · T1 (θ1 ) · · · )) (k times) and T1k (θ1 ) := θ1k .
It turns out that the trajectories which change mode infinite number of times
converge to the trajectories starting from certain directions on H, which are
called fixed directions as defined below.
Definition 2. Let θi∗ be a fixed point of T1 (θ1 ) or equivalently T1 (θ1∗ ) for i =
1, 2. Then, vi (θi∗ ) is called a fixed direction. A fixed direction vi (θi∗ ) is said to
be attractive in an interval Ii containing θi∗ if for any θi in Ii and for every
ε > 0 there exists a positive integer k such that |Tik (θi ) − θI∗ | < ε for i = 1, 2.
38
If Ii consists of only one point θi∗ , then the fixed point is said to be
repulsive.
Furthermore, all the trajectories which change mode infinite number of times
are stable if and only if the trajectories starting from the fixed directions are
stable. Therefore, BPLS are GAS if and only if the following hold.
i ) The real eigenvalues of both modes are negative.
ii ) The trajectories starting from all the fixed directions converge to the origin
as t → ∞.
The stability of the trajectories starting from fixed directions can be assessed by
calculating the convergence rate of each fixed direction. Convergence rates can
only be calculated via a numerical algorithm outlined in Eldem and Şahan []. (6)
The basic issue here is to prove the uniqueness of fixed directions in an given
interval. This is done for only a limited number of classes of BPLS.
This is a joint work with Gökhan Şahan and Işil Öner
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
Imura, J.I., and van der Schaft, A. Characterization of well-posedness of piecewise-linear
systems, IEEE Transactions on Automatic Control, V.45, N.9, 2000, pp.1600-1619.
Carmona, V., Freire, E., Ponce, E. and Torres, F. Bifurcation of invariant cones in piecewise
linear homoge- nous systems, International Journal of Bifurcation and Chaos (IJBC), V.15,
Issue: 8, (2005), pp.2469-2484.
Carmona, V., Freire, E., Ponce, E. and Torres, F., The continuous matching of two stable
linear systems can be unstable, Discrete and Continuous Dynamic Systems, Vol: 16 Issue:
3 pp. 689-703, 2006,
Çamlibel M.K.,Well-posed Bimodal Piecewise Linear Systems do not Exhibit Zeno Behavior, Proc. Of the 17th IFAC Congress Seoul, Korea, (2008), pp. 7973-7978.
Çamlibel M.K., Heemels W.P.M.H., Schumacher J.M., Algebraic necessary and sufficient
conditions for the controllability of conewise linear systems, IEEE Trans. Automatic Control, 53(3), (2008), pp. 762-774
Thuan L.Q., Çamlibel M.K., On the existence, uniqueness and the nature of CarathÈodory
and Filippov solutions for bimodal piecewise affine systems, System and Control Letters,
68, (2014), pp.76ñ85.
Iwatani, Y., and Hara, S. Stability tests and stabilization for piecewise linear systems based
on poles and zeros of subsystems, Automatica, V.42, 2006, pp.1685-1695.
Şahan G. and Eldem V., Well posedness Conditions for Bimodal Piecewise Affine Systems,
Systems & Control Letters, 2015, Vol.83, pp.9-18.
Çamlibel, M. K., Heemels, W. P. M. H., & Schumacher, J.M. (2003). Stability and controllability of planar bimodal complementarity systems. 42th IEEE Conference on Decision
and Control, Hawaii, USA, 9-12 Dec 2003 (pp. 1651 - 1656 ).
Çamlibel, M. K., Heemels, W. P. M. H., & Schumacher, J. M. (2008). A full characterization of stabilizability of bimodal piecewise linear systems with scalar inputs. Automatica,
44 (5), 1261ñ1267.
Eldem V., Şahan G., Structure and Stability of Bimodal Systems in R3 : Part I, Applied
and Computational Math. An Int. Journal, (2014),Vol. 13, No: 2, pp. 206-229.
39
[12] Eldem V., and Şahan G., The Effect of Coupling Conditions on the Stability of Discontinuous Bimodal Systems, submitted, under review.
[13] Eldem V. and I. Öner, A note on the stability of bimodal systems in R3 with discontinuous
vector fields, International Journal of Control, Vol 88, No. 4, pp. 729-744, 2015.
Contact address: [email protected]
Mike Field
Asynchronous networks and event driven dynamics.
Abstract: Real-world networks in physics, biology and technology often exhibit
dynamics that cannot be satisfactorily reproduced using network models given
by smooth dynamical systems and a fixed network topology. Asynchronous networks constitute a framework for the study of network dynamics where nodes can
evolve independently of one another, be constrained, stop, and later restart, and
where the interaction between different components of the network may depend
on time, state, and stochastic effects. Typically, dynamics is piecewise smooth
and associations can be made with Filippov systems. In this talk, we emphasize the notion of a functional asynchronous network, discuss the phenomenon of
dynamical locks, and present a theorem about the spatiotemporal factorization
of the dynamics for a class of deadlock free functional asynchronous networks of
feedforward type. This results allows a reductionist and modular approach to
dynamics on complex networks. We conclude by describing some of the interesting questions that arise in the bifurcation theory of functional asynchronous
networks.
This is a joint work with Christian Bick, Exeter.
Contact address: [email protected]
Laura Gardini
Bifurcation structure in a family of discontinuous linear-power maps.
Abstract: The piecewise smooth map de.ned by two functions, fL (x) and fR (x),
as follows:
(
fL (x) = ax + µ if x ≤ 0
(1)
x → fµ (x) =
fR (x) = bxz + µ if x > 0
where a, b, z are real parameters and µ > 0; has been investigated in the continuous case z > 0 as it is related to several applied systems. We recall that the
case z = 1/2 leads to the square-root nonlinearity typical in Nordmark’ systems
and grazing bifurcations [8]. The piecewise linear case with z = 1 leads to the
continuous skew tent map, whose dynamics are now well known (see e.g. [3,
11]). The power z = 3/2 was considered in [2] and in [1] where the normal-form
40
mapping of sliding bifurcations is derived (leading to the map in (1) with power
z = 3/2, z = 2 and z = 3, related to different cases of sliding bifurcations).
In all the cases mentioned above, the power z takes positive values. However,
also the case with real power z < 0 has been recently analyzed, and this leads
to a discontinuous map, with a vertical asymptote, whose dynamic properties
and bifurcations are very much different from those occurring in the continuous
case. In discontinuous systems the classi.cation of the possible different results
of a BCB is still to be investigated, as well as the use of the PWL map as a
normal form, and especially in maps with a vertical asymptote new phenomena
arise, which are still to be understood. It was .rst considered in [9] where, besides
the cases z > 0, the authors extend the analysis to the discontinuous case with
z < 0. The particular case with z = −1/2 is also considered in [10]. However,
the main results in the case z < 0 have been shown in some recent papers ([4, 6,
7, 5]). It is easy to see that system (1) can be reduced to the study of the cases
with µ = 1, µ = −1 or the simplest one with µ = 0. In the cited works the case
with µ = 1 is considered, and it is shown that the bifurcation structure has still
open problems. We shall propose those problems as well as those occurring in
the interesting case with µ = −1.
References
[1] di Bernardo M Kowalczyk P and Nordmark A B 2002 Physica D vol 170 p 170–175.
[2] Dankowicz H and Nordmark A B 2000 Physica D vol 136 280–302.
[3] Y.L. Maistrenko, V.L. Maistrenko, and L.O. Chua, Cycles of Chaotic Intervals in a TimeDelayed Chua.s Circuit, Int. J. Bifurcation Chaos 3 (1993), p. 1557–1572.
[4] Makrooni R Abbasi N Pourbarat M and Gardini L 2015 Chaos, Solitons & Fractals vol 77
p 310–318.
[5] Makrooni R Gardini L and Sushko I 2015 Int. Journal of Bifurcation and Chaos (to appear)
[6] Makrooni R Khellat F and Gardini L 2015 Journal of Difference Equations and Applications
DOI:10.1080/10236198.2015.1045893.
[7] Makrooni R Khellat F and Gardini L 2015 Journal of Difference Equations and Applications
DOI:10.1080/10236198.2015.1046855.
[8] Nordmark A B 1997 Physical Review E vol 55 p 266–270.
