Alanna Hoyer-Leitzel
Research Statement
[email protected]
1
Introduction
The field of dynamical systems has its origins in the abstraction of the first ordinary differential
equations and historical problems posed by Newton and Euler. The laws governing mechanical
systems, natural processes, and even social or economic systems have the assumption that future
states of the system can be determined by the present state. Poincaré and Lyaponov took these
historic mathematical problems and developed a new qualitative theory using rich and complex
tools from many fields such as geometry, analysis, and topology.
My research focuses on two areas in dynamical systems. The first area is motivated by problems
in climate and ecology. Through modeling an ecological system as a dynamical system, the goal
is to define and measure the system’s behavior when it is perturbed. This research examines the
interaction between discrete disturbance and basins of attraction in a dynamical system, including
when those basins of attraction are changing at critical rates that may cause the system to tip
to another state. The second area is Hamiltonian mechanical systems, specifically the n-vortex
problem. It is closely related to the famous n-body problem and uses many of the fundamental
tools of dynamical systems. My research in the n-vortex problem also incorporates tools from
algebraic geometry, and extends to applications of geophysical vortices. Both of these interests in
dynamical systems are well suited for research projects with undergraduate students.
2
Resilience and Tipping Points
One reason I am motivated to study resilience and tipping points is a case study of Caribbean
coral reefs in the book Resilience Thinking by Brian Walker and David Salt [19]. At the top of
page 68, there is a figure titled “A Two-Dimensional Representation of a Ball-in-a-Basin Model of a
Coral Ecosystem Changing over Time with Overfishing” which contains four pictures of a potential
function for a differential equation going through a very traditional saddle-node bifurcation, and a
caption describing that the potential function changes with respect to a parameter for overfishing,
after which disturbance from hurricanes can easily kick the ball out of the valley.
There are two phenomena occurring in this case study: the first is the resilience of the system
to repeated disturbance, and the second is the tipping of the state of the system from one basin
of attraction to the next. I am studying these two phenomena separately, with the future goal of
combining them in one model and quantifying their interactions.
2.1
Resilience to repeated disturbance
Resilience is becoming increasingly important to sustainability thinkers, especially with growing
awareness of human impact on climate and ecological systems. The wide ranging interest in resilience has led to numerous definitions, and those interested in influencing policy are hoping resilience can be understood as quantitative, rather than just a qualitative property. This proves an
interesting task within dynamical systems theory.
Roughly defined, resilience is the capacity of a system to absorb disturbance and still retain
structure and function. How this is interpreted is highly context dependent [5]. Figure 1 illustrates
various definitions of resilience. Disturbance can be applied to either the state space variables or to
the parameters of a system. If a single state space perturbation is applied to the system, resilience
is defined as the width of the basin of attraction [10, 18] or the distance between an attracting
equilibrium point and boundary of the basin of attraction [4, 6]. However, in many ecological
Alanna Hoyer-Leitzel
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Figure 1: Various definitions of resilience
applications of resilience, disturbances are often reoccurring. In my research I consider repeated
disturbances to state space variables.
A repeated discrete perturbation to state space - a kick - is applied to a dynamical system
ẋ = f (x) at regular intervals of time τ . In collaboration with the Resilience Focus Group of the
MCRN (Mathematics and Climate Research Network), I study the “flow-kick system,” a discrete
time-τ plus kick k map: Φ(x) = φτ (x) + k where φτ (x) is the flow of the continuous dynamical
system. To understand resilience is to see if solutions retain original structure and function, which
is usually interpreted as invariance of the domain of attraction to perturbation.
Analysis of the flow-kick system for a linear underlying n-dimensional vector field ẋ = Ax
has determined that the fixed points x̄ = (I − eτ A )−1 k of the flow-kick system will have the same
stability characteristics as the equilibrium point at the origin in the original system. For 1D systems,
analysis is geometrically intuitive assuming that ẋ = f (x) is a gradient vector field. New invariant
sets containing new fixed points can be calculated using a bounded on the size of the vector field
with respect to the size of the ratio of the kick size to flow time. The maximum size for a kick
that will not push solutions out of a basin of attraction is given as a function F (τ ), the maximum
“flow back” down the basin of attraction over time τ . These results are proved in an undergraduate
honors thesis project by a student I co-mentored at Bowdoin College, Stephen Ligtenburg [11].
