Wolf-Jürgen Beyn

Chapter 4 – Numerical Continuation, and Computation of Normal Forms

This chapter describes numerical continuation methods for analyzing the solution behavior of the dynamical system. Time-integration of a dynamical system gives much insight into its solution behavior. However, once a solution type has been computed—for example, a stationary solution (equilibrium) or a periodic solution (cycle)—then continuation methods become very effective in determining the dependence of this solution on the parameter α. Once a co-dimension-1 bifurcation has been located, it can be followed in two parameters—that is, with α e ℝ 2 . However, in many cases, detection of higher co-dimension bifurcations requires computation of certain normal forms for equations restricted to center manifolds at the critical parameter values. Pseudo-arclength continuation method allows the continuation of any regular solution, including folds. Geometrically, it is the most natural continuation method. The periodic solution continuation method is very suitable for numerical computations, and it is not difficult to establish the Poincare continuation with the help of it.

On the numerical approximation of phase portraits near stationary points

We show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one-step methods or multi-step methods satisfying a strong root condition. This means that any continuous trajectory can be approximated by an appropriate discrete trajectory, and vice versa, to the correct order of convergence and uniformly on arbitrarily large time intervals. In particular, the stable and unstable manifolds of the discretization converge to their continuous counterparts.

Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

AbstractIn this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.

The Numerical Computation of Homoclinic Orbits for Maps

Transversal homoclinic orbits of maps are known to generate shift dynamics on a set with Cantor-like structure. In this paper a numerical method is developed for computation of the corresponding homoclinic orbits. They are approximated by finite-orbit segments subject to asymptotic boundary conditions. We provide a detailed error analysis including a shadowing-type result by which one can infer the existence of a transversal homoclinic orbit from a finite segment. This approach is applied to several examples. In some of them parameters appear and closed loops of homoclinic orbits are found by a path-following algorithm.

Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals

If a traveling wave is stable or unstable depends essentially on the spectrum of a differential operator P obtained by linearization. We investigate how spectral properties of the all-line operator P are related to eigenvalues of finite-interval . Here R is a. linear boundary operator, for which we will derive determinant conditions, and the x-interval is assumed to be sufficiently large. Under suitable assumptions, we show (a) resolvent estimates for large s; (b) if s is in the resolvent of the all-line operator P, then s is also in the resolvent for finite-interval BVPs; (c) eigenvalues of P lead to approximating eigenvalues on finite intervals. These results allow to study the stability question for traveling waves by investigating eigenvalues of finite-interval problems. We give applications to the FitzHugh-Nagumo system with small diffusion and to the complex Ginzburg-Landau equations.

Attractors of Reaction Diffusion Systems on Infinite Lattices

AbstractIn this paper, we study global attractors for implicit discretizations of a semilinear parabolic system on the line. It is shown that under usual dissipativity conditions there exists a global (Zu,Zρ)-attractor

Continuation of Low-Dimensional Invariant Subspaces in Dynamical Systems of Large Dimension

A

Stability and Multiplicity of Solutions to Discretizations of Nonlinear Ordinary Differential Equations

in the sense of Babin-Vishik and Mielke-Schneider. Here Zρ is a weighted Sobolev space of infinite sequences with a weight that decays at infinity, while the space Zu carries a locally uniform norm obtained by taking the supremum over all Zρ norms of translates. We show that the absorbing set containing