[9] Qin Z Yang J Banerjee S and Jiang G 2011 Discrete and Continuous Dynamical System,
Series B vol 16 p 547–567.
[10] Qin Z Zhao Y and Yang J 2012 Int. Journal of Bifurcation and Chaos vol 22 p 1250112.
[11] Sushko I and Gardini L 2010 Int. Journal of Bifurcation and Chaos vol 20 p 2045–2070.
Contact address: [email protected]
Paul Glendinning
Dimension, dynamics and bifurcations of attractors in piecewise
linear maps.
Abstract: Piecewise linear maps can have bifurcations to chaotic attractors
which have no obvious analogues in the smooth setting. Some of these will be
41
discussed with a particular emphasis on the structure of attractors, the existence
of high dimensional attractors and open questions.
Contact address: [email protected]
Albert Granados
The period adding bifurcation in one and n-dimensional piecewisesmooth maps.
Abstract: The period adding bifurcation, widely observed and reported in the
literature, typically appears in orientation preserving one-dimensional maps undergoing a discontinuity. Such maps are obtained in applications where nonsmooth approach is desirable, such as neuroscience, power electronics, control
theory and engineering mechanics.
Despite the novelty provided in the recent years by the non-smooth perspective,
such bifurcation scenarios appear to be very similar to those reported in the past
in the context of homoclinic bifurcations and circle maps. In this talk we fill the
gap between both perspectives. We will review some theory developed in the late
80’s and extend it to provide sufficient conditions for the occurrence of the period
adding bifurcation in one-dimensional piecewise-smooth maps.
Finally, we discuss how these results can be used to study bifurcations of orientation preserving piecewise-smooth maps in Rn .
This is a joint work with Lluı́s Alsedà, Gemma Huguet and Maciej Krupa.
Contact address: [email protected]
John Hogan
2007 and all that.
Abstract: The last CRM meeting on Nonsmooth Systems was held JanuaryMarch 2007, organised by Mario di Bernardo, Gerard Olivar, Enric Fossas and
me. In this talk, I will look back at what we did in those heady days of youth,
what has happened since and speculate about what might happen in the next 9
years.
Contact address: [email protected]
Luigi Iannelli
On dissipative time varying monotone evolution equations.
Abstract: In this talk we are going to discuss about a particular class of time
dependent evolution equations defined over a finite dimensional space, let’s say
Rn . Specifically, we consider a time invariant dynamic linear system Σ that has
42
some set-valued feedback interconnection and that is forced by exogenous inputs.
The linear system is the following:
(1a)
ẋ(t) = Ax(t) + Bz(t) + u(t)
(1b)
w(t) = Cx(t) + Dz(t) + v(t),
where x ∈ Rn is the state, (u, v) ∈ Rn × Rm are the exogenous inputs and
z, w ∈ Rm are the external variables that are interconnected through a set-valued
mapping M : Rm ⇒ Rm
w ∈ M(−z).
(2)
The class of systems we deal with is such that the following assumptions are
satisfied:
H1 The linear system Σ(A, B, C, D) is passive.
H2 The operator M is maximal monotone(see [1]).
H3 The exogenous inputs u(t), v(t) are bounded signals defined over R+ .
The special case with v(t) ≡ 0 has been investigated in [2]. When v(t) 6= 0
the feedback interconnection between w and z depends on time and, thus, the
dynamic evolution is dictated by a time varying set-valued mapping. It is not
difficult to show that the system (1) can be rewritten in the following way
(3)
ẋ(t) ∈ −Ht (x) + u(t),
where
(4)
Ht (x) = −Ax + B(M + D)−1 (Cx + v(t)),
and
(5)
Cdom(Ht ) + v(t) ⊆ im(M + D) ∀t
Note that for any fixed time instant t̄ assumptions 3 and 3 imply Ht̄ being a setvalued maximal monotone operator (m.m.o). In this sense dynamics of system (3)
are described by a time varying dissipative evolution equation. Such class of quite
general evolution equations has been investigated by Vladimirov in [3] and by
Kunze and Marques in [4]. However, since they deal with very general evolution
equations, their results require quite strong assumptions on the behavior of the
time-varying operator Ht that cannot be easily proved in general. On the other
hand system (1) represents a wide class of systems of much interest in engineering
applications. In particular we would like to focus on the power electronics field
where all passive linear circuits interconnected with nonlinear solid state devices
(like diodes, transistors, MOSFETs, etc.) whose characteristics are monotone
functions that can be “idealized” (or better analyzed) as monotone set-valued
mappings, can be cast in such a framework.
43
In this talk we will exploit the particular structure of the time-varying m.m.o.
Ht with the aim of studying the following time-stepping scheme:
(6a)
xhk = (I − hA)−1 xhk−1 + h(I − hA)−1 Bzkh + h(I − hA)−1 uhk
(6b)
zkh ∈ −(M + Gh )−1 C(I − hA)−1 xhk−1 + hC(I − hA)−1 uhk + vkh ,
with
(7)
Gh = D + hC(I − hA)−1 B.
It will be shown that when the least norm solution is chosen in the feasible
set of (6b), under the further assumption of linear growth behavior of the M
operator1, the time stepping scheme (6a) is consistent and sequences {xhk }, {zkh }
converge to a solution of the dissipative evolution equation (3) as the time step
h goes to zero.
This is a joint work with Kanat M. Çamlibel.
References
[1] R. T. Rockafellar and R. Wets, Variational analysis. Berlin: Springer Verlag, 1998.
[2] K. M. Çamlibel and J. M. Schumacher, Linear passive systems and maximal monotone mappings Mathematical Programming, 2014. [Online]. Available: http://link.
springer.com/article/10.1007/s10107-015-0945-7
[3] A. A. Vladimirov, Nonstationary dissipative evolution equations in a Hilbert space. Nonlinear Analysis: Theory, Methods & Applications, vol. 17, no. 6, pp. 499–518, Jan. 1991.
[4] M. Kunze and M. D. P. M. Marques, BV Solutions to Evolution Problems with TimeDependent Domains. Set-Valued Analysis, vol. 5, no. 1, pp. 57–72, 1997.
Contact address: [email protected]
Jun Jiang
Switching sensitive and insensitive responses in a piecewise smooth
nonlinear planar motion systems.
Abstract: In this work the response characteristics of two-degree-of-freedom
non-autonomous piecewise smooth nonlinear system, which consists of a linear
subsystem and a nonlinear subsystem, are investigated. The piecewise smooth
system is used to model an isotropic rotor-to-stator rubbing systems and possesses some specific features: (1) the swithching surface is defined as a deflection
magnitude of two displacement coordinates of the system; (2) when a periodic
trajectory of the system begins to touch the switching surface, it touches switching surface at all points to make it different from gazing in the usual non-smooth
systems; (3) there is no periodic motion formed by crossing the switching surface
between the two subsystems. So some techniques developed for the bifurcation
analysis of equilibriums or periodic solutions to the non-smooth systems are not
1Essentially,
it means that the least norm solution in M(w) is linearly bounded by kwk.
44
directly applicable to the present system. Thus, according to the dynamical characteristics of subsystems, this work tries to classify the parameter regions into
the switching sensitive and insensitive regions. In the switching intensitive regions, the responses of the whole non-smooth system can be determined through
the analysis of the responses of the two subsystems. In the switching sensitive
regions, the occurrence of some responses of the whole non-smooth system can
be explained based on the dynamical characteristics of subsystems, but the occurrence of some other responses is not well understood.
This is a joint work with Ling Hong.
Contact address: [email protected]
Kristian U. Kristiansen
On the interpretation of the piecwise smooth visible-invisible two-fold
singularity in R3 using regularization and blowup.
Abstract: In this talk, we demonstrate how geometric singular perturbation theory and blowup methods can be applied to provide a deterministic interpretation
of forward ambiguity in piecewise smooth systems. In particular, for the visibleinvisible two-fold we will show that the Sotomayor and Teixeira regularization
X , under generic conditions, possesses a limit cycle Γ attracting a large set of
initial conditions. Blowup allow us to resolve the singular limit Γ0 and therefore
obtain a closed cycle for the piecewise smooth system that passes through the
two-fold singularity. The regularization therefore ”selects” a distinguished orbit
among all the candidates. The result is independent of the monotonic regularization function φ ∈ C k , k < ∞. In addition, we will discuss future research
directions.
