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SIAM J. on Applied Dynamical Systems

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2011

Volume 10, Issue 4, pp. 1213-1553


Collective Behaviors in Two-Dimensional Systems of Interacting Particles

Joceline Lega

SIAM J. Appl. Dyn. Syst. 10, pp. 1213-1231 (19 pages)

Online Publication Date: October 04, 2011

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This article presents results of molecular dynamics simulations that show the emergence of collective behaviors in a two-dimensional system of particles (hard disks) interacting through a properly chosen collision rule. The particles, which are of finite size and are in free flight between collisions, are not self-propelled. They tumble randomly like bacteria and interact only when they collide, not through continuous potential forces. This work therefore indicates that interactions at the microscopic level, which occur only locally and discretely both in time and space, are sufficient to lead to large-scale macroscopic behaviors. Order parameters that capture and quantify the formation of collective behaviors are introduced and used to describe how the choice of collision rule affects the steady state dynamics of the system, by comparing the outcome to the standard case of elastic collisions. This work was motivated by recent results on the dynamics of bacterial colonies. Possible applications of the present approach to other systems are also discussed.

Shared Inputs, Entrainment, and Desynchrony in Elliptic Bursters: From Slow Passage to Discontinuous Circle Maps

Guillaume Lajoie and Eric Shea-Brown

SIAM J. Appl. Dyn. Syst. 10, pp. 1232-1271 (40 pages)

Online Publication Date: October 25, 2011

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What input signals will lead to synchrony versus desynchrony in a group of biological oscillators? This question connects with both classical dynamical systems analyses of entrainment and phase locking and with emerging studies of stimulation patterns for controlling neural network activity. Here, we focus on the response of a population of uncoupled, elliptically bursting neurons to a common pulsatile input. We extend a phase reduction from the literature to capture inputs of varied strength, leading to a circle map with discontinuities of various orders. In a combined analytical and numerical approach, we apply our results to both a normal form model for elliptic bursting and to a biophysically based neuron model from the basal ganglia. We find that, depending on the period and amplitude of inputs, the response can appear either chaotic (with provably positive Lyapunov exponent for the associated circle maps) or periodic with a broad range of phase-locked periods. Throughout, we discuss the critical underlying mechanisms, including slow-passage effects through Hopf bifurcation, the role and origin of discontinuities, and the impact of noise.

Periodically Forced Hopf Bifurcation

Yanyan Zhang and Martin Golubitsky

SIAM J. Appl. Dyn. Syst. 10, pp. 1272-1306 (35 pages)

Online Publication Date: October 27, 2011

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We study a periodically forced system of ODEs near a point of Hopf bifurcation, where the forcing is pure harmonic, has small amplitude $\varepsilon$, and has forcing frequency $\omega_F$ that is near the Hopf frequency $\omega_H$. In this system we vary the forcing frequency and determine all small amplitude periodic solutions to the forced system that have frequency $\omega_F$. In other words, we determine how the number of $\frac{2\pi}{\omega_F}$-periodic solutions to the forced system changes with $\omega_F$. This problem is complicated because of the existence of three small parameters: the amplitude of the forcing $\varepsilon$, the deviation of the bifurcation parameter from the point of Hopf bifurcation $\lambda$, and the relative deviation of the forcing frequency from the Hopf frequency $\omega=\frac{\omega_F-\omega_H}{\omega_F}$. Our results are presented in terms of bifurcation diagrams of amplitude of periodic solution versus $\omega$ for fixed $\varepsilon$ and $\lambda$. We assume that the unforced system has a supercritical Hopf bifurcation at $\lambda=0$ and that the coefficient of the cubic term of the Hopf bifurcation normal form is $\gamma_R+i\gamma_I$. We find that the qualitative form of the bifurcation diagrams depends on $\gamma=\gamma_I/\gamma_R$. For example, if $\lambda<0$ (so that the equilibrium is stable and there are no small amplitude periodic solutions in the unforced system), then multiplicity of periodic solutions of the forced system occurs in the bifurcation diagrams precisely when $|\gamma|>\sqrt{3}$.