Additionally, I’ve considered the specific case of a flow-kick system for the underlying vector
field ẋ = x − x3 + µ. This particular system was chosen because it exhibits bistability or alternative
stable states, and saddle-node bifurcations, both important characteristics in resilience theory.
Taking the system with µ = 0, it is easy to compute the exact bound for resilience F (τ ). As
τ → ∞, F (τ ) → k = 1, which is the distance from stable to unstable equilibrium points. This
aligns with the definition of resilience given for a single state-space disturbance [6]. As τ → 0, F (τ )
is concave and bounded by the tangent line at τ = 0, k = (µ − µ0 )τ , where µ0 is the bifurcation
value for the underlying system. See Figure 2a. This bound measures the distance to bifurcation,
and we see an interesting interaction between the multiple definitions of resilience and disturbance
to either state or parameters spaces. The qualitative behavior of F (τ ) and these aysmptotic and
tangential bounds can be continued to other values of µ. See Figure 2b. Moreover, for disturbances
that are not regular but are contained in some compact set of (k, τ, µ), the flow-kick system will be
resilient.
Future Work. There are many generalizations of the flow-kick system, and the flow-kick
system is by no means a complete understanding of resilience to repeated disturbance. Future
work will continue in collaboration with the Resilience working group of MCRN and a group of
independent study students at Bowdoin College. The group has three immediate lines of inquiry.
Alanna Hoyer-Leitzel
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(a) The shaded region is the set of kick sizes k
for each flow time τ for which a system is not
resilient, bounded by the maximum flow function
F (τ ). The two bounds as τ → ∞ and τ → 0 are
shown as dotted lines.
(b) Bounds for the maximum flow function at
any value of µ are given by two surfaces. The
flow-kick system will not be resilient for values
(k, τ ) above these surfaces.
Figure 2: Resilience regions for ẋ = x − x3 + µ.
First is to look for case studies on which to apply the flow-kick system model. Second, I plan on
examining one-dimensional systems with alternative stable states and different types of parameters,
through co-mentoring a student at Bowdoin College who is designing and implementing numerical
experiments to explore the relationships between the type of parameter (additive, multiplicative,
etc) and the asymptotic and tangential bounds of resilience. Third, I plan to extend results about
fixed points and invariant sets to higher dimensional systems. Starting with two-dimensional system
and then generalizing to n-dimensional systems, I will answer the question, is it possible to predict
the stability of a new fixed point of the flow-kick map from the underlying vector field? For a fixed
point x̄ = Φ(x̄) = φτ (x̄) + k, the linearization of the flow-kick system is a linear nonautonomous
system u̇ = A(t)u where A(t) = Dx φτ |x=x̄ . Studying solutions of such nonautonomous systems
and generalizations to disturbances that are not discrete would involve highly computational work.
The usual method for analyzing stability uses a continuous QR decomposition of the fundamental
matrix solution of u̇ = A(t)u to compute approximations of Lyaponov exponents.
It is not surprising to me that the study of resilience comes to need nonautonomous theory for
dynamical systems as the laws and relationships modeled in dynamical systems change over time.
In the next section, more background is given for nonautonomous stability and bifurcation theory.
2.2
Tipping and Bifurcations
Tipping points are a particularly popular way of describing large scale, dramatic changes to Earth’s
climate. However, climate science and other fields have developed many related but different definitions for tipping points. Mathematically, a system is said to “tip” when there’s a sudden, irreversible
change [1]. This has been interpreted to mean behavior like that which includes saddle-node bifurcations and hysteresis, but can also occur without a bifurcation. Tipping has been characterized
into types: bifurcation tipping, noise-induced tipping such as in stochastically perturbed dynamical systems, and rate-induced tipping [2]. An example of rate-induced tipping in global climate
applications is given in the paper by Wieczorek et al.[20] on the compost bomb instability where
they show that it is not that atmospheric temperature which causes the system to tip, but the
rate of global warming. My research is specific to the relationship of rate-induced tipping and
nonautonomous bifurcations.
