Showing papers in "Siam Journal on Applied Dynamical Systems in 2003"
TL;DR: The first main result shows that robust patterns of synchrony are equivalent to the combinatorial condition that an equivalence relation on cells is "balanced" and shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissiblevector fields for a new coupled cell network, the "quotient network."
Abstract: A coupled cell system is a network of dynamical systems, or "cells," coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only mechanism that can create such states in a coupled cell system and show that it is not. The key idea is to replace the symmetry group by the symmetry groupoid, which encodes in- formation about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph—the dynamical systems with the corresponding internal dynamics and couplings—are precisely those that are equivariant under the symmetry groupoid. A pattern of synchrony is "robust" if it arises for all admissible vector fields. The first main result shows that robust patterns of synchrony (invariance of "polydiagonal" subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an equivalence relation on cells is "balanced." The second main result shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled cell network, the "quotient network." The existence of quotient networks has surprising implications for synchronous dynamics in coupled cell systems.
327 citations
TL;DR: The effects of the balance parameter b and the kernel g(x) on the solitary wave structures are studied and their interactions analytically for $
u=0$ and numerically for small or zero viscosity are investigated.
Abstract: We investigate the following family of evolutionary 1+1 PDEs that describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids: \[ m_t\ +\ \underbrace{\ \ um_x\ \ }_{\text{convection}}\ +\ \underbrace{\ \ b\,u_xm\ \ }_{\text{stretching}}\ =\ \underbrace{\ \
u\,m_{xx}\ }_{\text{viscosity}} \quad\text{with}\quad u=g*m. \] Here u=g*m denotes $u(x)=\int_{-\infty}^\infty g(x-y)m(y)\,dy$. This convolution (or filtering) relates velocity u to momentum density m by integration against the kernel g(x). We shall choose g(x) to be an even function so that u and m have the same parity under spatial reflection. When $
u=0$, this equation is both reversible in time and parity invariant. We shall study the effects of the balance parameter b and the kernel g(x) on the solitary wave structures and investigate their interactions analytically for $
u=0$ and numerically for small or zero viscosity. This family of equations admits the classic Burg...
295 citations
TL;DR: This work derives touchdown-to-touchdown Poincare maps under the common assumption of negligible gravitational effects during the stance phase and investigates the domains of attraction of symmetric periodic gaits and bifurcations from the branches of stable gaits in terms of nondimensional parameters.
Abstract: The spring-loaded inverted pendulum (SLIP), or monopedal hopper, is an archetypal model for running in numerous animal species. Although locomotion is generally considered a complex task requiring sophisticated control strategies to account for coordination and stability, we show that stable gaits can be found in the SLIP with both linear and ``air'' springs, controlled by a simple fixed-leg reset policy. We first derive touchdown-to-touchdown Poincare maps under the common assumption of negligible gravitational effects during the stance phase. We subsequently include and assess these effects and briefly consider coupling to pitching motions. We investigate the domains of attraction of symmetric periodic gaits and bifurcations from the branches of stable gaits in terms of nondimensional parameters.
232 citations
TL;DR: This work develops partial differential equation methods to study the dynamics of pattern formation in partial integro-differential equations defined on a spatially extended domain and predicts the exact number of peaks that form to comprise a multipeak solution of the full PDE.
Abstract: We develop partial differential equation (PDE) methods to study the dynamics of pattern formation in partial integro-differential equations (PIDEs) defined on a spatially extended domain. Our primary focus is on scalar equations in two spatial dimensions. These models arise in a variety of neuronal modeling problems and also occur in material science. We first derive a PDE which is equivalent to the PIDE. We then find circularly symmetric solutions of the resultant PDE; the linearization of the PDE around these solutions provides a criterion for their stability. When a solution is unstable, our analysis predicts the exact number of peaks that form to comprise a multipeak solution of the full PDE. We illustrate our results with specific numerical examples and discuss other systems for which this technique can be used.
173 citations
TL;DR: The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for colli- sions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem.
Abstract: Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for colli- sions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of vari- ational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated.
169 citations
TL;DR: The application of the transcription method is described to compute an optimal low thrust transfer from an Earth orbit using a sparse nonlinear programming algorithm with a discretization of the trajectory dynamics.