Scaling in Singular Perturbation Problems: Blowing Up a Relaxation Oscillator

Ilona Kosiuk and Peter Szmolyan

SIAM J. Appl. Dyn. Syst. 10, pp. 1307-1343 (37 pages)

Online Publication Date: October 27, 2011

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A detailed geometric analysis of the Goldbeter–Lefever model of glycolytic oscillations is given. In suitably scaled variables the governing equations are a planar system of ordinary differential equations depending singularly on two small parameters $\varepsilon$ and $\delta$. In [L. Segel and A. Goldbeter, J. Math. Biol., 32 (1994), pp. 147–160] it was argued that a limit cycle of relaxation type exists for $\varepsilon\ll\delta\ll1$. The existence of this limit cycle is proved by analyzing the problem in the spirit of geometric singular perturbation theory. The degeneracies of the limiting problem corresponding to $(\varepsilon,\delta)=(0,0)$ are resolved by a novel variant of the blow-up method. It is shown that repeated blow-ups lead to a clear geometric picture of this fairly complicated two-parameter multiscale problem.

2D Phase Diagram for Minimizers of a Cahn–Hilliard Functional with Long-Range Interactions

Rustum Choksi, Mirjana Maras, and J. F. Williams

SIAM J. Appl. Dyn. Syst. 10, pp. 1344-1362 (19 pages)

Online Publication Date: November 08, 2011

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This paper presents a two-dimensional investigation of the phase diagram for global minimizers to a Cahn–Hilliard functional with long-range interactions. Based upon the $H^{-1}$ gradient flow, we introduce a hybrid numerical method to navigate through the complex energy landscape and access an accurate depiction of the ground state of the functional. We use this method to numerically compute the phase diagram in a (finite) neighborhood of the order-disorder transition. We demonstrate a remarkably strong agreement with the standard asymptotic estimates for stability regions based upon a small parameter measuring perturbation from the order-disorder transition curve.

Passive and Self-Propelled Locomotion of an Elastic Swimmer in a Perfect Fluid

Alexandre Munnier

SIAM J. Appl. Dyn. Syst. 10, pp. 1363-1403 (41 pages)

Online Publication Date: November 29, 2011

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In this paper we are interested in studying the free motion of a hyperelastic body (also called a swimmer) immersed in a perfect fluid. We derive the Euler–Lagrange equations from the least action principle of Lagrangian mechanics and prove that they are well-posed when the number of elastic modes is finite. The recourse to a strain energy density function in the modeling allows many different constitutive equations for the hyperelastic material to be considered. We perform numerical simulations, aiming to study passive locomotion (i.e., locomotion at zero energy cost). As a first quite surprising result, we observe that the swimmer does not even have to be elastic to experience passive locomotion in its idealized environment. Indeed, we provide an example of a deformable (but nonelastic) swimmer, for which the fluid-body system behaves as an oscillating mechanical system. The shape changes caused solely by the hydrodynamical forces on the body's boundary turn out to be periodic strokes resulting in locomotion. This phenomenon can be seen as a generalization, to deformable bodies, of the famous D'Alembert paradox [J. l. R. D'Alembert, Essai d'une nouvelle théorie de la résistance des fluides, Paris, 1752], claiming that the drag force is zero on a rigid solid moving with constant velocity. Many other examples of passive locomotion, involving different types of hyperelastic swimmers, are studied. Special attention is devoted to the study of energy and impulse exchanges between the fluid and the body. In the last section, we assume that the swimmer has the ability to modify its shape by means of internal forces. We prove that in this case the equations of motion are still well-posed, and we illustrate again with numerical simulations that, starting from rest, self-propelled locomotion can be achieved.

Synchrony-Breaking Bifurcation at a Simple Real Eigenvalue for Regular Networks 1: 1-Dimensional Cells

Ian Stewart and Martin Golubitsky

SIAM J. Appl. Dyn. Syst. 10, pp. 1404-1442 (39 pages)

Online Publication Date: December 06, 2011

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We study synchrony-breaking local steady-state bifurcation in networks of dynamical systems when the critical eigenvalue is real and simple, using singularity theory to transform the bifurcation into normal form. In a general dynamical system, a generic steady-state local bifurcation from a trivial state is transcritical. In the presence of symmetry, a pitchfork is also possible generically. Network structure introduces constraints that may change the generic behavior. We consider regular networks, in which all cells have the same type and all arrows have the same type, and every cell receives inputs from the same number of arrows. A characterization of all smooth admissible maps permits a singularity-theoretic analysis based on Liapunov–Schmidt reduction. Assuming that the cells have 1-dimensional internal dynamics, we give conditions on the critical eigenvectors of the linearization and its transpose that determine when a generic bifurcation is transcritical, pitchfork, or more degenerate. We prove that for all regular $n$-cell networks, such bifurcations are generically $n$-determined. In the path-connected case, this is improved to $(n-1)$-determined. In bidirectional networks, generic bifurcation is transcritical or pitchfork, but the role of symmetry is minor. In the general case, degenerate cases can occur: the network must have at least 4 cells (5 in the path-connected case). We give examples of networks for which generic bifurcations are degenerate, including a 6-cell network with a normal form that is determined only at degree 6 and a path-connected 5-cell network with a normal form that is determined only at degree 4.