Rate-induced tipping has not been defined generally, but has been characterized in 1-dimensional
systems with specific parameter shifts [1]. For the system ẋ = f (x, λ(t)), tipping will occur if
λ(t) passes through a bifurcation value of the autonomous system ẋ = f (x, λ). However, in the
case where λ(t) is asymptotically constant and does not pass through a bifurcation point of the
parameterized system, tipping has been shown to occur if λ(t) varies at a critical rate. In other
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words, if we consider the extended phase space
λ̇ = λ0 (t)
ẋ = f (x, λ)
tipping corresponds to heteroclinic connections. Recent work on rate-induced tipping [1] uses pullback attractors in nonautonomous dynamical systems, quasi-static solutions that are attracting in
backwards time. Pullback attractors play a large part in bifurcation theory for nonautonomous
systems [16], which seems to indicate a relationship between tipping and nonautonomous bifurcations.
However, bifurcations in nonautonomous systems are a new idea in mathematics [9, 16], and are
often characterized as merging two solutions with different stability behaviors. Stability behavior for
a fundamental matrix solution X(t) of the linearized nonautonomous dynamical system ẋ = A(t)x
is assessed using Lyaponov exponents
λj (Xej ) = lim sup 1t ln |Xej |
t→∞
where ej is the set of standard basis vectors. If all Lyaponov exponents are negative, then the
equilibrium solution x = 0 is asymptotically stable. Bifurcations are characterized by sign changes
in the Lyaponov exponents for equilibrium solutions.
Unfortunately, it is often analytically intractable to solve for the fundamental matrix solution
X(t), and a computational approximation is needed instead. A continuous QR decomposition of
X(t) where Q(t) is an orthogonal matrix and R(t) is an upper triangular matrix, can be used to
define the upper triangular matrix B(t) = QT (t)A(t)Q(t) − QT Q̇. This is a Lyaponov transform
and preserves the Lyaponov exponents so that one can compute Lyaponov exponents as λj =
Z
1 t
lim
Bjj (τ )dτ . Finite time Lyaponov exponents can be found by removing the limit and
t→∞ t 0
calculating this quantity for any time T > 0 [7, 17].
My research starts with creating a computational program to examine the criteria for when a
rate-induced tipping point and nonautonomous bifurcation coincide. Work was started at the AMS
MRC on Differential Equations, Sea Ice, and Probability in June 2015. The research group started
with a simple 1D example with asymptotically constant parameter λ(t)
ẋ = −(x − λ + 12 )(x − λ)(x − λ − 21 )
λ̇ = aλ(1 − λ).
We see that for a critical rate a ≈ 4.5, tipping occurs. In this example, tipping can be realized as
a global bifurcation involving a heteroclinic connection. This critical value of a also exhibits sign
changes for finite time Lyaponov exponents.
Additional computations have been done on the compost bomb instability [20] that models the
response of peatlands to global warming. Peatlands sequester carbon at low temperatures and
release carbon into the atmosphere at high temperatures. The system models the relationship
between the carbon sequestered in peatlands C and the soil temperature T
εṪ = Cr(T ) −
λ
A (T
− Ta )
Ċ = Π − Cr(t)
Ṫa = ν > 0
where r(T ) = r0 exp(αT ), and Ta is atmospheric temperature. We find the critical rate and tipping
point correlates topologically to solutions on the stable manifold of a folded saddle, not unlike a
heteroclinic connection in the extended phase space. Moreover, the finite time Lyapnov exponents
also change sign at this critical value of vc .
Future work. I plan to refine computational methods for calculating nonautonomous bifurcations as an indicator of tipping. For example, for more general systems where λ(t) may
not be asymptotically constant, the proposed indicators of nonautonomous bifurcations (finitetime Lyponov exponents) will not be useful because they will indicate solutions will becoming
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infinite, rather
R t+h than indicating a bifurcation. Consideration of sequences of Steklov averages
Sj (t) = h1 t Bjj (τ )dτ or of Lyaponov vectors (directions of growth which can be calculated
from the Q(t) matrix in the QR decomposition) may be more robust indicators of bifurcations and
tipping.
In low dimensional examples, it is possible to analytically determine critical rates for tipping.
However, this will be impossible in higher dimensional problems. Having a computational tool and
second indicator for tipping would be incredibly useful, and highly applicable to climate science in
particular.
3
Relative Equilibria in the N -Vortex Problem
The n-vortex problem models the interactions of point vortices in an inviscid, incompressible
fluid. The n-vortex problem was historically formulated by Kirkhoff to describe electromagnetic
flows. Lord Kelvin proposed a point-vortex model as a now debunked model of electrons in an
atom. However, the n-vortex problem is still relevant for its models of electron configurations in
a Malmberg-Penning Trap, and for modeling hurricanes and other geophysical vortices, and for
vortices in ideal fluids, like the atmospheres of gas giant planets such as Jupiter and Saturn. To
each point vortex in the plane, there is an associated strength, called the circulation, written Γi .