This is a joint work with S. John Hogan.
Contact address: [email protected]
Claude Lacoursière
Variational time stepping for nonsmooth analytical system dynamics.
Abstract: We present a discrete-time Lagrangian description nonsmooth systems dynamics which includes multibody dynamics subject to frictional contacts,
electronics and hydraulics. For the latter two domains, nonsmooth components
include diodes, transistors, op-amps, switches and relays, as well as check, relief and load sensing valves. The resulting time-stepping scheme requires solving
Linear Complementarity Problems. Though we can show that some of these are
P0 and that others are copositive, as is the case for dry friction for rigid bodies [Anitescu and Potra(1997)], some appear to be neither. The existence of a
solution is therefore undetermined at this time. Similar non-smooth models have
been presented previously [Acary and Brogliato(2008)] but not from the unified
framework of analytical systems dynamics [Layton(1998)] principle as done here.
45
Using q and q̇ as generalized coordinates and velocities of the entire system, we
write T (q, q̇) and U (q) for total kinetic and potential energy, and R(q, q̇) and dissipation potentials. The latter are assumed sub-differentiable and generate forces
f = −∂R/∂ q̇ and dissipate at the rate q̇ · f . Holonomic constraints g(q) = 0 with
Jacobian G = ∂g/∂q can now be used to couple subsystems such as in the case
of a hydraulic piston where the stroke displacement is proportional to the volume of fluid in the chamber, subject to a maximum displacement. We also have
non-holonomic constraints A(q, q̇)q̇ = 0, where we understand = and 5 as componentwise equalities or inequalities as applies. Constraints are then represented
with a Lagrangian term λT g(q) and pseudo-potentials β̇ T A(q, q̇)q̇. Defining new
generalized coordinages including these variables with q̃ T = (q T , λT , β T )T , the
latter two being massless, d’Alembert’s principle then reads
Z T
Z T
˙
˙ +
˙ = 0, where f (q̃, q̃)
˙ = − ∂R(q̃, q̃) ,
(1) δ
dsL(q̃, q̃)
dsδ q̃(s)f (q̃, q̃)
∂ q̃˙
0
0
˙
and where δ q̃ is an infinitesimal variation of the trajectory (q̃(t), q̃(t)),
t ∈ [0, T ]
satisfying the inequality constraints. The Differential Algebraic Equations (DAE)s
of motion are well known but can be singular such as in the case of electrical systems since only inductance have non-zero “mass”. Our discretization solves that
problem.
We use a discrete-time variational integrators [Marsden and West(2001)] consisting of discretizing Eqn. (1) directly. This is also valid for the nonsmooth
case [Leine et al.(2009)Leine, Aeberhard, and Glocker]. To do this, the following
is used.
Z T
N
N Z (k+1)h
X
X
˙ =
˙ =
L d (q̃k , q̃k+1 ), and
dq̃L(q̃, q̃)
S[q̃] = dsL(q̃, q̃)
kh
0
k=0
k=0
(2)
Z
h
(+)
(−)
dsf · δ q̃(s) = fd (q̃0 , q̃1 )δ q̃0 + fd (q̃0 , q̃1 )δ q̃1
0
Performing variations of qi , i = 0, 1, . . . , N , the discrete-time equations are
(3)
(+)
(−)
D1 L d (qk , qk+1 ) + D2 L d (qk−1 , qk ) + fd (qk , qk+1 ) + fd (qk−1 , qk ) + GT λ + AT ν = 0
0 5 λ ⊥ g(qk+1 ) = 0, and 0 5 ν ⊥ A(qk+1 , qk )
(qk+1 − qk )
= 0.
h
Rh
(−)
(+)
0
Using q̇ = (qk+1 −qk )/h, 0 dsU (q) ≈ hU ( q1 +q
) fd (q1 , q0 ) = 0, and fd (q0 , q1 ) =
2
−qk )/h)
− ∂R(qk+1 ,(q∂k+1
, we obtain semi-implicit integrators and in such cases, we
q̇
have “numerical mass” for all elements. Pure potential elements such as capaci2 2 U (q)
tors now have pseudo masses M̃p = h4 ∂∂q∂q
T and purely dissipative elements such
46
2
as resistors have M̃d = h ∂∂q̇∂Rq̇T . The stepping equations then become


 

T
T
 M̃ −Gk −Ak  q̇k+1  M q̇k + hf 


 

G 0




 , with
=
0
0  λ  
ζ
 k



 

(4)
Ak0
0
0
ν
ρ
0 5 λ ⊥ ζ =, 0 5 ν ⊥ ρ = 0 and
qk+1 = qk + hq̇k+1 .
Analysis of convergence, stability as well as additional details of regularization
and constraint stabilization are found elsewhere [Lacoursière(2007)].
Spice
Spice
Nonsmooth
Spice
Nonsmooth
Figure 1. Diode bridge simulations.
Nonsmooth
Introducing constitutive laws c(q̃) = 0 in this framework is done via halfLegendre or Frenet [De Saxce and Feng(1998)] transforms which yield generally
(5)
R = β̇c(q̃), with 0 5 β̇ ⊥ c(q̃) = 0.
Transforming standard constitutive laws to this form is straight forward. The
main issue now is to understand how to model different components and to prove
whether or not the resulting complementarity problem in Eqn. (4) is solvable.
This is guaranteed if M̃ is symmetric and positive definite and Ak0 = Ak and
Gk = Gk0 , but this symmetry is broken in active electronics components for
instance.
To illustrate the multidomain couplings, consider Faraday’s law for an ideal
motor. The back EMF is vE = βω where ω is the angular velocity of the motor,
and β is a constant. The power dissipated in the electrical circuit is then Rf i2 /2
where Rf is the armature resistance and i is the current. For power balance
we need R = i(βω − v0 ) + Rf i2 /2 = 0. Following the previous rules, we have
the necessary equations coupling the two systems, the torque then being τ =
∂R/∂ω = βi as needed.
With regards to feasibility, we show that the Bipolar Junction Transistor (BJT)
can be modeled in terms of two coupled simple diodes using the Ebers-Moll
Spi
Non
smo
47
model. The change of variable however breaks the symmetry between the Kirchhoff’s current and voltage law matrices. This is nevertheless P but not copositive. Given that dry frictional contact problems are solvable because they have
a copositive formulation [Anitescu and Potra(1997)], questions remain as to the
existence of solutions. A numerical validation of our techniques is presented elsewhere [Sjöström(2012)]. Below is but just one result involving a diode bridge
modeled in this fashion showing the advantage of the method.
We will present several non-smooth, non-ideal models of electronic, mechanical,
and hydraulic components, some of which requires piecewise linear complementarity formulations, as well as describe properties of the corresponding complementarity problems. We also discuss practical issues related to nonsmooth electrical circuit simulations where there is a possibility that a switching component
disconnects the circuit into parts.
The advantage of our system dynamics formulation is to provide a natural
way to couple subsystems via kinematic boundary condition and power transfer
balance, and to allow systematic use of variational time-stepping methods for
nonsmooth systems.
This is a joint work with Tomas Sjöström.
References
[Acary and Brogliato(2008)] Vincent Acary and Bernard Brogliato. Numerical Methods for
Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics, volume 35 of
Series:Lecture Notes in Applied and Computational Mechanics. Springer-Verlag, 2008.
[Anitescu and Potra(1997)] Mihai Anitescu and F. A. Potra. Formulating dynamic multi-rigidbody contact problems with friction as solvable linear complementarity problems. Nonlinear
Dynamics, 14:231–247, 1997.
[De Saxce and Feng(1998)] G. De Saxce and Z. Q. Feng. The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms. Mathematical and Computer Modelling, 28(4–8):225–245, AUG 1998. ISSN
0895-7177.
[Lacoursière(2007)] Claude Lacoursière. Ghosts and Machines: Regularized Variational Methods for Interactive Simulations of Multibodies with Dry Frictional Contacts. PhD thesis,
Dept. of Computing Science, Umeå University, June 2007.
[Layton(1998)] Richard A. Layton. Principles of Analytical System Dynamics. Mechanical Engineering Series. Springer-Verlag, Berlin, 1998.
