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SIAM J. on Applied Dynamical Systems

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2008

Volume 7, Issue 4, pp. 1131-1611


The Geometry of Slow Manifolds near a Folded Node

M. Desroches, B. Krauskopf, and H. M. Osinga

SIAM J. Appl. Dyn. Syst. 7, pp. 1131-1162 (32 pages) | Cited 4 times

Online Publication Date: October 13, 2008

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This paper is concerned with the geometry of slow manifolds of a dynamical system with one fast and two slow variables. Specifically, we study the dynamics near a folded-node singularity, which is known to give rise to so-called canard solutions. Geometrically, canards are intersection curves of two-dimensional attracting and repelling slow manifolds, and they are a key element of slow-fast dynamics. For example, canard solutions are associated with mixed-mode oscillations, where they organize regions with different numbers of small oscillations. We perform a numerical study of the geometry of two-dimensional slow manifolds in the normal form of a folded node in $\mathbb{R}^3$. Namely, we view the part of a slow manifold that is of interest as a one-parameter family of orbit segments up to a suitable cross-section. Hence, it is the solution of a two-point boundary value problem, which we solve by numerical continuation with the package AUTO. The computed family of orbit segments is used to obtain a mesh representation of the manifold as a surface. With this approach we show how the attracting and repelling slow manifolds change in dependence on the eigenvalue ratio $\mu$ associated with the folded-node singularity. At $\mu = 1$ two primary canards bifurcate and secondary canards are created at odd integer values of $\mu$. We compute 24 secondary canards to investigate how they spiral more and more around one of the primary canards. The first sixteen secondary canards are continued in $\mu$ to obtain a numerical bifurcation diagram.

Asymptotics of a Slow Manifold

J. Vanneste

SIAM J. Appl. Dyn. Syst. 7, pp. 1163-1190 (28 pages)

Online Publication Date: October 13, 2008

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Approximately invariant elliptic slow manifolds are constructed for the Lorenz–Krishnamurthy model of fast-slow interactions in the atmosphere. As is the case for many other two-time-scale systems, the various asymptotic procedures that may be used for this construction diverge, and there are no exactly invariant slow manifolds. Valuable information can however be gained by capturing the details of the divergence: this makes it possible to define exponentially accurate slow manifolds, identify one of these as optimal, and predict the amplitude and phase of the fast oscillations that appear for trajectories started on it. We demonstrate this for the Lorenz–Krishnamurthy model by studying the slow manifolds obtained using a power-series expansion procedure. We develop two distinct methods to derive the leading-order asymptotics of the late coefficients in this expansion. Borel summation is then used to define a unique slow manifold, regarded as optimal, which is piecewise analytic in the slow variables. This slow manifold is not analytic on a Stokes surface: when slow solutions cross this surface, they switch on exponentially small fast oscillations through a Stokes phenomenon. We show that the form of these oscillations can be recovered from the Borel summation. The approach that we develop for the Lorenz–Krishnamurthy model has a general applicability; we sketch how it generalizes to a broad class of two-time-scale systems.

Novel Vehicular Trajectories for Collective Motion from Coupled Oscillator Steering Control

Margot Kimura and Jeff Moehlis

SIAM J. Appl. Dyn. Syst. 7, pp. 1191-1212 (22 pages)

Online Publication Date: October 24, 2008

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We consider a model for vehicle motion coordination for three vehicles that uses coupled oscillator steering control. Prior work on such models has focused primarily on sinusoidal coupling functions, which typically give behavior in which individual vehicles move either in straight lines or in circles. We show that other, more exotic trajectories are possible when more general coupling functions are considered. Such trajectories are associated with periodic orbits in the steering control subsystem. The proximity of these periodic orbits to heteroclinic bifurcations allows for a detailed characterization of the properties of the vehicular trajectories.

A Hamiltonian Analogue of the Meandering Transition

Claudia Wulff

SIAM J. Appl. Dyn. Syst. 7, pp. 1213-1246 (34 pages)

Online Publication Date: October 24, 2008

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In this paper a Hamiltonian analogue of the well-known meandering transition from rotating waves to modulated rotating and modulated traveling waves in systems with the Euclidean symmetry of the plane is presented. In non-Hamiltonian systems, for example, in spiral wave dynamics, this transition is a Hopf bifurcation in a corotating frame, as external parameters are varied, and modulated traveling waves occur only at certain resonances. In Hamiltonian systems, for example, in systems of point vortices in the plane, the conserved quantities of the system, angular and linear momentum, are natural bifurcation parameters. Depending on the symmetry properties of the momentum map, either modulated traveling waves do not occur, or, in contrast to the dissipative case, modulated traveling waves are the typical scenario near rotating waves, as momentum is varied. Systems with the symmetry group of a sphere and with the Euclidean symmetry group of three-dimensional space are also treated.

