Bifurcations in piecewise-smooth systems
Chris Budd
What is a piecewise-smooth system?
Map
F
(
x
)if
H
(
x
)

0

1
x

f
(
x
)

Heartbeats or

F
(
x
)if
H
(
x
)

0
2

Poincare maps
Flow
F
(
x
) ifH
(
x
)

0

dx
1


F
(
x
) ifH
(
x
)

0
dt
2

Hybrid
dx

F
(x
) if H
(x
)
0
,
dt
x
R
(x
) if H
(x
)
0
.
Rocking block,
friction, Chua
circuit
Impact or
control systems
PWS Flow
PWS Sliding Flow
Hybrid
Key idea …
The functions F
1(x),F
2(x) or one of their nth
derivatives, differ when


x



x
:H
(x
)
0
Discontinuity set
Interesting discontinuity induced bifurcations occur when
limit sets of the flow/map intersect the discontinuity set
Why are we interested in them?
• Lots of important physical systems are
piecewise-smooth: bouncing balls, Newton’s
cradle, friction, rattle, switching, control
systems, DC-DC converters, gear boxes …
• Piecewise-smooth systems have behaviour
which is quite different from smooth systems
and is induced by the discontinuity: period
adding
• Much of this behaviour can be analysed, and
new forms of discontinuity induced
bifurcations can be studied: border collisions,
grazing bifurcations, corner collisions.
Will illustrate the behaviour of piecewise
smooth systems by looking at
• Maps
• Hybrid impacting systems
Some piecewise-smooth maps
Linear, discontinuous


x

,x

0
1
x

f
(
x
)


x


1
,x

0
2

Square-root, continuous

 
 

x
,x

,
x

f
(
x
)


,x

.
 x
Both maps have fixed points over certain
ranges of 
Border collision bifurcations occur when
for certain parameter values the fixed
points intersect with the discontinuity
set
Get exotic dynamics close to these
parameter values
Dynamics of the piecewise-linear map
01,2 1
Homoclinic orbit
Fixed point
Fixed point
Period adding Farey sequence

Dynamics of the piecewise-linear map
01 12
Period adding Farey sequence
Chaotic

Square-root map

 
 

x
,x

,
x

f
(
x
)


,x

.
 x
Map arises in the study of grazing
bifurcations of flows and hybrid systems
Infinite stretching when
Fixed point at
x
x0 if
0
0 
1
4
Period adding
1
2
 
4
3
Chaos

2
  1
3
Immediate jump to robust
chaos
Partial period adding
Get similar behaviour in higher-dimensional
square-root maps



Ax

M
,C
(
Ax

M
)

N

H
(
x
)

0
,


x

f
(
x
,)


M

By
,H
(
x
)

0
,
 Ax

y


H
(
x
)
.


Map [Nordmark] also arises naturally in the
study of grazing in flows and hybrid systems.
If A has complex eigenvalues we
see discontinuous transitions
between periodic orbits
If A has real eigenvalues we see
similar behaviour to the 1D map
Impact oscillators: a canonical hybrid system
obstacle
x   x  x  cos(t ), x   ,
x  rx ,
x  .
Periodic dynamics
Experimental
Analytic
Chaotic dynamics
Complex domains of attraction of periodic orbits
Regular and discontinuity induced bifurcations
Regular and discontinuity induced bifurcations as
as
parameters vary.
parameters vary
Period doubling
Grazing
Grazing occurs when periodic orbits intersect
the obstacle tanjentially
x 
x 
Observe grazing bifurcations identical to the
dynamics of the two-dimensional square-root map
  0.01
Transition to a periodic orbit
Non-impacting
orbit
x

Period-adding
 2
Local analysis of a Poincare map associated
with a grazing periodic orbit shows that this
map has a locally square-root form, hence the
observed period-adding and similar behaviour
Poincare map associated with a grazing
periodic orbit of a piecewise-smooth flow
typically is smoother (eg. Locally order 3/2
or higher) giving more regular behaviour
Systems of impacting oscillators can have
even more exotic behaviour which arises when
there are multiple collisions. This can be
described by looking at the behaviour of the
discontinuous maps
CONCLUSIONS
•
Piecewise-smooth systems have interesting
dynamics
• Some (but not all) of this dynamics can be
understood and analysed
• Many applications and much still to be
discovered


1
Parameter range for simple periodic orbits 0
1
2

Fractions 1/n

Fractions (n-1)/n