[Leine et al.(2009)Leine, Aeberhard, and Glocker] R. I. Leine, U. Aeberhard, and C. Glocker.
Hamilton’s principle as variational inequality for mechanical systems with impact. Journal of Nonlinear Science, 19(6):633–664, Dec 2009. ISSN 0938-8974. doi: {10.1007/
s00332-009-9048-z}.
[Marsden and West(2001)] J. E. Marsden and M. West. Discrete mechanics and variational
integrators. Acta Numer., 10:357–514, 2001. ISSN 0962-4929.
[Sjöström(2012)] Tomas Sjöström. Discrete time variational mechanics of multidomain systems:
Applications to coupled electronic, hydraulic, and multibody systems. Master’s thesis, Dept.
of Physics, Umeå university, 2012.
Contact address: [email protected]
48
Julie Leifeld
Homoclinic bifurcation in a climate application.
Abstract: Collision of equilibria with a splitting manifold has been locally studied, but might also be a contributing factor to global bifurcations. In particular
a boundary collision can be coincident with collision of a virtual equilibrium
with a periodic orbit, giving an analogue to a homoclinic bifurcation. This type
of bifurcation is demonstrated in a nonsmooth climate application. There are
fundamental topological differences between the nonsmooth bifurcation and a
classical homoclinic bifurcation, implying that the structure seen in the model is
not a limiting case of a smooth homoclinic bifurcation. I will describe the nonsmooth bifurcation structure, highlighting these differences. I will also discuss a
smooth bifurcation structure for which the nonsmooth homoclinic bifurcation is
a limiting case.
Contact address: [email protected]
Jaume Llibre
Averaging theory for computing periodic solutions of nonsmooth
differential systems with applications.
Abstract: Recently I with some coauthors have extended some of the results of
the classical averaging theory for studying the periodic solutions of the smooth
differential systems to the nonsmooth ones. In this talk we shall present some of
these new results together with several applications. The talk will use results of
the following papers:
- J. Llibre, D.D. Novaes and M.A. Teixeira, On the birth of limit cycles for
non-smooth dynamical systems, Bull. des Sciences Mathématiques 139
(2015), 229-244.
- J. Llibre, A.C. Mereu and D.D. Novaes, Averaging theory for discontinuous piecewise differential systems, to appear in J. Differential Equations.
- J. Llibre, M.A. Teixeira and I.O. Zeli, Birth of limit cycles for a class of
continuous and discontinuous differential systems in (d+2)-dimension, to
appear in Dynamical Systems: An international J. B.R. de Freitas,
- J. Llibre and J.C. Medrado, Limit cycles of continuous and discontinuous
piecewise-linear differential systems in R3 , preprint, 2015.
- J. Llibre and D.D. Novaes, On the continuation of periodic solutions in
discontinuous dynamical systems, preprint, 2015.
Contact address: [email protected]
49
Tamás G. Molnár
Higher-order estimation of the amplitude of regenerative machine
tool vibrations.
Abstract: Chatter is a large-amplitude self-excited vibration during machining,
which involves intermittent loss of contact between the tool and the workpiece.
Correspondingly, the dynamics of the cutting process involves switching between
two vector fields representing cutting and free oscillation of the tool. Chatter
develops from a subcritical Hopf bifurcation, by which an unstable periodic orbit
emerges in the vicinity of the linearly stable equilibrium. The amplitude of the
unstable periodic orbit increases for decreasing bifurcation parameter. At a critical value of the parameter, the periodic orbit touches the switching line and a kind
of grazing bifurcation takes place. Our aim is to improve the analytical approximation derived in [1] for this grazing bifurcation point by providing higher-order
estimations for the amplitude of the unstable periodic orbit as a function of the
bifurcation parameter. Acknowledgements: This work was supported by the Hungarian National Sci- ence Foundation under grant OTKA-K105433. The research
leading to these results has received funding from the European Research Council
under the European Union’s Seventh Framework Programme (FP/2007-2013) /
ERC Advanced Grant Agreement n. 340889.
This is a joint work with Tamás Insperger, Gábor Stépán.
References
[1] Dombóvári, Z., Barton, D. A. W., Wilson, R. E., and Stépán, G. (2011) On the global
dynamics of chatter in the orthogonal cutting model. International Journal of Non-Linear
Mechanics, 46(1), pp. 330–338.
Contact address: [email protected]
Karin Mora
Non-smooth Hopf-type bifurcations arising from impact/friction
contact events.
Abstract: The analysis of single degree of freedom systems with impacts is key
to understanding the effects of discontinuities on the dynamics of such systems.
However, higher dimensional systems can exhibit more complex dynamics as will
be shown in this talk, exemplified by a mechanical problem.
The novel dynamics observed in a nonlinear system comprised of a rotor colliding with the rotor housing will be presented. The focus of this analysis is the
investigation of the discontinuity induced bifurcations arising in such systems.
The simplified Föppl/Jeffcott rotor with clearance and mass unbalance is modelled by a two degree of freedom impact-friction oscillator. In such systems the
reset map describing the impact law introduces nonlinearities and coupling terms.
In experiments, two types of motion have been observed: no contact and repeated
50
instantaneous impact. How these are affected by the system’s parameters, such
as damping and stiffness, is revealed by using analytical and numerical piecewisesmooth dynamical systems methods. The global analysis exploits the rotational
symmetry to study periodic orbits with and without impact and their coexistence. In fact, by studying the impact map, we show that these types of motion
arise at a novel non-smooth Hopf-type bifurcation from a boundary equilibrium
bifurcation point for certain parameter values. We present criteria for the existence of smooth and non-smooth bifurcations, which are an essential step towards
achieving a better understanding of systems with discontinuities in general.
This is a joint work with Chris Budd (University of Bath, UK), Paul Glendinning
(University of Manchester, UK), and Patrick Keogh (University of Bath, UK).
Contact address: [email protected]
Gerard Olivar
Challenges from system dynamics to complexity and piecewisedeterministic Markov processes: market modeling .
Abstract: This talk proposes a general economics model for the supply and
demand of a commodity in a domestic market, when investments are required
for supporting it. Starting from System Dynamics, we recover a well-known
model. Then, we improve the mathematical equations in order to be precise at
the simulation level.
The model is shown as a system of piecewise-smooth differential equations.
Piecewise smooth and hybrid dynamical systems have been increasingly used
in Engineering and Applied Sciences. More recently, these systems appeared
also in Economics and Social Science, mainly in Sustainability Development,
Bioeconomics and new knowledge areas. Theoretical work mainly deals with the
problem of one surface dividing the state space into two different regions since,
usually, a more complicated problem can be locally reduced to this situation.
When two switching surfaces are taken into account, also the generic case is
considered, where surfaces intersect transversally. Papers where more surfaces are
considered do not abound in the literature since the number of different regions
increases exponentially and the analysis becomes quite cumbersome. However,
many applications lie on this multi-surface situation and one must mostly rely
on the numerics, like in this talk.
Several nonsmooth bifurcations have been reported in the literature by the
authors. They are the fingerprint of an intrinsic complex system. When several
markets are connected, complex networks (in the dynamics and structure) naturally appear. Finally, stochasticity is introduced in the system in order to model
the risk aversion of investment agents. This is done through Markov Chains.
This combination of deterministic paths and stochasticity leads to the so-called
51
Piecewise-Deterministic Markov Processes. Depending on the risk behavior, simulations show several decision patterns.
This is a joint work with Johnny Valencia.
Contact address: [email protected]
Yizhar Or
Analysis of foot slippage in simple theoretical models of dynamic
legged locomotion in sagittal plane.
Abstract: Dynamic legged locomotion has been a subject of extensive research in
biology, robotics and biomechanics. Simple theoretical models that describe the
leg-body dynamics in sagittal plane have been proposed and investigated, such as
compass biped robot and SLIP (spring-loaded inverted pendulum). Vast majority
of these models assume that the stance foot makes stationary contact with the
ground without slippage. Nevertheless, in some situation of slippery surfaces
and high velocities, foot slippage becomes inevitable, and may have a substantial
influence on the dynamic behavior. In this talk, we present our recent efforts to
incorporate slippage effects under Coulomb’s friction model into three classical
model of passive and controlled dynamic walking - the rimless wheel, compass
biped, and SLIP. Calculation of contact forces at the stance foot under no-slip
periodic solutions of these models reveals that they often require impractically
large friction. Thus, stick-slip transitions should be incorporated into the hybrid
dynamics. We numerically explore the evolution and co-existence of periodic
solutions under changes in Coulomb’s friction coefficient, and investigate their
effect on stability and performance measures such as average speed and energetic
cost. It is found that slippage at ground impact or during stance typically results
in a slight decrease in open-loop stability but may also lead to a significant
reduction in energetic cost.