Traveling Waves and Synchrony in an Excitable Large-Scale Neuronal Network with Asymmetric Connections

William C. Troy

SIAM J. Appl. Dyn. Syst. 7, pp. 1247-1282 (36 pages) | Cited 3 times

Online Publication Date: October 24, 2008

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We study (i) traveling wave solutions, (ii) the formation and spatial spread of synchronous oscillations, and (iii) the effects of variations of threshold in a system of integro-differential equations which describe the activity of large-scale networks of excitatory neurons on spatially extended domains. The independent variables are the activity level $u$ of a population of excitatory neurons which have long range connections, and a recovery variable $v$. In the integral component of the equation for $u$ the firing rate function is the Heaviside function, and the coupling function $w$ is positive. Thus, there is no inhibition in the system. There is a critical value of the parameter $\beta$ ($\beta_{*} > 0$) that appears in the equation for $v$, at which the eigenvalues $\mu^{\pm}$ of the linearization of the system around the rest state $(u,v) = (0,0)$ change from real to complex. We focus on the range $\beta > \beta_{*}$, where $\mu^{\pm}$ are complex, and analyze properties of wave fronts and 1-pulse and 2-pulse waves when the connection function $w$ is asymmetric. For wave fronts we demonstrate how an initial stimulus evolves into two solutions which propagate in opposite directions with different speeds and shapes. For 1-pulse waves our main theoretical result (Theorem theorem1aaa) shows that there is a range of $\beta > \beta_{*}$ where two families of waves exist, each consisting of infinitely many solutions. The waves in these two families also propagate in opposite directions with different speeds and shapes. There is a critical value $\theta^{*} > 0$ such that if $\theta > \theta^{*}$, then 1-pulse waves can propagate only in one direction. In addition, there is a second critical $\beta$ value, $\beta^{*} > \beta_{*}$, where bulk oscillations come into existence and the system becomes bistable. When $\beta \geq \beta_{*}$ we show how an initial stimulus evolves into a solution with large amplitude oscillations that spread out uniformly from the point of stimulus. The asymmetry in $w$ causes the rate of spread of the “region of synchrony” to be more rapid to the right of the point of stimulus than to the left. When $\theta > \theta^{*}$ we construct a “unidirectional” circuit where synchronization in one region can trigger synchronization in a distant, second region. However, when synchronization is initially triggered in the second region, it cannot spread to the first region.

TC-HAT ($\widehat{TC}$): A Novel Toolbox for the Continuation of Periodic Trajectories in Hybrid Dynamical Systems

Phanikrishna Thota and Harry Dankowicz

SIAM J. Appl. Dyn. Syst. 7, pp. 1283-1322 (40 pages) | Cited 5 times

Online Publication Date: October 31, 2008

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This paper describes the underlying formulation and functionality of the newly developed software program $\widehat{\text{{\sc tc}}}$ (“tc-hat”), to perform bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, often referred to as hybrid dynamical systems. Boundary-value-problem formulations corresponding to single- and two-parameter continuations of periodic trajectories and selected associated codimension-one bifurcations in such systems are presented. Finally, the capabilities of the program are illustrated by performing bifurcation analysis of a few example hybrid dynamical systems.

Realization of Critical Eigenvalues for Scalar and Symmetric Linear Delay-Differential Equations

P.-L. Buono and V. G. LeBlanc

SIAM J. Appl. Dyn. Syst. 7, pp. 1323-1354 (32 pages)

Online Publication Date: October 31, 2008

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This paper studies the link between the number of critical eigenvalues and the number of delays in certain classes of delay-differential equations. There are two main results. The first states that for $k$ purely imaginary numbers which are linearly independent over the rationals, there exists a scalar delay-differential equation depending on $k$ fixed delays whose spectrum contains those $k$ purely imaginary numbers. The second result is a generalization of the first result for delay-differential equations which admit a characteristic equation consisting of a product of $s$ factors of scalar type. In the second result, the $k$ eigenvalues can be distributed among the different factors. Since the characteristic equation of scalar equations contain only exponential terms, the proof exploits a toroidal structure which comes from the arguments of the exponential terms in the characteristic equation. Our second result is applied to delay coupled $\mathbf{D}_n$-symmetric cell systems with one-dimensional cells. In particular, we provide a general characterization of delay coupled $\mathbf{D}_n$-symmetric systems with an arbitrary number of delays and cell dimension.

