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SIAM J. on Applied Dynamical Systems

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2012

Volume 11, Issue 1, pp. 1-566


Breathers and Q-Breathers: Two Sides of the Same Coin

T. Penati and S. Paleari

SIAM J. Appl. Dyn. Syst. 11, pp. 1-30 (30 pages)

Online Publication Date: January 10, 2012

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We construct, and approximate from the continuum, two-parameter families of time periodic, small amplitude, localized solutions, for both the focusing and defocusing finite discrete nonlinear Schrödinger models, with Dirichlet boundary conditions. Within such families, depending on the parameters, both real space localization (breathers) and Fourier space localization (Q-breathers) are present. For the former type of solutions, convergence to the ground state of the focusing infinite chain is also proved; for the latter, a description of the localization properties is given, and some numerical results on the difference between the focusing and defocusing cases are explained. The proofs are based on continuation tools, ideas from the finite element methods, and techniques of convergence of variational problems.

An Algebraic Approach to Reverse Engineering Finite Dynamical Systems Arising from Biology

Alan Veliz-Cuba

SIAM J. Appl. Dyn. Syst. 11, pp. 31-48 (18 pages)

Online Publication Date: January 10, 2012

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Finite dynamical systems have been used successfully in modeling biological processes. When certain regulatory mechanisms of a biological system or a model are unknown it is important to be able to identify the best model with the available data. In this context, reverse engineering of finite dynamical systems from partial information is an important problem. While this problem has been studied in the past, there are currently no algorithms that can predict the signs of the interactions. In this paper we propose a framework and algorithms to reverse engineer the possible signed wiring diagrams of a finite dynamical system from data. The algorithm consists of encoding all possible wiring diagrams using ideals and algebraic sets and choosing those that are minimal using the primary decomposition and the irreducible components.

Noise-Induced Behaviors in Neural Mean Field Dynamics

Jonathan Touboul, Geoffroy Hermann, and Olivier Faugeras

SIAM J. Appl. Dyn. Syst. 11, pp. 49-81 (33 pages)

Online Publication Date: January 13, 2012

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The collective behavior of cortical neurons is strongly affected by the presence of noise at the level of individual cells. In order to study these phenomena in large-scale assemblies of neurons, we consider networks of firing-rate neurons with linear intrinsic dynamics and nonlinear coupling, belonging to a few types of cell populations and receiving noisy currents. As the number of neurons tends to infinity, asymptotic equations (mean field equations) are rigorously derived based on a probabilistic approach. These equations are implicit on the probability distribution of the solutions, which generally makes their direct analysis difficult. However, in our case, the solutions are Gaussian, and their moments satisfy a closed system of nonlinear ordinary differential equations (ODEs), which are much easier to study than the original stochastic network equations, and the statistics of the empirical process uniformly converge towards the solutions of these ODEs. Based on this description, we analytically and numerically study the influence of noise on the collective behaviors, and compare these asymptotic regimes to simulations of the network. We observe that the mean field equations provide an accurate description of the solutions of the network equations for network sizes as small as a few hundred neurons. In particular, we observe that the level of noise in the system qualitatively modifies its collective behavior, producing, for instance, synchronized oscillations of the whole network, desynchronization of oscillating regimes, and stabilization or destabilization of stationary solutions. These results shed new light on the role of noise in shaping collective dynamics of neurons, and give us clues for understanding similar phenomena observed in biological networks.

Efficient Automation of Index Pairs in Computational Conley Index Theory

Rafael Frongillo and Rodrigo Treviño

SIAM J. Appl. Dyn. Syst. 11, pp. 82-109 (28 pages)

Online Publication Date: January 24, 2012

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We present new methods of automating the construction of index pairs, essential ingredients of discrete Conley index theory. These new algorithms are further steps in the direction of automating computer-assisted proofs of semiconjugacies from a map on a manifold to a subshift of finite type. We apply these new algorithms to the standard map at different values of the perturbative parameter $\varepsilon$ and obtain rigorous lower bounds for its topological entropy for $\varepsilon\in[.7,2]$.