Contact address: [email protected]
Petri T Piiroinen
Discontinuity geometry – an approach to discover the landscape of
impact oscillators.
Abstract: A topological approach, named discontinuity geometry, for analysis
and visualization of the dynamics of periodically-forced impact oscillators was
introduced by Chillingworth [1]. This method was further explored in [2, 3] to
specifically analyse grazing bifurcations, chatter and saddle-node bifurcations.
In this talk I will introduce the idea behind discontinuity geometry and how
the corresponding geometric objects can be visualised in Matlab. The talk will
also highlight how typical features of impacting systems such as grazing and
chatter are expressed in the discontinuity-geometry landscape. I will end the
52
talk by describing some open questions and challenges from both numerical and
analytical perspectives.
References
[1] D. R. J. Chillingworth, “Discontinuity geometry for an impact oscillator”, Dynamical Systems 17(4), pp. 389-420, 12/2002.
[2] D. R. J. Chillingworth, “Dynamics of an impact oscillator near a degenerate graze”, Nonlinearity 23(11), 05/2010.
[3] N. Humphries and P. T. Piiroinen, “A discontinuity-geometry view of the relationship between saddle-node and grazing bifurcations”, Physica D Nonlinear Phenomena 241(22).
11/2012.
Contact address: [email protected]
Camille Poignard
Oscillations for non monotonic smooth negative feedback systems
bounded by two hybrid systems.
Abstract: Negative feedback circuits are a recurrent motif in regulatory biological networks, strongly linked to the emergence of oscillatory behavior. The
theoretical analysis of the existence of oscillations is a difficult problem and typically involves many constraints on the monotonicity of the activity functions.
Here, we study the occurrence of periodic solutions in an n-dimensional class of
negative feedback systems defined by smooth vector fields with a window of not
necessarily monotonic activity.
Our method consists in circumscribing the smooth system by two hybrid systems of a specific type, called piecewise linear systems. We can show that each
of these hybrid systems has a periodic solution. It can then be shown that the
smooth negative feedback system also has a periodic orbit, inscribed in the topological solid torus constructed from the two piecewise linear orbits.
The interest of our approach, using both hybrid and continuous formalism, lies
in: first, adopting a general class of functions, with a non monotonicity window,
which permits a better fitting between theoretical models and experimental data,
and second, establishing a more accurate location for the periodic solution, which
is useful for computational purposes in high dimensions.
As an illustration, a model for the “Repressilator” (a synthetic biological system of three genes) is analyzed and compared to real data, and shown to admit
a periodic orbit, for a range of activity functions.
Submission for a talk, done by JL Gouzé or C Poignard.
This is a joint work with Madalena Chaves and Jean-Luc Gouzé.
Contact address: [email protected]
53
Enrique Ponce
The boundary focus in planar Filippov systems: a wolf in sheep’s
clothing .
Abstract: Planar Filippov differential systems with a straight line as the discontinuity manifold in the so called focus-fold singularity are revisited. This
co-dimension two critical configuration appears when there is a collision between
an invisible tangency (fold) from one side and a boundary focus, which represents
the transition from invisible to visible tangency, from the other side.
The analysis is made mainly in a piecewise linear context and the complete local
unfolding of the focus-fold singularity is provided in a convenient two-parameter
setting. A universal piecewise linear canonical form is proposed and exploited in
order to show the existence of a missing, narrow parametric sector for which two
small crossing limit cycles coexist.
The hidden subtleties of the boundary focus giving rise to special difficulties in
its analysis are emphasized throughout. This is a joint work with Emilio Freire
and Francisco Torres, see references [1, 2].
References
[1] E. Freire, E. Ponce and F. Torres, A general mechanism to generate three limit cycles in
planar Filippov systems with two zones, Nonlinear Dynamics (2014) 78:251-263.
[2] E. Freire, E. Ponce and F. Torres, On the critical crossing cycle bifurcation in planar Filippov
systems, J. Differential Equations 259 (2015) 7086-7107.
Contact address: [email protected]
Thibaut Putelat
Nonlinear dynamics of localized frictional slip.
Abstract: Inhomogeneous frictional sliding between solid bodies is ubiquitous
and characterised by multiple spatio-temporal scales, the most compelling example being earthquakes. A comprehensive understanding of the physical mechanisms that determine the diversity of the frictional slip pattern formation that
has been reported over the years is still lacking, partly because of incomplete
mathematical modelling. Considering rate-and-state friction models with nonmonotonic spinodal (i.e., N-shape) velocity dependence, we present recent advances in the nonlinear dynamics of distributed frictional rup- ture of a thin
elastic slab; noticeably with regards to the occurrence of frictional travelling
self-healing ‘slip pulses’. We show that these arise from a homoclinic global bifurcation of travelling periodic slip patterns as the applied shear stress increases.
Such slip pulses are anchored at the equilibrium saddle point lying on the lowvelocity-strengthening branch of the steady-state friction curve. Interestingly, the
existence of a high velocity strengthening branch in spinodal friction also allows
the existence of ‘stick pulse’ which corresponds to a narrow travelling ‘stick’ zone.
54
Along the bifurcated branch, travelling wave trains of slip pulses develop from a
canard explosion, which can lead to relaxation oscillations. Finally, heteroclinic
connections corresponding to travelling fronts promoting the slab acceleration
are also possible. This plethora of behaviours strongly depends on the analytical
details of the friction model and may shed new light on the dynamics of earthquake ruptures, in particular with respect to the recent field evidence according
to which seismic slip localizes along a fault patch that is partially locked during
the interseismic period.
Contact address: [email protected]
Tere M. Seara
Regularization of a planar Filippov vector field near a visible-invisible
fold point.
Abstract: In this talk we will study the regularization of planar a Filippov vector
field near a visible-invisible fold point. We will consider generic unfoldings of this
singularity and study their regularizations. We will pay special attention to a
Hopf bifurcation of the smooth vector field that does not exist in the Filippov
one. We will show the existence of a cannard point in the regularized system
and show how its existence is the cause of the disappearence of the periodic orbit
which arises in the Hopf bifurcation.
This is a joint work with Carles Bonet and Juliana F. Larrosa.
Contact address: [email protected]
David J.W. Simpson
Border-collision bifurcations: Myths, facts and open problems.
Abstract: The collision of a fixed point with a switching manifold in a piecewisesmooth continuous map, known as a border-collision bifurcation, can give rise to
a seemingly endless zoo of complicated dynamics. An understanding of these
dynamics, which are described merely by piecewise-linear continuous maps, is
one of the most fundamental problems in nonsmooth bifurcation theory. In this
talk I will review some key facts, dispel some common myths, and provide a list
of pertinent open problems for future research.
Piecewise-smooth maps are used to model discrete-time systems with switching events, and arise as return maps of piecewise-smooth systems of differential
equations. In the latter context, border-collision bifurcations correspond to certain discontinuity-induced bifurcations of limit cycles. Examples include so-called
grazing-sliding bifurcations, corner-collisions, and event collisions in systems with
time-delayed switching.
55
By expanding a piecewise-smooth map about a border-collision bifurcation,
and truncating the expansion to leading order, we obtain a piecewise-linear map
that approximates the dynamics. In N -dimensions, this map can be written as
(
AL xi + bµ , si ≤ 0
(6)
xi+1 =
,
AR xi + bµ , si ≥ 0
where si = eT1 xi denotes the first component of xi ∈ RN , AR = AL + CeT1 for
some C ∈ RN , b ∈ RN , and µ ∈ R is a parameter. Structurally stable dynamics
of (6) with µ 6= 0 are exhibited by the original map near the border-collision
bifurcation.