Singular Hopf Bifurcation in Systems with Two Slow Variables

John Guckenheimer

SIAM J. Appl. Dyn. Syst. 7, pp. 1355-1377 (23 pages) | Cited 7 times

Online Publication Date: October 31, 2008

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Hopf bifurcations have been studied intensively in two dimensional vector fields with one slow and one fast variable [É. Benoît et al., Collect. Math., 31 (1981), pp. 37–119; F. Dumortier and R. Roussarie, Mem. Amer. Math. Soc., 121 (577) (1996); W. Eckhaus, in Asymptotic Analysis II, Lecture Notes in Math. 985, Springer-Verlag, Berlin, 1983, pp. 449–494; M. Krupa and P. Szmolyan, SIAM J. Math. Anal., 33 (2001), pp. 286–314; J. Guckenheimer, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Sci. Ser. II Math. Phys. Chem. 137, Kluwer, Dordrecht, The Netherlands, 2004, pp. 295–316]. Canard explosions are associated with these singular Hopf bifurcations [S. M. Baer and T. Erneux, SIAM J. Appl. Math., 46 (1986), pp. 721–739; S. M. Baer and T. Erneux, SIAM J. Appl. Math., 52 (1992), pp. 1651–1664; B. Braaksma, J. Nonlinear Sci., 8 (1998), pp. 457–490; Y. Lijun and Z. Xianwu, J. Differential Equations, 206 (2004), pp. 30–54], manifested by a very rapid growth in the amplitude of periodic orbits. There has been less analysis of Hopf bifurcations in slow-fast systems with two slow variables where singular Hopf bifurcation occurs simultaneously with type II folded saddle-nodes [A. Milik and P. Szmolyan, in Multiple-Time-Scale Dynamical Systems, IMA Vol. Math. Appl. 122, Springer-Verlag, New York, 2001, pp. 117–140; M. Wechselberger, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 101–139]. This work contributes to our understanding of these Hopf bifurcations in five ways: (1) it computes the first Lyapunov coefficient of the bifurcation in terms of a normal form, (2) it describes global features of the flow that constrain the types of trajectories found in the system near the bifurcation, (3) it identifies codimension two bifurcations that occur as coefficients in the normal form vary, (4) it exhibits complex solutions that occur in the vicinity of the bifurcation for some values of the normal form coefficients, and (5) it identifies singular Hopf bifurcation as a mechanism for the creation of mixed-mode oscillations. A subtle aspect of the normal form is that terms of higher order contribute to the first Lyapunov coefficient of the bifurcation in an essential way.

Eulerian Equilibria of a Gyrostat in Newtonian Interaction with Two Rigid Bodies

J. A. Vera

SIAM J. Appl. Dyn. Syst. 7, pp. 1378-1396 (19 pages)

Online Publication Date: October 31, 2008

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In this paper the noncanonical Hamiltonian dynamics of a gyrostat in the three body problem will be examined. By means of geometric-mechanics methods we will study the approximate dynamics that arises when we develop the potential in Legendre series and truncate the series to the second harmonics. Some relative equilibria, called Eulerian, of the dynamics of a gyrostat in Newtonian interaction with two rigid bodies will be studied. Taking advantage of the results obtained in previous papers, working on the reduced problem, we will study the bifurcations of these relative equilibria. The instability of Eulerian relative equilibria if the gyrostat is close to a sphere is proven. The rotational Poisson dynamics of the gyrostat placed in an Eulerian equilibrium and the study of the nonlinear stability of some equilibria are considered. The analysis is done in vectorial form avoiding the use of canonical variables and the tedious expressions associated with these variables. In this way, the classic results on equilibria of the three body problem, many of them obtained by other authors who had made used of more classic techniques, are generalized.