Switching in Mass Action Networks Based on Linear Inequalities

Carsten Conradi and Dietrich Flockerzi

SIAM J. Appl. Dyn. Syst. 11, pp. 110-134 (25 pages)

Online Publication Date: January 26, 2012

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Many biochemical processes can successfully be described by dynamical systems allowing some form of switching when, depending on their initial conditions, solutions of the dynamical system end up in different regions of state space (associated with different biochemical functions). Switching is often realized by a bistable system (i.e., a dynamical system allowing two stable steady state solutions) and, in the majority of cases, bistability is established numerically. In our view, this approach is too restrictive. On the one hand, due to predominant parameter uncertainty, numerical methods are generally difficult to apply to realistic models originating in systems biology. On the other hand, switching already arises with the occurrence of a saddle-type steady state (characterized by a Jacobian where exactly one eigenvalue is positive and the remaining eigenvalues have negative real part). Consequently we derive conditions based on linear inequalities that allow the analytic computation of states and parameters where the Jacobian derived from a mass action network has a defective zero eigenvalue so that—under certain genericity conditions—a saddle-node bifurcation occurs. Our conditions are applicable to general mass action networks involving at least one conservation relation; however, they are only sufficient (as infeasibility of linear inequalities does not exclude defective zero eigenvalues).

Canard-Like Explosion of Limit Cycles in Two-Dimensional Piecewise-Linear Models of FitzHugh–Nagumo Type

Horacio G. Rotstein, Stephen Coombes, and Ana Maria Gheorghe

SIAM J. Appl. Dyn. Syst. 11, pp. 135-180 (46 pages)

Online Publication Date: January 26, 2012

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We investigate the mechanism of abrupt transition between small- and large amplitude oscillations in fast-slow piecewise-linear (PWL) models of FitzHugh–Nagumo (FHN) type. In the context of neuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials (spikes), respectively. The minimal model that shows such phenomena has a cubic-like nullcline (for the fast equation) with two or more linear pieces in the middle branch and one piece in the left and right branches. Simpler models with only one linear piece in the middle branch or a discontinuity between the left and right branches (McKean model) show a single oscillatory mode. As the number of linear pieces increases, PWL models of FHN type approach smooth FHN-type models. For the minimal model we investigate the bifurcation structure; we describe the mechanism that leads to the abrupt, canard-like transition between subthreshold oscillations and spikes; and we provide an analytical way of predicting the amplitude regime of a given limit cycle trajectory which includes the approximation of the canard critical control parameter. We extend our results to models with a larger number of linear pieces. Our results for PWL-FHN-type models are consistent with similar results for smooth FHN-type models. In addition, we develop tools that are amenable for the investigation of a variety of related, and more complex, problems including forced, stochastic, and coupled oscillators.

Canonical Discontinuous Planar Piecewise Linear Systems

Emilio Freire, Enrique Ponce, and Francisco Torres

SIAM J. Appl. Dyn. Syst. 11, pp. 181-211 (31 pages)

Online Publication Date: January 31, 2012

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The family of Filippov systems constituted by planar discontinuous piecewise linear systems with two half-plane linearity zones is considered. Under generic conditions that amount to the boundedness of the sliding set, some changes of variables and parameters are used to obtain a Liénard-like canonical form with seven parameters. This canonical form is topologically equivalent to the original system if one restricts one's attention to orbits with no points in the sliding set. Under the assumption of focus-focus dynamics, a reduced canonical form with only five parameters is obtained. For the case without equilibria in both open half-planes we describe the qualitatively different phase portraits that can occur in the parameter space and the bifurcations connecting them. In particular, we show the possible existence of two limit cycles surrounding the sliding set. Such limit cycles bifurcate at certain parameter curves, organized around different codimension-two Hopf bifurcation points. The proposed canonical form will be a useful tool in the systematic study of planar discontinuous piecewise linear systems, in which this paper is a first step.

A Fast-Slow Analysis of the Dynamics of REM Sleep

Cecilia G. Diniz Behn and Victoria Booth

SIAM J. Appl. Dyn. Syst. 11, pp. 212-242 (31 pages)

Online Publication Date: January 31, 2012

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Waking and sleep states are regulated by the coordinated activity of a number of neuronal populations in the brainstem and hypothalamus whose synaptic interactions compose a sleep-wake regulatory network. Physiologically based mathematical models of the sleep-wake regulatory network contain mechanisms operating on multiple time scales including relatively fast synaptic-based interactions between neuronal populations, and much slower homeostatic and circadian processes that modulate sleep-wake temporal patterning. In this study, we exploit the naturally arising slow time scale of the homeostatic sleep drive in a reduced sleep-wake regulatory network model to utilize fast-slow analysis to investigate the dynamics of rapid eye movement (REM) sleep regulation. The network model consists of a reduced number of wake-, non-REM (NREM) sleep-, and REM sleep-promoting neuronal populations with synaptic interactions reflecting the mutually inhibitory flip-flop conceptual model for sleep-wake regulation and the reciprocal interaction model for REM sleep regulation. Network dynamics regularly alternate between wake and sleep states as governed by the slow homeostatic sleep drive. By varying a parameter associated with the activation of the REM-promoting population, we cause REM dynamics during sleep episodes to vary from suppression to single activations to regular REM-NREM cycling, corresponding to changes in REM patterning induced by circadian modulation and observed in different mammalian species. We also utilize fast-slow analysis to explain complex effects on sleep-wake patterning of simulated experiments in which agonists and antagonists of different neurotransmitters are microinjected into specific neuronal populations participating in the sleep-wake regulatory network.