It is straight-forward to characterise fixed points and period-two solutions of
(6), but a complete classification of more complicated dynamics in terms of the
parameters of the map is unachievable (except with N = 1). Our best hope
is to explain general features and unfold codimension-one and two (and higher)
points. For instance, (6) can exhibit chaos that is robust in the sense that as parameters are varied windows of periodicity do not arise. Mode-locking regions of
(6) commonly exhibit a chain structure that can be understood using a symbolic
framework. Also, it was recently shown that (6) has codimension-three points at
which infinitely many attractors coexist. Yet many features of border-collision
bifurcations remain unexplained, and many aspects of the dynamics of humble
piecewise-linear maps remain to be understood.
Contact address: [email protected]
Pascal Stiefenhofer
Analysis of nonsmooth periodic orbits in dynamical systems with
discontinuous right hand side.
Abstract: This paper considers a nonsmooth dynamical system described by
a two dimensional autonomous ordinary differential equation with discontinuous
right hand side given by
ẋ = f (x)
1
2
2
with f ∈ C (R \ {(x1 , 0)}, R ). We consider the case where f is discontinuous
for x2 = 0. We have
(
f + (x) if x2 > 0
ẋ± = f ± (x) =
f − (x) if x2 < 0.
The paper provides necessary conditions for existence, uniqueness, and exponentially asymptotically stability of a nonsmooth periodic orbit. Moreover, it
provides a formula for a part of its basin of attraction. The advantage of the
theory is that we do not need to calculate the periodic orbit in order to show
above properties. The theorem is demostrated by showing local contraction of
two adjacent soultuions over the smooth and jumping parts of the periodic orbits.
56
This requires definining a metric function. Together with a converse theorem the
theory provides the conditions for the development of numerical methods.
This is a joint work with Peter Giesl.
Contact address: [email protected]
Iryna Sushkoa
2D Border collision normal form and smale horseshoe construction.
Abstract: The 2D Border Collision Normal Form F : R2 → R2 is given by two
linear maps FL and FR which are de.ned in two half planes denoted L and R:
(
FL (x, y) if (x, y) ∈ L,
F : (x, y) 7→=
FR (x, y) if (x, y) ∈ R,
where
FL :
x
!
7−→
FR :
y
!
,
L = {(x, y) : x ≤ 0},
,
R = {(x, y) : x > 0},
−δL x
y
x
τL x + y + x
!
7−→
τR x + y + x
!
−δR x
and τL , τR are the traces and δL , δR are the determinants of the Jacobian matrix
of the map F in the left and right halfplanes, i.e., in L and R, respectively. The
dynamics of map F is nowadays quite intensively studied by many researchers
not only due to its appearance in several applications, but also in order to classify
border collision bifurcations in generic 2D piecewise smooth maps.
Our aim is to discuss the construction of a Smale horseshoe in map F; associated
with transverse homoclinic points. Recall that the original horseshoe (Smale,
1963) is constructed for diffeomorphisms, while F is a nonsmooth (piecewise
linear) map. Particular attention is paid to noninvertible case. Transformations
of basin boundaries related to homoclinic bifurcations are also discussed.
This is a joint work with L. Gardini.
Contact address: [email protected]
Antonio E. Teruel
Folded nodes, canards and mixed-mode oscillations in 3D piecewiselinear systems.
Abstract: New advances in 3D piecewise-linear (PWL) slow-fast systems are
presented. In particular, a complete comparison with the smooth case near folded
singularities is shown: singular phase portraits, singular weak and strong canards
57
and control of the number of maximal canards are obtained in a way that is
entirely compatible with the smooth case. Furthermore, by using the previous
analysis we present a minimal model displaying periodic canard induced mixed
mode oscillations near a PWL folded node.
This is a joint work with Mathieu Desroches, Antoni Guillamon, Enrique
Ponce, Rafel Prohens, Serafim Rodrigues.
Contact address: [email protected]
Joan Torregrosa
Limit cycles in piecewise planar systems via ET-systems with
accuracy .
Abstract: The Poincaré-Pontriaguin-Melnikov theory in the plane, or equivalently the averaging theory, can be used to provide lower bounds for the number
of limit cycles that can have a planar polynomial system. Recently this theory
has been developed and applied to piecewise planar systems. In particular it
is used to find upper bounds for the number of limit cycles that can bifurcate
from the period annulus of a point of center type. This technique is based in the
study of the number of simple zeros of the so-called Abelian integrals. We will
present some new results using first and second order perturbations of concrete
piecewise polynomial families. ET-systems and ET-systems with accuracy will
be used to get precise upper bounds for that number of limit cycles up to second
order perturbation.
Contact address: [email protected]
Catalina Vich
Estimation of synaptic conductances in a McKean neuron model .
Abstract: To understand the flow of information in the brain, some computational strategies have been developed in order to estimate the synaptic conductances impinging on a single neuron directly from its membrane potential (see [3]
and [4] for instance). Despite these existent strategies that give circumstantial
solutions, they all present the inconvenience that the estimation can only be done
in subthreshold activity regimes, that is, when the neuron is not spiking. The
main constraint to provide strategies for the oscillatory regimes is related to the
nonlinearity of the input-output curve since most of the methods rely on an a
priori linear relationship, which is no longer true in spiking regimes.
In this work, we aim at giving a first proof of concept to address the estimation
of synaptic conductances when the neuron is spiking. For this purpose, we use
a simplified model of neuronal activity given by the slow-fast piecewise linear
system
Cdv/dt = f (v) − w − w0 + I − gsyn (v − vsyn ),
dw/dt = v − γw − v0 ,
58
where v is the membrane potential, w is an auxiliary variable, f(v) is an N-shaped
3-zone piecewise linear function, gsyn is the synaptic conductance, 0 < C < 0.1 is
the capacitance, I is the external current, which will be taken constant here, and
the rest of parameters are related to conductance properties and combinations of
membrane reversal potentials. This model is called the McKean model (see [1],
among others) and it allows an exact knowledge of the nonlinear f − I curve.
As a first step, under suitable conditions to ensure the existence of a periodic orbit, we are able to find out an approximated function for the period,
T (I, C, gsyn ), using a new approach that allows us to improve the approximations
done so far (see [2], among others). Since this function results to be monotone
with respect to gsyn , we are able to infer a steady synaptic conductance from the
cell’s oscillatory activity with relative errors of order C. Finally, we extend the
results to a more realistic case, where we present a proof of concept to estimate
the full time course of the conductances in spiking regimes, provided that they
vary slowly in time.
This is a joint work with Antoni Guillamon, Rafel Prohens, Antonio E. Teruel.
References
[1] Coombes S (2008) Neuronal networks with gap junctions: A study of piecewise linear planar
neuron models. SIAM Journal of Applied Dynamical Systems 7(3):1101.1129.
[2] Fernandez-Garcia S, Desroches M, Krupa M, Clement F (2015) A multiple time scale coupling of piecewise linear oscillators. application to a neuroendocrine system. SIAM Journal
on Applied Dynamical Systems 14(2):643.673
[3] Lankarany M, Zhu W-P, Swamy S, Toyoizumi T (2013) Inferring trial-to-trial excitatory
and inhibitory synaptic inputs from membrane potential using Gaussian mixture Kalman
filtering. Frontiers in Computational Neuroscience 7(109).
[4] Rudolph M, Piwkowska Z, Badoual M, Bal T, Destexhe A (2004) A method to estimate
synaptic conductances from membrane potential fluctuations. Journal of Neurophysiology
91(6):2884.2896.
Contact address: [email protected]
Marian Wiercigroch
Grazing induced bifurcations: Innocent or dangerous?
Abstract: In this lecture I will examine nature of subtle phenomenon such grazing bifurcations occurring in non-smooth systems. I will start with linear oscillators undergoing impacts with secondary elastic supports, which have been studied
experimentally and analytically for near-grazing conditions [1]. We discovered a
narrow band of chaos close to the grazing condition and this phenomenon was
observed experimentally for a range of system parameters. Through stability
analysis, we argue that this abrupt onset to chaos is caused by a dangerous bifurcation in which two unstable period-3 orbits, created at “invisible” grazing
collide [2].
59
The experimentally observed bifurcations are explained theoretically using
mapping solutions between locally smooth subspaces. Smooth as well as nonsmooth bifurcations are observed, and the resulting bifurcations are often as an
interplay between them. In order to understand the observed bifurcation scenarios, a global analysis has been undertaken to investigate the influence of stable
and unstable orbits which are born in distant bifurcations but become important
at the near-grazing conditions [3]. A good degree of correspondence between the
experiment and theory fully justifies the adopted modelling approach.