Snaking of Multiple Homoclinic Orbits in Reversible Systems

J. Knobloch and T. Wagenknecht

SIAM J. Appl. Dyn. Syst. 7, pp. 1397-1420 (24 pages) | Cited 3 times

Online Publication Date: November 07, 2008

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We study $N$-homoclinic orbits near a heteroclinic cycle in a reversible system. The cycle is assumed to connect two equilibria of saddle-focus type. Using Lin's method, we establish the existence of infinitely many $N$-homoclinic orbits for each $N$ near the cycle. In particular, these orbits exist along snaking curves, thus mirroring the behavior of 1-homoclinic orbits. The general analysis is illustrated by numerical studies for a Swift–Hohenberg system.

Hysteresis in a Rotating Differentially Heated Spherical Shell of Boussinesq Fluid

Gregory M. Lewis and William F. Langford

SIAM J. Appl. Dyn. Syst. 7, pp. 1421-1444 (24 pages)

Online Publication Date: November 07, 2008

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A mathematical model of convection of a Boussinesq fluid in a rotating spherical shell is analyzed using numerical computations guided by bifurcation theory. The fluid is differentially heated on its inner spherical surface, with the temperature increasing from both poles to a maximum at the equator. The model is assumed to be both rotationally symmetric about the polar axis and reflectionally symmetric across the equator. This work is an extension to spherical geometry of previous work on the differentially heated rotating annulus. The spherical geometry is motivated by applications to planetary atmospheres. As the temperature gradient increases from zero, large Hadley cells extending from equator to poles form immediately. For larger temperature differences, two or three convection cells appear in each hemisphere. An organizing center is shown to exist, at which two saddle-node bifurcations come together in a codimension-2 hysteresis bifurcation (or cusp) point, providing a mechanism for hysteretic transitions between different cell patterns as the temperature gradient is varied.

Stable Synchrony in Globally Coupled Integrate-and-Fire Oscillators

Yu-Chuan Chang and Jonq Juang

SIAM J. Appl. Dyn. Syst. 7, pp. 1445-1476 (32 pages) | Cited 2 times

Online Publication Date: December 03, 2008

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A model of integrate-and-fire oscillators is studied. In the special case of identical oscillators, the model was first proposed and analyzed by Mirollo and Strogatz [SIAM J. Appl. Math., 50 (1990), pp. 1645–1662]. We assume, as in Mirollo and Strogatz's model, that each oscillator $x_i$ evolves according to a map $f_i$. Our main results are to demonstrate that the concavity structure of $f_i$ plays an important role in determining whether Peskin's second conjecture holds true. Specifically, the following statements are proved. First, the system of convex oscillators (i.e., $f”_i< 0$ for all $i$), in general, synchronizes when the oscillators are not quite identical. Second, the system of a certain class of concave oscillators (i.e., $f”_i> 0$ for all $i$) will not achieve synchrony for initial conditions in a set of positive measure when the oscillators are nearly identical. Third, the system of concave oscillators may achieve synchrony under certain sufficient conditions, provided that the oscillators are not quite nonidentical and that its concavity is small.

Algorithms for Rigorous Entropy Bounds and Symbolic Dynamics

Sarah Day, Rafael Frongillo, and Rodrigo Treviño

SIAM J. Appl. Dyn. Syst. 7, pp. 1477-1506 (30 pages) | Cited 3 times

Online Publication Date: December 03, 2008

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The aim of this paper is to introduce a method for computing rigorous lower bounds for topological entropy. The topological entropy of a dynamical system measures the number of trajectories that separate in finite time and quantifies the complexity of the system. Our method relies on extending existing computational Conley index techniques for constructing semiconjugate symbolic dynamical systems. Besides offering a description of the dynamics, the constructed symbol system allows for the computation of a lower bound for the topological entropy of the original system. Our overall goal is to construct symbolic dynamics that yield a high lower bound for entropy. The method described in this paper is algorithmic and, although it is computational, yields mathematically rigorous results. For illustration, we apply the method to the Hénon map, where we compute a rigorous lower bound of 0.4320 for topological entropy.