Selection of Ground States in the Zero Temperature Limit for a One-Parameter Family of Potentials

A. T. Baraviera, R. Leplaideur, and A. O. Lopes

SIAM J. Appl. Dyn. Syst. 11, pp. 243-260 (18 pages)

Online Publication Date: February 02, 2012

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For the subshift of finite type $\Sigma=\{0,1,2\}^{\mathbb{N}}$ we study the convergence and the selection at temperature zero of the Gibbs measure associated to a non–locally constant Hölder potential which admits exactly two maximizing ergodic measures. These measures are Dirac measures at two different fixed points, and the potential is flatter at one of these two fixed points. We prove that there always is convergence but not necessarily to the Dirac measure at the point where the potential is the flattest. This is contrary to what was expected in light of the analogous problem in Aubry-Mather theory [N. Anantharaman et al., Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), pp. 513–528]. This is also contrary to the finite range case where the equilibrium state converges to the equi-barycenter of the two Dirac measures. Moreover, we emphasize the unexpected behavior of the Gibbs measure: the eigenmeasure selects one Dirac measure (at the point where the potential is the flattest), and the eigenfunction selects the other one (at the point where the potential is the sharpest).

Localized States in an Extended Swift–Hohenberg Equation

John Burke and Jonathan H. P. Dawes

SIAM J. Appl. Dyn. Syst. 11, pp. 261-284 (24 pages)

Online Publication Date: March 01, 2012

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Recent work on the behavior of localized states in pattern-forming partial differential equations has focused on the traditional model Swift–Hohenberg equation which, as a result of its simplicity, has additional structure; it is variational in time and conservative in space. In this paper we investigate an extended Swift–Hohenberg equation in which nonvariational and nonconservative effects play a key role. Our work concentrates on aspects of this much more complicated problem. First we carry out the normal form analysis of the initial pattern-forming instability that leads to small-amplitude localized states. Next we examine the bifurcation structure of the large-amplitude localized states. Finally, we investigate the temporal stability of one-peak localized states. Throughout, we compare the localized states in the extended Swift–Hohenberg equation with the analogous solutions to the usual Swift–Hohenberg equation.

Existence of a Center Manifold in a Practical Domain around $L_1$ in the Restricted Three-Body Problem

Maciej J. Capiński and Pablo Roldán

SIAM J. Appl. Dyn. Syst. 11, pp. 285-318 (34 pages)

Online Publication Date: March 06, 2012

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We present a method of proving existence of center manifolds within specified domains. The method is based on a combination of topological tools, normal forms, and rigorous computer-assisted computations. We apply our method to obtain a proof of a center manifold in an explicit region around the equilibrium point $L_1$ in the Earth–Sun planar restricted circular three-body problem.

Resonances and Twist in Volume-Preserving Mappings

H. R. Dullin and J. D. Meiss

SIAM J. Appl. Dyn. Syst. 11, pp. 319-349 (31 pages)

Online Publication Date: March 08, 2012

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The phase space of an integrable, volume-preserving map with one action and $d$ angles is foliated by a one-parameter family of $d$-dimensional invariant tori. Perturbations of such a system may lead to chaotic dynamics and transport. We show that near a rank-one, resonant torus these mappings can be reduced to volume-preserving “standard maps.” These have twist only when the image of the frequency map crosses the resonance curve transversely. We show that these maps can be approximated—using averaging theory—by the usual area-preserving twist or nontwist standard maps. The twist condition appropriate for the volume-preserving setting is shown to be distinct from the nondegeneracy condition used in (volume-preserving) KAM theory.

Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit

Kendrick M. Shaw, Young-Min Park, Hillel J. Chiel, and Peter J. Thomas

SIAM J. Appl. Dyn. Syst. 11, pp. 350-391 (42 pages)

Online Publication Date: March 13, 2012

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Rhythmic behaviors in neural systems often combine features of limit-cycle dynamics (stability and periodicity) with features of near heteroclinic or near homoclinic cycle dynamics (extended dwell times in localized regions of phase space). Proximity of a limit cycle to one or more saddle equilibria can have a profound effect on the timing of trajectory components and response to both fast and slow perturbations, providing a possible mechanism for adaptive control of rhythmic motions. Reyn [“Generation of limit cycles from separatrix polygons in the phase plane” in Geometrical Approaches to Differential Equations, Lecture Notes in Math. 810, Springer, New York, 1980, pp. 264–289] showed that for a planar dynamical system with a stable heteroclinic cycle (or separatrix polygon), small perturbations satisfying a net inflow condition will generically give rise to a stable limit cycle (see also [J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed., Appl. Math. Sci. 42, Springer-Verlag, Berlin, 1990]). Here we consider the asymptotic behavior of the infinitesimal phase response curve (iPRC) for examples of two systems satisfying Reyn's inflow criterion, (i) a smooth system with a chain of four hyperbolic saddle points and (ii) a piecewise linear system corresponding to local linearization of the smooth system about its saddle points. For system (ii), we obtain exact expressions for the limit cycle and the iPRC as functions of a parameter $\mu>0$ representing the distance from a heteroclinic bifurcation point. In the $\mu\to 0$ limit, we find that perturbations parallel to the unstable eigenvector direction in a piecewise linear region lead to divergent phase response, as previously observed [E. Brown, J. Moehlis, and P. Holmes, Neural Comput., 16 (2004), pp. 673–715]. In contrast to previous work, we find that perturbations parallel to the stable eigenvector direction can lead to either divergent or convergent phase response, depending on the phase at which the perturbation occurs. In the smooth system (i), we show numerical evidence of qualitatively similar phase specific sensitivity to perturbation. Having the exact expression for the iPRC for the piecewise linear system allows us to investigate its stability under diffusive coupling. In addition, we qualitatively compare iPRCs obtained for systems (i) and (ii) to iPRCs for the Morris–Lecar equations near a bifurcation from limit cycles to a saddle-homoclinic orbit.

Clustering Conditions and the Cluster Formation Process in a Dynamical Model of Multidimensional Attracting Agents

F. De Smet and D. Aeyels

SIAM J. Appl. Dyn. Syst. 11, pp. 392-415 (24 pages)

Online Publication Date: March 20, 2012

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We consider a multiagent clustering model where each agent belongs to a multidimensional space. We investigate its long term behavior, and we prove emergence of clustering behavior in the sense that the velocities of the agents approach asymptotic values, independently of the initial conditions; agents with equal asymptotic velocities are said to belong to the same cluster. We propose a set of relations governing these asymptotic velocities. These results are compared with results obtained earlier for the model with agents belonging to a one-dimensional space and are then explored for the case of an infinite number of agents. For the particular case of a spherically symmetric configuration of an infinite number of agents a rigorous analysis of the relations governing the asymptotic velocities is possible, assuming that a continuity property established for the finite case remains true for the infinite case. This leads to a characterization of the onset of cluster formation in terms of the evolution of the cluster size with varying coupling strength. A remarkable point is that the cluster formation process depends critically on the dimension of the agent state space; considering the cluster size as an order parameter, the cluster formation in the one-dimensional case may be seen as a second-order phase transition, while the multidimensional case is associated with a first-order phase transition. We provide bounds for the critical coupling strength at the onset of the cluster formation, and we illustrate the results with two examples in three dimensions.

A Saddle in a Corner—A Model of Collinear Triatomic Chemical Reactions

L. Lerman and V. Rom-Kedar

SIAM J. Appl. Dyn. Syst. 11, pp. 416-446 (31 pages)

Online Publication Date: March 20, 2012

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A geometrical model which captures the main ingredients governing atom-diatom collinear chemical reactions is proposed. This model is neither near-integrable nor hyperbolic, yet it is amenable to analysis using a combination of the recently developed tools for studying systems with steep potentials and the study of the phase space structure near a center-saddle equilibrium. The nontrivial dependence of the reaction rates on parameters, initial conditions, and energy is thus qualitatively explained. Conditions under which the phase space transition state theory assumptions are satisfied and conditions under which they fail are derived.