Similar phenomena were observed for a rotor system with bearing clearances,
which was analysed numerically [4] and experimentally [5]. To gain further insight
into the system dynamics we have used a path following method to unveil complex
bifurcation structures often featuring dangerous co-existing attractors.
References
[1] Ing, J., Pavlovskaia, E.E., Wiercigroch, M. and Banerjee, S. 2008 Philosophical Transactions
of the Royal Society - Part A 366, 679-704. Experimental study of impact oscillator with
one sided elastic constraint.
[2] Banerjee, S., Ing, J., Pavlovskaia, E., Wiercigroch, M. and Reddy, R. 2009 Physical Review
E 79, 037201. Invisible grazing and dangerous bifurcations in impacting systems.
[3] Ing, J., Pavlovskaia, E., Wiercigroch, M. and Banerjee, S. 2010 International Journal of
Bifurcation and Chaos 20(11), 3801-3817. Complex dynamics of bilinear oscillator close to
grazing.
[4] Páez Chávez, J. and Wiercigroch, M. 2013 Communications in Nonlinear Science and Numerical Simulation 18, 2571-2580. Bifurcation analysis of periodic orbits of a non-smooth
Jeffcott rotor model.
[5] Páez Chávez, J., Vaziri Hamaneh, V. and Wiercigroch, M. 2015 Journal of Sound and
Vibration 334, 86-97. Modelling and experimental verification of an asymmetric Jeffcott
rotor with radial clearance.
Contact address: [email protected]
61
4. Abstracts of the Posters
Elena Bossolini
Non-robustness of non-smooth systems to regularization.
Abstract: Non-smooth systems can be studied through regularization. For Filippov systems this process allows one to recover the sliding region as the critical
manifold of the singular system.
However, non-Filippov systems may not be robust to regularization. In particular, the reduced problem may have solutions that do not appear in the original
non-smooth system. We show the fragility of such systems by considering the 1 12
degree of freedom oscillator that describes the brake-pad interaction with stiction friction. Under certain conditions there appears a canard solution in the
regularized problem that is not present in the original non-smooth problem.
We also study the behaviour of periodic solutions interacting with the canard
numerically by continuation of a parameter.
Contact address: [email protected]
Juan Castillo
Global connections in a class of discontinuous piecewise linear
systems.
Abstract: In this work, we give families of three-dimensional discontinuous
piecewise linear systems which have heteroclinic and homoclinic connections. The
systems are formed by two pieces separated by a switching plane so that each
system has a unique equilibrium point in its region of definition. To construct
the heteroclinic connection we consider for simplicity two real saddles and for the
homoclinic connection a real saddle and a center. Families can be found taking
advantage of a normal form for the class of discontinuous piecewise linear systems
with a unique two-fold singularity. We also discussed about bifurcations that
occur by perturbing the connections such the focus-center-limit cycle-homoclinic
bifurcation, a phenomenon that is only possible in nonsmooth systems.
Contact address: [email protected]
Christian Erazo
Dynamic cell-to-cell mapping for computing basins of attraction in
bimodal Filippov systems.
Abstract: Filippov systems are often used for modeling mechanical, electrical
and biological systems. Different numerical approaches have been developed for
investigating their complex dynamics, by playing direct numerical simulations or
by computing bifurcation diagrams. Less attention has been given in the literature to the problem of computing numerically basins of attraction in Filippov
62
systems. Some examples of previous works based on the use of Lyapunov based
methods can be found in [1] but are typically very conservative. Here, we present
an algorithm based on the Simple Cell Mapping (SCM) method [2] which exploits
the event-driven integration routine proposed in [3] that can cope with the presence of sliding solutions and automatically correct for possible numerical drifts.
Our algorithm encompasses a dynamic selection of the cells. Specifically, after
an initial application of SCM, layers of cells are added and examined iteratively.
The mapping information is stored and used at each iteration, such that integrations for just the extra cells are performed. Moreover, a refinement stage is used
to obtain a better resolution of the basin boundary. The aim of this poster will
be to discuss this novel algorithm and present some illustrative examples. The
ouput of the algorithm that was implemented in Matlab is shown in Fig. 1 where
the basin is derived of a stable equilibrium of a bimodal Filippov system of the
form
(
F1 (x), x ∈ S1
(1)
ẋ =
F2 (x), x ∈ S2
with a stable sliding surface Σ̂ = {x ∈ R2 : x1 + x2 = 0} and the two vector
fields




−x1 + x2 
−x1 + x2 
(2)
F1 (x) = 
.
 , F2 (x) = 
3x2 + 10
3x2 − 10
References
[1] Hetel, L.; Fridman, E.; Floquet, T., “Variable Structure Control With Generalized Relays:
A Simple Convex Optimization Approach”, in IEEE Transactions on Automatic Control,
vol.60, no.2, pp.497–502, 2015.
[2] J. A. W. van der Spek. Cell Mapping Methods: Modifications and Extensions. PhD Thesis,
Eindhoven University of Technology, Eindhoven, 1994
[3] Piiroinen, P. T. & Kuznetsov, Y. A.. An event-driven method to simulate Filippov systems
with accurate computing of sliding motions. ACM Transactions on Mathematical Software
(TOMS), 34(3), 13, 2008.
This is a joint work with M. di Bernardo, M. Homer, P.T. Piiroinen.
Contact address: [email protected]
Davide Fiore
Incremental stability of bimodal Filippov systems: analysis and
control .
Abstract: Incremental stability has been established as a powerful tool to prove
convergence in nonlinear dynamical systems [1]. An effective approach to obtain sufficient conditions for incremental stability comes from contraction theory
63
Figure 2. Output window of the cell-to-cell mapping algorithm
applied to investigate the basin of attraction of the origin of system
(1)-(2).
[5]. More specifically, incremental exponential stability over a given forward
invariant set is guaranteed if some matrix measure of the system Jacobian matrix is uniformly negative in that set for all time. Hence classical contraction
analysis requires the system vector field to be continuously differentiable Several
results have been presented in the literature to extend contraction analysis to
non-differentiable vector fields, e.g. [4]. In this poster we present that our recent
work [3], by using results on regularization of switched dynamical systems from
Sotomayor and Teixeira [6], derives conditions to ensure the Filippov system to
be contracting. We then discuss a switching control strategy to either locally or
globally incrementally stabilize a class of nonlinear dynamical systems [2]. Following this design procedure we derive a control action that is active only where
the open-loop system is not sufficiently incrementally stable and thus the required
control effort is reduced. The theoretical derivations are illustrated by a control
design example where the problem is to choose a switched feedback control input
to incrementally stabilize systems of the form ẋ = f (x) + g(x)u.
References
[1] D. Angeli. A Lyapunov approach to incremental stability properties. IEEE Transactions on
Automatic Control, 47(3):410–421, 2002.
[2] M. di Bernardo and D. Fiore. Incrementally stabilizing switching control via contraction
theory. arXiv preprint arXiv:1510.08368,2015.
64
[3] M. di Bernardo, D. Fiore and S. J. Hogan. Contraction analysis of switched Filippov systems
via regularization. arXiv preprint arXiv:1507.07126,2015.
[4] M. di Bernardo, D. Liuzza, and G. Russo. Contraction analysis for a class of nondifferentiable systems with applications to stability and network synchronization. SIAM Journal
on Control and Optimization, 52(5):3203–3227, 2014.
[5] W. Lohmiller and J.-J. E. Slotine. On contraction analysis for non-linear systems.Automatica, 34(6):683–696, 1998.
[6] J. Sotomayor and M. A. Teixeira. Regularization of discontinuous vector fields. In Proc. of
International Conference on Differential Equations, Lisboa, pages 207–223, 1996.
Contact address: [email protected]
Wilker Thiago Resende Fernandes
Investigation of isochronous center conditions for a family of vector
fields.
Abstract: Assuming that a given vector field has a singular point is known to
be a center we shall present a computational algebra method for determining
whether or not it is isochronous, that is, whether or not every periodic orbit in
a neighborhood of the origin has the same period. The main goal of this poster
is to introduce this computational algebra method and apply them in a family
of vector fields in C2 . The investigated family is a Z2 –equivariant system with
two elementary focus. The necessary and sufficient conditions for the existence
of the bi-center are investigated. The results in this poster make part of the PhD
investigation of one of the author.