Let zi (t) = (xi (t), yi (t)) ∈ R2 be the position of the n point vortices. The flow of the system is
Γi ẋi =
∂H
∂yi
Γi ẏi = −
∂H
∂xi
where H is the Hamiltonian function
H(zi ) = −
n
X
Γi Γj log |zi − zj |.
i,j=1,i<j
My research focuses on relative equilibria solutions of the (N + 1)-vortex problem. Relative
equilibria solutions to the n-vortex problem where the shape of the configurations of n vortices
is fixed while the vortices rotate with constant angular velocity around a fixed center of vorticity,
which is analogous to center of mass. These solutions are fixed points when the equations of motion
are written in a rotating coordinate system. The (N + 1)-vortex problem is defined by one large
central vortex of strength Γ0 = 1, and N smaller vortices of strength Γi = µi where µi ∈ R for
i = 1, ..., N , and 0 < << 1. Models of hurricanes and electron configurations use a large central
vortex surrounded by a number of small vortices, much like Maxwell’s famous model of planetary
rings. Following the work done by Hall in [8], Moeckel in [12], and Barry et al in [3], I considered
the problem of relative equilibria in the limit as → 0. Relative equilibria of the (1 + N )-vortex
problem are defined as the limits as of relative equilibria in the (N + 1)-vortex problem, where
> 0, as opposed to looking at the restricted problem of N infinitesimally small vortices, in the
vortex-equivalent of N central force problems. In the limit, the large vortex is at the origin, and
the small vortices are arranged on the unit circle. The first theorem gives the limiting positions of
the small vortices on the unit circle and the second theorem gives conditions for linear stability of
relative equilibria.
Theorem 1. The configuration of N vortices (r, θ) = (1, ..., 1, θ1 , ...θN ) is a relative equilibrium of
the (1 + N )-vortex problem if and only if θ is a critical point of the function
X
V (θ) = −
µi µj [cos(θi − θj ) + 12 log(2 − 2 cos(θi − θj ))].
i<j
Alanna Hoyer-Leitzel
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(a) Stereographic Projection of Bifurcation Diagram for the (1+3)-Vortex Problem. The bifurcation curve for traditional bifurcations f is shown in red. For context, the curve
where µ1 + µ2 + µ3 = 0 (black circle) and the
three curves (blue, green, orange) where µi = 0,
i = 1, 2, 3 are included. The pink circle in 3a
is the curve h = µ1 µ2 + µ1 µ3 + µ2 µ3 = 0 where
there is a bifurcation at the singularity in V , corresponding to all 3 small vortices colliding.
(b) Numerical Result on Linear Stability
in the (1+3)-Vortex Problem. Overlaid on
3a, are the dotted grey, orange, and black curves,
which correspond to the three conditions for linear stability. The shaded regions correspond to
values of (µ1 , µ2 , µ3 ) where there is at least one
stable family of relative equilibria.
Figure 3: (1+3)-Vortex Problem Pictures
Theorem 2. Let µ = Diag(µ1 , µ2 , ..., µN ). Let (ρ , φ ) be a sequence of relative equilibria of the
(N + 1)-vortex problem which converges to a relative equilibrium (ρ, φ) = (1, ..., 1, φ1 , ..., φN ) of the
(1 + N )-vortex problem as → 0, and let φ be a nondegenerate critical point of V . For sufficiently
small, (ρ , φ ) is nondegenerate and is linearly stable if and only if µ−1 Vθθ has N − 1 nonzero
positive eigenvalues.
In the case when N = 2, the critical points of V are collinear and equilateral triangle configurations for all values of the circulation parameters (µ1 , µ2 ), which is as we expect since the
(1 + 2)-vortex problem is a subcase of the 3-vortex problem.
The lowest dimensional non-trivial case is when N = 3, a subcase of the 4-vortex problem.