Abstract: The direct transcription or collocation method has demonstrated notable success in the solution of trajectory optimization and optimal control problems. This approach combines a sparse nonlinear programming algorithm with a discretization of the trajectory dynamics. A challenging class of optimization problems occurs when the spacecraft trajectories are characterized by thrust levels that are very low relative to the vehicle weight. Low thrust trajectories are demanding because realistic forces, due to oblateness, and third-body perturbations often dominate the thrust. Furthermore, because the thrust is so low, significant changes to the orbits require very long duration trajectories. When a collocation method is applied to a problem of this type, the resulting nonlinear program is very large, because the trajectories are long, and very nonlinear because of the perturbing forces.This paper describes the application of the transcription method to compute an optimal low thrust transfer from an Earth orbit t...
140 citations
TL;DR: A hybrid system consisting of the dynamics of the trajectories on the slow manifold coupled with "jumps" at the folds in the critical manifold to approximate the fast subsystem leads to an understanding of the bifurcations in the periodic orbits of the forced van der Pol system.
Abstract: The forced van der Pol oscillator has been the focus of scientific scrutiny for almost a century, yet its global bifurcation structure is still poorly understood. In this paper, we present a hybrid system consisting of the dynamics of the trajectories on the slow manifold coupled with "jumps" at the folds in the critical manifold to approximate the fast subsystem. The global bifurcations of the fixed points and periodic points of this hybrid system lead to an understanding of the bifurcations in the periodic orbits (without canards) of the forced van der Pol system.
137 citations
TL;DR: This paper extends the reduced system to account for canards, trajectory segments that follow the unstable portion of the slow manifold in the forced van der Pol oscillator, and concludes with computations of return maps and periodic orbits in the full three dimensional flow that are compared with the computations and analysis of the reduction system.
Abstract: This is the second in a series of papers about the dynamics of the forced van der Pol oscillator (J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2(2 003), pp. 1-35). The first paper described the reduced system, a two dimensional flow with jumps that reflect fast trajectory segments in this vector field with two time scales. This paper extends the reduced system to account for canards, trajectory segments that follow the unstable portion of the slow manifold in the forced van der Pol oscillator. This extension of the reduced system serves as a template for approximating the full nonwandering set of the forced van der Pol oscillator for large sets of parameter values, including parameters for which the system is chaotic. We analyze some bifurcations in the extension of the reduced system, building upon our previous work in (J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2(2 003), pp. 1-35). We conclude with computations of return maps and periodic orbits in the full three dimensional flow that are compared with the computations and analysis of the reduced system. These comparisons demonstrate numerically the validity of results we derive from the study of canards in the reduced system.
86 citations
TL;DR: Many applications give rise to dynamical systems in the form of a vector field with a phase space of moderate dimension, Examples are the Lorenz equations, mechanical and other oscillators, and mod...
Abstract: Many applications give rise to dynamical systems in the form of a vector field with a phase space of moderate dimension. Examples are the Lorenz equations, mechanical and other oscillators, and mod...
73 citations
TL;DR: In this article, the authors study the flow of an incompressible viscous fluid through a long tube with compliant walls, where the flow is governed by a given time-dependent pressure drop between the inlet and the outlet boundary.
Abstract: We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given time-dependent pressure drop between the inlet and the outlet boundary. The pressure drop is assumed to be small, thereby introducing creeping flow in the tube. Stokes equations for incompressible viscous fluid are used to model the flow, and the equations of a curved, linearly elastic membrane are used to model the wall. Due to the creeping flow and to small displacements, the interface between the fluid and the lateral wall is linearized and supposed to be the initial configuration of the membrane. We study the dynamics of this coupled fluid-structure system in the limit when the ratio between the characteristic width and the characteristic length tends to zero. Using the asymptotic techniques typically used for the study of shells and plates, we obtain a set of Biot-type visco-elastic equations for the effective pressure and the effective displacements. The approximation is rigorously justified through a weak convergence result and through the error estimates for the solution of the effective equations modified by an outlet boundary layer. Applications of the model problem include blood flow in small arteries. We recover the well- known law of Laplace and obtain new improved models that hold in cases when the shear modulus of the vessel wall is not negligible and the Poisson ratio is arbitrary.
71 citations
TL;DR: This article develops a theory for the semistrong interaction of pulses in a class of singularly perturbed coupled reaction-diffusion equations that includes the (generalized) Gierer--Meinhardt, Gray--Scott, Schnakenberg, and Thomas models, among others.
Abstract: Pulse-pulse interactions play central roles in a variety of pattern formation phenomena, including self-replication. In this article, we develop a theory for the semistrong interaction of pulses in a class of singularly perturbed coupled reaction-diffusion equations that includes the (generalized) Gierer--Meinhardt, Gray--Scott, Schnakenberg, and Thomas models, among others. Geometric conditions are determined on the reaction kinetics for whether the pulses in a two-pulse solution attract or repel, and ODEs are derived for the time-dependent separation distance between their centers and for their wave speeds. In addition, conditions for the existence of stationary two-pulse solutions are identified, and the interactions between stationary and dynamically evolving two-pulse solutions are studied. The theoretical results are illustrated in a series of examples. In two of these, which are related to the classical Gierer--Meinhardt equation, we find that the pulse amplitudes blow up in finite time. Moreover, ...