Numerical Bifurcation Study of Superconducting Patterns on a Square

Nico Schlömer, Daniele Avitabile, and Wim Vanroose

SIAM J. Appl. Dyn. Syst. 11, pp. 447-477 (31 pages)

Online Publication Date: March 22, 2012

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This paper considers the extreme type-II Ginzburg–Landau equations that model vortex patterns in superconductors. The nonlinear PDEs are solved using Newton's method, and properties of the Jacobian operator are highlighted. Specifically, this paper illustrates how the operator can be regularized using an appropriate phase condition. For a two-dimensional square sample, the numerical results are based on a finite-difference discretization with link variables that preserves the gauge invariance. For two exemplary sample sizes, a thorough bifurcation analysis is performed using the strength of the applied magnetic field as a bifurcation parameter and focusing on the symmetries of this system. The analysis gives new insight into the transitions between stable and unstable states, as well as the connections between stable solution branches.

Polychromatic Solitary Waves in a Periodic and Nonlinear Maxwell System

Dmitry E. Pelinovsky, Gideon Simpson, and Michael I. Weinstein

SIAM J. Appl. Dyn. Syst. 11, pp. 478-506 (29 pages)

Online Publication Date: March 22, 2012

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We consider the one-dimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite time-scales, the coupled evolution of backward and forward waves is governed by nonlocal equations of resonant nonlinear geometrical optics. For the special class of solutions which are periodic in the fast phase, these equations are equivalent to an infinite system of nonlinear coupled mode equations, the so-called extended nonlinear coupled mode equations, or xNLCME. Numerical studies support the existence of long-lived spatially localized coherent structures, featuring a slowly varying envelope and a train of carrier shocks. In this paper we explore, by analytical, asymptotic, and numerical methods, the existence and properties of spatially localized structures of the xNLCME system for the case where the refractive index profile consists of a periodic array of Dirac delta functions. We consider, in particular, the limit of small amplitude solutions with frequencies near a spectral band edge. In this case, stationary xNLCME is well approximated by an infinite system of coupled, stationary, nonlinear Schrödinger (NLS) equations, the extended nonlinear Schrödinger system, xNLS. We embed xNLS in a one-parameter family of equations, xNLS$^\epsilon$, which interpolates between infinitely many decoupled NLS equations ($\epsilon=0$) and xNLS ($\epsilon=1$). Using bifurcation methods we show existence of solutions for a range of $\epsilon\in(-\epsilon_0,\epsilon_0)$ and, by a numerical continuation method, establish the continuation of certain branches all the way to $\epsilon=1$. Finally, we perform time-dependent simulations of a truncated xNLCME and find the small-amplitude near–band edge gap solitons to be robust to both numerical errors and the NLS approximation.

Bistability in a Leaky Integrate-and-Fire Neuron with a Passive Dendrite

Michael A. Schwemmer and Timothy J. Lewis

SIAM J. Appl. Dyn. Syst. 11, pp. 507-539 (33 pages)

Online Publication Date: March 22, 2012

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We examine the influence of dendritic load on the firing dynamics of a spatially extended leaky integrate-and-fire (LIF) neuron that explicitly includes spiking dynamics. We obtain an exact analytical solution for this model and use it to derive a return map that completely captures the dynamics of the system. Using the map, we find that dendritic properties can significantly change the firing dynamics of the system. Under certain conditions, the addition of the dendrite can change the LIF model from type 1 excitability to type 2 excitability and induce bistability between periodic firing and the quiescent state. We identify the mechanism that causes the periodic behavior in the bistable regime as somatodendritic ping-pong. Furthermore, we use the return map to fully explore the model parameter space in order to find regions where this bistable behavior occurs. We then give physical interpretations of the dependence of the bistable behavior on model parameters. Finally, we demonstrate that the simpler two-compartment model displays qualitatively similar dynamics to the more complicated ball-and-stick model.

A Contraction Argument for Two-Dimensional Spiking Neuron Models

Eric Foxall, Roderick Edwards, Slim Ibrahim, and P. van den Driessche

SIAM J. Appl. Dyn. Syst. 11, pp. 540-566 (27 pages)

Online Publication Date: March 27, 2012

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A number of two-dimensional spiking neuron models that combine continuous dynamics with an instantaneous reset have been introduced in the literature. The models are capable of reproducing a variety of experimentally observed spiking patterns and also have the advantage of being mathematically tractable. Here an analysis of the transverse stability of orbits in the phase plane leads to sufficient conditions on the model parameters for regular spiking to occur. The application of this method is illustrated by three examples, taken from existing models in the neuroscience literature. In the first two examples the model has no equilibrium states, and regular spiking follows directly. In the third example there are equilibrium points, and some additional quantitative arguments are given to prove that regular spiking occurs.
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