Here the number of critical points of V is dependent on the values of µi . By writing the equations
for degenerate critical points of V as a system of polynomial equations, it is possible to use a
Groebner basis that eliminates the position variables and that gives a bifurcation curve in the
circulation parameters (µ1 , µ2 , µ3 ). Additionally, a bifurcation happens at a nondegenerate critical
point at the singularity of V , where all three small vortices collide. Figure 3a shows a stereographic
projection of these bifurcation curves from the unit sphere in the parameter space to the plane.
Using the Sturm algorithm for counting roots, there are 5 components (counted radially) in this
diagram. Barry et al found that for equal µi and sufficiently small, there are exactly 14 relative
equilibria up to rotations. This corresponds to the center most region of the bifurcation diagram.
Using the Hermite method for root counting, I found the following result.
Theorem 3. There are 14, 10, or 8 relative equilibria in the (1 + 3)-vortex problem. It follows that
for sufficiently small, there are 14, 10, or 8 relative equilibria in the 4-vortex problem for vortices
with circulations Γ0 = 1, Γi = µi , i = 1, 2, 3.
Alanna Hoyer-Leitzel
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Additionally it is easy to use a Groebner basis and the partial derivatives of V to prove a result
about the symmetry of configurations in the (1 + 3)-vortex problem.
Theorem 4. If a relative equilibrium of the (1 + 3)-vortex problem is symmetric, then two of the
infinitesimal vortices have equal weight i = j , i 6= j.
Results on linear stability can be found numerically. Unfortunately, a similar technique using
a Groebner basis on the equations for two positive non-zero eigenvalues of µ−1 Vθθ is not computationally possible on a computer with reasonable amounts of memory. Instead, the conditions for
stability can be examined in the configuration space, rather than the circulation parameter space.
The null-space of the matrix µ−1 ∇V , which is equivalent to the set of critical points for V , is linear
in µ1 , µ2 , and µ3 and is one dimensional. Thus configurations of the three small vortices on the unit
circle correspond to a unique (up to scaling) set of circulation parameters. Using the null-space
map, it is possible to numerically examine regions of stability in the parameter space after first
looking at the regions in the configuration space. These are pictured in Figure 3b.
Future Work. Immediate lines of inquiry involve generalizing to the (1 + 4)-vortex problem
and applying the Sturm Algorithm to count relative equilibria for specific combinations of µi = ±1,
and to the (2 + N )-case with two large vortices and N small vortices.
Another related problem of relative equilibria is clusters of vortices. In the theorems above and
in [3], it is assumed that the positions of the point vortices are bounded away from each other and
that vortices will not coalesce in the limit as → 0. If vortices did coalesce, this configuration is
called a cluster. Some work has been done by O’Neil in [14] and [15] on the problem of vortex
clusters of relative equilibria. However, the problem of vortex collisions in general is relatively open
[13]. Clusters small masses with two or more large masses are studied in the n-body problem, and
so techniques for those problems may carry over to the n-vortex problem.
I also work with Anna Barry at the University of British Colombia on a generalization of the
classical vortex problem to the system
ẋi = −C(α)
X
ẏi = −C(α)
X
Γj
yi − yj
|zi − zj |4−α
Γj
xi − xj
|zi − zj |4−α
j6=i
j6=i
where 0 ≤ α ≤ 3. The cases when α = 1, 2, 3 all correspond to known physical scenarios. We are
working to see if results for the classical case when α = 2 for the (1 + N )-vortex problem extend
to some interval around α = 2, as well as Roberts [cite gareth’s paper] might generalize.
4
Undergraduate Research Projects
As part of my research in resilience, I have co-mentored two students doing independent study
projects. After defining the problem of resilience and a kick-flow system for these students, they
were able to ask astute questions and design some experiments of their own. For example, one
student is exploring the relationship between the steepness of a potential and the resilience of
a system, and another student is looking at the difference in the behavior of a kick-flow system
with different sized kicks and flow times, but with the same total perturbation. Using or adapting
the Matlab code developed by a collaborator in the resilience group, they are able to test their
ideas. The students are able to build intuition and make conjectures about resilience. Meanwhile,
through discussions and reading, the students learn the mathematical tools and formalism necessary
for proving their conjectures.
The Resilience Group and MCRN offer many opportunities for students to work on climate
problems, which are relevant and pressing problems to students. Additionally, they are able to
Alanna Hoyer-Leitzel
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collaborate with researchers at other universities and colleges. If a student wants to explore a
climate question with which I’m not familiar, I have resources to connect them to someone who
does.