TL;DR: A two-step strategy is proposed for the computation of singularities in nonlinear PDEs using a Fourier spectral method and the epsilon algorithm to sum the Fourier series.
Abstract: A two-step strategy is proposed for the computation of singularities in nonlinear PDEs. The first step is the numerical solution of the PDE using a Fourier spectral method; the second step involves numerical analytical continuation into the complex plane using the epsilon algorithm to sum the Fourier series. Test examples include the inviscid Burgers and nonlinear heat equations as well as a transport equation involving the Hilbert transform. Numerical results, including Web animations that show the dynamics of the singularities in the complex plane, are presented.
TL;DR: In this paper, the Hamiltonian formalism for relative equilibria is developed, a nonexistence theorem for relative equilibrium is proved, and several results on relative periodic orbits are established, including as corollaries classical results on vortex streets.
Abstract: Point vortices on a cylinder (periodic strip) are studied geometrically. The Hamiltonian formalism is developed, a nonexistence theorem for relative equilibria is proved, equilibria are classified when all vorticities have the same sign, and several results on relative periodic orbits are established, including as corollaries classical results on vortex streets and leapfrogging.
TL;DR: This study is the first example of a two-parameter bifurcation study, including bIfurcations of periodic and connecting orbits, of a delay system, made possible by new numerical continuation tools implemented in the package DDE-BIFTOOL.
Abstract: We present a detailed bifurcation analysis of a single-mode semiconductor laser subject to phase-conjugate feedback, a system described by a delay differential equation. Codimension-one bifurcation curves of equilibria and periodic orbits and curves of certain connecting orbits are presented near the laser's locking region in the two-dimensional parameter plane of feedback strength and pump current. We identify several codimension-two bifurcations, including a double-Hopf point, Belyakov points, and a T-point bifurcation, and we show how they organize the dynamics. This study is the first example of a two-parameter bifurcation study, including bifurcations of periodic and connecting orbits, of a delay system. It was made possible by new numerical continuation tools, implemented in the package DDE-BIFTOOL, and showcases their usefulness for the study of delay systems arising in applications.
TL;DR: This work considers applications to dynamic pitchfork bifurcation, pattern formation below the threshold of stability, and transient dynamics of stochastic PDEs near this deterministic bifurancations.
Abstract: For systems of partial differential equations (PDEs) with locally cubic nonlinearities, which are perturbed by additive noise, we describe the essential dynamics for small solutions. If the system is near a change of stability, then a natural separation of time-scales occurs and the amplitudes of the dominant modes are given on a long time-scale by a stochastic ordinary differential equation. We consider applications to dynamic pitchfork bifurcation, pattern formation below the threshold of stability, and transient dynamics of stochastic PDEs near this deterministic bifurcations.
TL;DR: This paper addresses questions with respect to both the general class of smooth mappings $\phi$ and the subclass of delay coordinate mappings about an arbitrary compact set A in $\mathbb{R}^{n}$.
Abstract: Takens, Ruelle, Eckmann, Sano, and Sawada launched an investigation of images of attractors of dynamical systems. Let A be a compact invariant set for a map f on $\mathbb{R}^{n}$ and let $\phi : \mathbb{R}^{n} \to \mathbb{R}^{m}$ be a "typical" smooth map, where n > m. When can we say that A and $\phi (A)$ are similar, based only on knowledge of the images in $\mathbb{R}^{m}$ of trajectories in A? For example, under what conditions on $\phi (A)$ (and the induced dynamics thereon) are A and $\phi (A)$ homeomorphic? Are their Lyapunov exponents the same? Or, more precisely, which of their Lyapunov exponents are the same? This paper addresses these questions with respect to both the general class of smooth mappings $\phi$ and the subclass of delay coordinate mappings.In answering these questions, a fundamental problem arises about an arbitrary compact set A in $\mathbb{R}^{n}$. For $x \in A$, what is the smallest integer d such that there is a C1 manifold of dimension d that contains all points of A that lie...
TL;DR: Bressloff, Cowan, Golubitsky, Thomas, and Wiener generalize this class of models to include the orientation tuning of neurons in V1 and the Hubel and Wiesel hypercolumns.