Contact address: [email protected]
Ricardo Miranda
Chaos in piecewise smooth vector fields on compact surfaces.
Abstract: We study the global dynamics of piecewise smooth vector fields defined in the two dimensional torus and sphere. We provide conditions under these
families exhibits periodic and dense trajectories and we describe some global bifurcations. We also study its minimal sets and characterize the chaotic behavior
of the piecewise smooth vector fields defined in torus and sphere.
This is a joint work with Durval J Tonon (UFG/Brazil).
Contact address: [email protected]
65
Chara Pantazi
Quadratic systems with a singular curve of degree 3 .
Abstract: The study of polynomial differential systems with invariant algebraic
curves/ surfaces is a very interesting problem with many applications in physics,
dynamic populations, neuroscience, cosmology, etc. Some times the existence of
invariant algebraic curves/ surfaces provides useful information about the integrability of the system. In particular for planar polynomial differential systems,
the existence of invariant algebraic curves helps to understand the qualitative
behavior of the dynamic system. Here, in this talk we are going to deal with
planar quadratic differential systems having a singular invariant curve of degree
three. We are going to explain how the existence of this curve determines the
qualitative behavior of the corresponding dynamical system and we are going to
present the phase portraits in the Poincaré disk.
In the work of [1] the authors present a classification of all real quadratic
systems having one real invariant algebraic curve of degree 3 such that its complex
irreducible factors satisfy some generic conditions (and also have a first integral).
Hence, singular curves are not considering in [1]. The goal of this presentation is
to complete the study of quadratic systems with an invariant algebraic curve of
degree 3 that is irreducible and singular.
References
[1] J. Llibre, J.S. Pérez del Rı́o and J.A. Rodrı́guez, Phase portraits of a new class of integrable
quadratic vector fields, Dynamics of Continuous, Discrete and Impulsive systems, 7, (2000),
595–616.
Contact address: [email protected]
Zsolt Verasztó
Hardware-in-the-loop test of stick-slip phenomena: model, analysis,
experiment.
Abstract: The topic of this study is the analysis of a one degree of freedom
(DoF) oscillatory system, subjected to a Stribeck-type friction force generated by
a moving surface of mixed dry and viscous friction. The stick-slip phenomenon,
which arises on partially lubricated surfaces, can be critical in many engineering systems. This study aims to show the theoretical considerations behind the
use of the hardware-in-the-loop (HIL) method and to present a possible experimental implementation. We compare the results of the local stability and global
dynamical analysis with the numerical results of the simulations implemented in
MATLAB. The mathematical model of a HIL experiment is set up, which takes
into account the delay of the digital control used in the experiments. After the
analytical and numerical investigation of the discrete model, we present the results of the measurements and draw conclusions regarding how applicable and
66
realistic the HIL experiments are in case of a non-trivial nonlinear dynamical
system performing stick-slip.
This is a joint work with Gábor Stépán.
Acknowledgements: The research leading to these results has received funding
from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Advanced Grant Agreement n. 340889.
Contact address: [email protected]
Si Mohamed Sah
Reconstructing non-smooth forces in atomic force micoscopy with
frequency combs.
Abstract: Intermodulation Atomic Force Microscopy (ImAFM) probes the nonsmooth force of an AFM tip as it taps on a surface, by analyzing the frequency
mixing products generated from a two-tone drive. A frequency comb of mixing
products is generated near resonance when the high-Q resonator is perturbed by
the nonlinearity. The slow-time behavior can be easily obtained from the measured data by down-shifting the comb so that it is centered at zero frequency. We
present a polynomial force reconstruction method which is tested on simulated
AFM data from a piecewise tip-surface force model. The method uses analytic expressions for the slow-time amplitude and phase evolution of the AFM cantilever
motion, obtained from time-averaging over the rapidly oscillating part of the
cantilever dynamics. A direct fit of the theoretical expressions to the simulated
data gives the best-fit parameters for the force model. The method combines
and complements previous work [1, 2] and it allows for computationally more
efficient parameter mapping with ImAFM. Results for the simulated piecewise
force model are compared with the reconstructed polynomial force and show a
good agreement.
This is a joint work with D. Forchheimer, R. Borgan, D. B. Haviland.
References
[1] Platz, D., Forchhheimer, D., Tholen, E. A. & Haviland, D. B., Interaction imaging with
amplitude-dependence force spectroscopy. Nat. Commun. 4, 1360 (2013).
[2] Forchheimer, D., Platz, D., Tholen, E. A. & Haviland, D., Model-based extraction of material properties in multifrequency atomic force microscopy. Phys. Rev. B 85, 195449 (2012).
Contact address: [email protected]
67
5. List of Participants
Mate Antali
Viktor Avrutin
Tamás Baranyai
Luis Benadero
Chris Bick
Elena Blokhina
Carles Bonet
Elena Bossolini
Mireille Broucke
Kanat Camlibel
Juan Andres Castillo
Alan Champneys
David Chillingworth
Alessandro Colombo
Manuel Dominguez-Pumar
Zoltan Dombóvári
Douglas Duarte
Abdelali El Aroudi
Vasfi Eldem
Christian Camilo Erazo
Marina Esteban
Wilker Fernandes
Michael Field
Davide Fiore
Laura Gardini
Paul Glendinning
Albert Granados
Toni Guillamon
Peter Harte
John Hogan
Gemma Huguet
Luigi Iannelli
Georgios Kafanas
Gabriella Keszthelyi
Mike R. Jeffrey
Jun Jiang
Universität Stuttgart
Universitat Rovira i Virgili
University College of Dublin
Universitat Politècnica de Catalunya
Technical University of Denmark
University of Toronto
Universidad de Sonora
University of Brist
University of Southampton
Politecnico di Milano
Universitat Politècnica de Catalunya
Universidade Estadual de Campinas
Universitat Rovira i Virgili
Okan University
Universidade de São Paulo
Università degli studi di Urbino
University of Manchester
Technical University of Denmark
Universitat Politècnica de Catalunya
University College Dublin
University of Bristol
Universitat Politècnica de Catalunya
University of Bristol
University of Bristol
State Key Lab. for Strength and Vibration
68
Kristian Uldall Krsitiansen
Alex Küronya
Rachel Kuske
Claude Lacoursière
J. Tomàs Lázaro
Julie Leifeld
Tere Martı́nez-Seara
Ricardo Miranda
Tamás G. Molnár
Karin Mora
Ehud Moshe
Gerard Olivar
Josep M. Olm
Yizhar Or
Chara Pantazi
Petri Piiroinen
Camille Poignard
Enrique Ponce
Rafel Prohens
Thibaut Putelat
Si Mohamed Sah
Gökhan Sahan
David Simpson
Sarah Spurgeon
Eoghan Staunton
Gábor Stépán
Iryna Sushko
Antonio E Teruel
J. Tomás
Joan Torregrosa
Francisco Torres
Peter Varkonyi
Zsolt Verasztó
Catalina Vich
Simon Webber
Marian Wiercigroch
Frankfurt University
The University of British Columbia
Umea University
Universitat Politècnica de Catalunya
University of Minnesota
Universitat Politècnica de Catalunya
Unicamp
Budapest University of Technology and Economics
University of Paderborn
Universidad Nacional de Colombia
Universitat Politècnica de Catalunya
Technion Israel Institute of Technology
Universitat Politècnica de Catalunya
National University of Ireland Galway
Inria Biocore
Universidad de Sevilla
Universitat de les Illes Balears
Laboratoire de Mécanique des Solides, E. Polytech.
Massey University
University of Kent
National University of Ireland Galway
Budapest University of Technology and Economics
National Academy of Sciences of Ukraine
Universitat de les Illes Balears
Universitat Politècnica de Catalunya
Universitat Autonòma de Barcelona
Universidad de Sevilla
Budapest University of Technology and Economics
Universitat de les Illes Balears
University of Bristol
University of Aberdeen
This list is automatically updated as participants confirm their registration.

Download Report

Open Problems in Nonsmooth Dynamics

Paperzz.com

Your Paperzz