Asymptotic Expansions of I-V Relations via a Poisson–Nernst–Planck System

Nicole Abaid, Robert S. Eisenberg, and Weishi Liu

SIAM J. Appl. Dyn. Syst. 7, pp. 1507-1526 (20 pages) | Cited 2 times

Online Publication Date: December 03, 2008

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We investigate higher order matched asymptotic expansions of a steady-state Poisson–Nernst–Planck (PNP) system with particular attention to the I-V relations of ion channels. Assuming that the Debye length is small relative to the diameter of the narrow channel, the PNP system can be viewed as a singularly perturbed system. Special structures of the zeroth order inner and outer systems make it possible to provide an explicit derivation of higher order terms in the asymptotic expansions. For the case of zero permanent charge, our results concerning the I-V relation for two oppositely charged ion species are (i) the first order correction to the zeroth order linear I-V relation is generally quadratic in V; (ii) when the electro-neutrality condition is enforced at both ends of the channel, there is NO first order correction, but the second order correction is cubic in V. Furthermore (Theoremsigmo), up to the second order, the cubic I-V relation has (except for a very degenerate case) three distinct real roots that correspond to the bistable structure in the FitzHugh–Nagumo simplification of the Hodgkin–Huxley model.

Separatrix Splitting in 3D Volume-Preserving Maps

Héctor E. Lomelí and Rafael Ramírez-Ros

SIAM J. Appl. Dyn. Syst. 7, pp. 1527-1557 (31 pages)

Online Publication Date: December 10, 2008

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We construct a family of integrable volume-preserving maps in $\mathbb{R}^3$ with a two-dimensional heteroclinic connection of spherical shape between two fixed points of saddle-focus type. In other contexts, such structures are called Hill's spherical vortices or spheromaks. We study the splitting of the separatrix under volume-preserving perturbations using a discrete version of the Melnikov method. First, we establish several properties under general perturbations. For instance, we bound the topological complexity of the primary heteroclinic set in terms of the degree of some polynomial perturbations. We also give a sufficient condition for the splitting of the separatrix under some entire perturbations. A broad range of polynomial perturbations verify this sufficient condition. Finally, we describe the shape and bifurcations of the primary heteroclinic set for a specific perturbation.

Electrical Waves in a One-Dimensional Model of Cardiac Tissue

Margaret Beck, Christopher K. R. T. Jones, David Schaeffer, and Martin Wechselberger

SIAM J. Appl. Dyn. Syst. 7, pp. 1558-1581 (24 pages) | Cited 1 time

Online Publication Date: December 10, 2008

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The electrical dynamics in the heart is modeled by a two-component PDE. Using geometric singular perturbation theory, it is shown that a traveling pulse solution, which corresponds to a single heartbeat, exists. One key aspect of the proof involves tracking the solution near a point on the slow manifold that is not normally hyperbolic. This is achieved by desingularizing the vector field using a blow-up technique. This feature is relevant because it distinguishes cardiac impulses from, for example, nerve impulses. Stability of the pulse is also shown, by computing the zeros of the Evans function. Although the spectrum of one of the fast components is only marginally stable, due to essential spectrum that accumulates at the origin, it is shown that the spectrum of the full pulse consists of an isolated eigenvalue at zero and essential spectrum that is bounded away from the imaginary axis. Thus, this model provides an example in a biological application reminiscent of a previously observed mathematical phenomenon: that connecting an unstable—in this case marginally stable—front and back can produce a stable pulse. Finally, remarks are made regarding the existence and stability of spatially periodic pulses, corresponding to successive heartbeats, and their relationship with alternans, irregular action potentials that have been linked with arrhythmia.

Canard Induced Mixed-Mode Oscillations in a Medial Entorhinal Cortex Layer II Stellate Cell Model

Horacio G. Rotstein, Martin Wechselberger, and Nancy Kopell

SIAM J. Appl. Dyn. Syst. 7, pp. 1582-1611 (30 pages) | Cited 10 times

Online Publication Date: December 17, 2008

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Stellate cells (SCs) of the medial entorhinal cortex (layer II) display mixed-mode oscillatory activity, subthreshold oscillations (small-amplitude) interspersed with spikes (large amplitude), at theta frequencies (8–12 Hz). In this paper we study the mechanism of generation of such patterns in an SC biophysical (conductance-based) model. In particular, we show that the mechanism is based on the three-dimensional canard phenomenon and that the subthreshold oscillatory phenomenon is intrinsically nonlinear, involving the participation of both components (fast and slow) of a hyperpolarization-activated current in addition to the voltage and a persistent sodium current. We discuss some consequences of this mechanism for the SC intrinsic dynamics as well as for the interaction between SCs and external inhibitory inputs.
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