Abstract: Mathematical studies of drug induced geometric visual hallucinations include three components: a model (or class of models) that abstracts the structure of the primary visual cortex V1; a mathematical procedure for finding geometric patterns as solutions to the cortical models; and a method for interpreting these patterns as visual hallucinations.Ermentrout and Cowan used the Wilson--Cowan equations to model the evolution of an activity variable a(x) that represents, for example, the voltage potential a of the neuron located at point x in V1. Bressloff, Cowan, Golubitsky, Thomas, and Wiener generalize this class of models to include the orientation tuning of neurons in V1 and the Hubel and Wiesel hypercolumns. In these models, $a({\mathbf x},\phi)$ represents the voltage potential a of the neuron in the hypercolumn located at x and tuned to direction $\phi$. The work of Bressloff et al. assumes that lateral connections between hypercolumns are anisotropic; that is, neurons in neighboring hypercolumns are ...
TL;DR: The results of the numerical exploration of the fractal structure found by S.P. Novikov in an elementary multivalued Poisson dynamical system on the 3-torus coming from the problem of the dependence of magnetoresistance on the direction of the magnetic field in a normal metal are presented.
Abstract: We present the results of our numerical exploration of the fractal structure found by S.P. Novikov in an elementary multivalued Poisson dynamical system on the 3-torus coming from the problem of the dependence of magnetoresistance on the direction of the magnetic field in a normal metal.
TL;DR: This work considers the dynamics of coorbital motion of two small moons about a large planet which have nearly circular orbits with almost equal radii as in Poincare's periodic orbits of the first kind.
Abstract: We consider the dynamics of coorbital motion of two small moons about a large planet which have nearly circular orbits with almost equal radii. These moons avoid collision because they switch orbits during each close encounter. We approach the problem as a perturbation of decoupled Kepler problems as in Poincare's periodic orbits of the first kind. The perturbation is large but only in a small region in the phase space. We discuss the relationship required among the small quantities (radial separation, mass, and minimum angular separation). Persistence of the orbits is discussed.
TL;DR: The synergetic elimination procedure is applied for the stable modes in nonlinear delay systems close to a dynamical instability and the normal form for the delay-induced Hopf bifurcation in the Wright equation is derived.
Abstract: We apply the synergetic elimination procedure for the stable modes in nonlinear delay systems close to a dynamical instability and derive the normal form for the delay-induced Hopf bifurcation in the Wright equation. The resulting periodic orbit is confirmed by numerical simulations.
TL;DR: It is shown that an amplitude-modulated laser is subject to large transient growth and it is demonstrated that at threshold the transient growth is precisely the Petermann excess noise factor for a laser governed by a nonnormal operator.
Abstract: We consider pulse formation dynamics in an actively mode-locked laser. We show that an amplitude-modulated laser is subject to large transient growth and we demonstrate that at threshold the transient growth is precisely the Petermann excess noise factor for a laser governed by a nonnormal operator. We also demonstrate an exact reduction from the governing PDEs to a low-dimensional system of ODEs for the parameters of an evolving pulse. A linearized version of these equations allows us to find analytical expressions for the transient growth below threshold. We also show that the nonlinear system collapses onto an appropriate fixed point, and thus in the absence of noise the ground-mode laser pulse is stable. We demonstrate numerically that, in the presence of a continuous noise source, however, the laser destabilizes and pulses are repeatedly created and annihilated.
TL;DR: In this article, a blow-up construction transforms the stability problem to the analysis of symmetry-breaking perturbations of Hamiltonian relative equilibria, and the stability follows by KAM theory rather than energy-momentum confinement.
Abstract: Purely rotational relative equilibria of an ellipsoidal underwater vehicle occur at nongeneric momentum where the symplectic reduced spaces change dimension. The stability of these relative equilibria under momentum changing perturbations is not accessible by Lyapunov functions obtained from energy and momentum. A blow-up construction transforms the stability problem to the analysis of symmetry-breaking perturbations of Hamiltonian relative equilibria. As such, the stability follows by KAM theory rather than energy-momentum confinement.
TL;DR: In this paper, the authors use concepts from dynamical systems theory to present a model-based method for quantifying the risk of impending cycle irregularity, such as episodes of damped oscillation and abrupt changes of cycle phase.
Abstract: Oscillating population data often exhibit cycle irregularities such as episodes of damped oscillation and abrupt changes of cycle phase. The prediction of such irregularities is of interest in applications ranging from food production to wildlife management. We use concepts from dynamical systems theory to present a model-based method for quantifying the risk of impending cycle irregularity.