Interaction of Canard and Singular Hopf Mechanisms in a Neural Model

R. Curtu and J. Rubin

SIAM J. Appl. Dyn. Syst. 10, pp. 1443-1479 (37 pages)

Online Publication Date: December 06, 2011

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We consider an ordinary differential equation model for neural competition, presented previously in the study of binocular rivalry, which features two adapting populations of neurons interacting through mutual inhibition. This model is known to exhibit a variety of dynamic regimes, including mixed-mode oscillations (MMOs) featuring alternating small- and large amplitude oscillations, depending on the value of an input parameter. In this work, we use geometric dynamical systems techniques to study the structure of the model in the singular limit as well as the emergence of MMOs in the perturbed system. In particular, exploiting a normal form calculation allows us to numerically compute a way-in/way-out function, which we use to elucidate the interaction of canard and singular Hopf mechanisms for small amplitude oscillations that occur as the input parameter approaches a critical value.

Tinkerbell Is Chaotic

Alexandre Goldsztejn, Wayne Hayes, and Pieter Collins

SIAM J. Appl. Dyn. Syst. 10, pp. 1480-1501 (22 pages)

Online Publication Date: December 08, 2011

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Shadowing is a method of backward error analysis that plays a important role in hyperbolic dynamics. In this paper, the shadowing by containment framework is revisited, including a new shadowing theorem. This new theorem has several advantages with respect to existing shadowing theorems: It does not require injectivity or differentiability, and its hypothesis can be easily verified using interval arithmetic. As an application of this new theorem, shadowing by containment is shown to be applicable to infinite length orbits and is used to provide a computer assisted proof of the presence of chaos in the well-known noninjective Tinkerbell map.

Continuity of Resetting a Pacemaker in an Excitable Medium

Bartłomiej Borek, Leon Glass, and Bart E. Oldeman

SIAM J. Appl. Dyn. Syst. 10, pp. 1502-1524 (23 pages)

Online Publication Date: December 13, 2011

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Pacemakers in excitable media generate waves that propagate outward from the pacemaker. Such waves of excitation are well known in biological and chemical systems such as nerves, the heart, and the Belousov–Zhabotinsky reaction. Stimuli delivered at a distant site from the pacemaker can reset the pacemaker, leading to a change in the timing of the pacemaker. The relation between stimulus timing and resultant resetting of the pacemaker is captured by phase resetting curves. The continuity of resetting curves has been investigated in both experiments and numerical models. We present theoretical results discussing conditions for continuity of resetting curves as the amplitude and phase of the stimulus varies. We also use continuation and shooting methods to analyze the continuity of resetting curves in simple mathematical models of cardiac and neural activity. Under continuous changes of stimulus parameters, resetting curves will be continuous unless a stimulus leads to dynamics that fall outside the basin of attraction of the pacemaker-driven excitable medium.

Two Degenerate Boundary Equilibrium Bifurcations in Planar Filippov Systems

Fabio Dercole, Fabio Della Rossa, Alessandro Colombo, and Yuri A. Kuznetsov

SIAM J. Appl. Dyn. Syst. 10, pp. 1525-1553 (29 pages)

Online Publication Date: December 13, 2011

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We contribute to the analysis of codimension-two bifurcations in discontinuous systems by studying all equilibrium bifurcations of 2D Filippov systems that involve a sliding limit cycle. There are only two such local bifurcations: (1) a degenerate boundary focus, which we call the homoclinic boundary focus (HBF), and (2) the boundary Hopf (BH). We prove that—besides local bifurcations of equilibria and pseudoequilibria—the universal unfolding of the HBF singularity includes a codimension-one global bifurcation at which a sliding homoclinic orbit to a pseudosaddle exists, while that of the BH singularity has a codimension-one bifurcation curve along which a cycle grazing occurs. We define two canonical forms, one for each singularity, to which a generic 2D Filippov system can be locally reduced by smooth changes of variables and parameters and time reparametrization. Explicit genericity conditions are also provided, as well as the asymptotics of the bifurcation curves in the two-parameter space. We show that both studied codimension-two bifurcations occur in a known 2D Filippov system modeling an ecosystem subject to on-off harvesting control, and we provide two Mathematica scripts that automatize all computations.
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