The n-vortex problem has open questions where students are able to find research projects.
The problem of vortex clusters is perfect for student exploration. Here again, a curious student can
ask many questions about clusters, and with the aid of some Mathematica code, the student can
experiment and make conjectures.
In general, dynamical systems is full of problems for undergraduate research as the components
of modeling, numerical approximation and analysis are accessible to undergraduate students, and
I am open to mentoring any dynamical systems project that interests a student. Undergraduate
research is about letting students explore an idea, examine what they know about mathematics,
learn a little more, and gently guiding them to a result. For most problems in dynamical systems,
all a student needs to get started is curiosity, an introductory differential equations class, linear
algebra or multivariable calculus, and a willingness to learn how to use a computer program, for
example Mathematica, Matlab, or XPP and Auto.
References
[1] Peter Ashwin, Clare Perryman, and Sebastian Wieczorek. Parameter shifts for nonautonomous systems in low dimension: Bifurcation-and rate-induced tipping. arXiv preprint
arXiv:1506.07734, 2015.
[2] Peter Ashwin, Sebastian Wieczorek, Renato Vitolo, and Peter Cox. Tipping points in open
systems: bifurcation, noise-induced and rate-dependent examples in the climate system. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 370(1962):1166–1184, 2012.
[3] Anna M Barry, Glen R Hall, and C Eugene Wayne. Relative equilibria of the (1+ n)-vortex
problem. Journal of nonlinear science, 22(1):63–83, 2012.
[4] Beatrix E Beisner, C Lisa Dent, and Stephen R Carpenter. Variability of lakes on the landscape:
roles of phosphorus, food webs, and dissolved organic carbon. Ecology, 84(6):1563–1575, 2003.
[5] Steve Carpenter, Brian Walker, J Marty Anderies, and Nick Abel. From metaphor to measurement: resilience of what to what? Ecosystems, 4(8):765–781, 2001.
[6] Lei Dai, Kirill S Korolev, and Jeff Gore. Relation between stability and resilience determines
the performance of early warning signals under different environmental drivers. Proceedings of
the National Academy of Sciences, 112(32):10056–10061, 2015.
[7] Luca Dieci and Erik S Van Vleck. Lyapunov and sacker–sell spectral intervals. Journal of
dynamics and differential equations, 19(2):265–293, 2007.
[8] G. R. Hall. Central configurations in the planar 1 + n body problem. preprint, 1988.
[9] JA Langa, JC Robinson, and A Suárez. Bifurcations in non-autonomous scalar equations.
Journal of Differential Equations, 221(1):1–35, 2006.
[10] Simon A Levin and Jane Lubchenco. Resilience, robustness, and marine ecosystem-based
management. Bioscience, 58(1):27–32, 2008.
[11] Stephen Ligtenburg. Mathematical notions of resilience: the effects of disturbance in onedimensional nonlinear systems. Honors Thesis, Bowdoin College, 2015.
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[12] Richard Moeckel. Linear Stability of Relative Equilibria with a Dominant Mass. Journal of
Dynamics and Differential Equations, 6(1), 1994.
[13] Paul K Newton. The N-vortex problem: analytical techniques, volume 145. Springer, 2001.
[14] Kevin A O’Neil. Singular continuation of point vortex relative equilibria on the plane and
sphere. Nonlinearity, 26(3):777, 2013.
[15] Kevin A ONeil. Clustered equilibria of point vortices.
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[16] Martin Rasmussen.
Springer, 2007.
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Attractivity and bifurcation for nonautonomous dynamical systems.
[17] Erik S Van Vleck. On the error in the product qr decomposition. SIAM Journal on Matrix
Analysis and Applications, 31(4):1775–1791, 2010.
[18] Brian Walker, Crawford S Holling, Stephen R Carpenter, and Ann Kinzig. Resilience, adaptability and transformability in social–ecological systems. Ecology and society, 9(2):5, 2004.
[19] Brian Walker and David Salt. Resilience Thinking: Sustaining Ecosystems and People in a
Changing World. Island Press, 2006.
[20] Sebastian Wieczorek, Peter Ashwin, Catherine M Luke, and Peter M Cox. Excitability in
ramped systems: the compost-bomb instability. In Proceedings of the Royal Society of London
A: Mathematical, Physical and Engineering Sciences, volume 467, pages 1243–1269. The Royal
Society, 2011.