Eusebius J. Doedel

NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (II): BIFURCATION IN INFINITE DIMENSIONS

A number of basic algorithms for the numerical analysis and control of bifurcation phenomena are described. The emphasis is on algorithms based on pseudoarclength continuation for ordinary differential equations. Several illustrative examples computed with the AUTO software package are included. This is Part II of the paper that appeared in the preceding issue [Doedel et al., 1991] and that mainly dealt with algebraic problems.

A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

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.

Numerical computation and continuation of invariant manifolds connecting fixed points

A numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in

Lecture Notes on Numerical Analysis of Nonlinear Equations

\mathbb{R}^n

Numerical computation of heteroclinic orbits

is given, extending the results of an earlier paper [Comput. Appl. Math., 26 (1989), pp. 159’170] by the authors. Basically, a boundary value problem on the real line is truncated to a finite interval. The method applies, in particular, to the computation of heteroclinic orbits. The emphasis is on the systematic computation of such orbits by continuation. Using the fact that the linearized operator of our problem is Fredholm in appropriate Banach spaces, the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval. Several applications are considered, including the computation of traveling wave solutions to reaction diffusion problems. Computations were done using the software package AUTO.

Continuation of periodic orbits in conservative and Hamiltonian systems

Numerical integrators can provide valuable insight into the transient behavior of a dynamical system. However, when the interest is in stationary and periodic solutions, their stability, and their transition to more complex behavior, then numerical continuation and bifurcation techniques are very powerful and efficient. The objective of these notes is to make the reader familiar with the ideas behind some basic numerical continuation and bifurcation techniques. This will be useful, and is at times necessary, for the effective use of the software Auto and other packages, such as XppAut [17], Content [24], Matcont [21], and DDE-Biftool [16], which incorporate the same or closely related algorithms. These lecture notes are an edited subset of material from graduate courses given by the author at the universities of Utah and Minnesota [9] and at Concordia University, and from short courses given at various institutions, including the Universite Pierre et Marie Curie (Paris VI), the Centre de Recherches Mathematiques of the Universite de Montreal, the Technische Universitat Hamburg-Harburg, and the Benemerita Universidad Autonoma de Puebla.

On the resonance structure in a forced excitable system

We give a numerical method for the computation of heteroclinic orbits connecting two saddle points in ℝ2. These can be computed to very high period due to an integral phase condition and an adaptive discretization. We can also compute entire branches (one-dimensional continua) of such orbits. The method can be extended to compute an invariant manifold that connects two fixed points in ℝ n . As an example we compute branches of traveling wave front solutions to the Huxley equation. Using weighted Sobolev spaces and the general theory of approximation of nonlinear problems we show that the errors in the approximate wave speed and in the approximate wave front decay exponentially with the period.

ELEMENTAL PERIODIC ORBITS ASSOCIATED WITH THE LIBRATION POINTS IN THE CIRCULAR RESTRICTED 3-BODY PROBLEM

We introduce and justify a computational scheme for the continuation of periodic orbits in systems with one or more first integrals, and in particular in Hamiltonian systems having several independent symmetries. Our method is based on a generalization of the concept of a normal periodic orbit as introduced by Sepulchre and MacKay [Nonlinearity 10 (1997) 679]. We illustrate the continuation method on some integrable Hamiltonian systems with two degrees of freedom and briefly discuss some further applications.

Global bifurcations of the Lorenz manifold

The dynamics of forced excitable systems are studied analytically and numerically with a view toward understanding the resonance or phase-locking structure. In a singular limit the system studied reduces to a discontinuous flow on a two-torus, which in turn gives rise to a set-valued circle map. It is shown how to define rotation numbers for such systems and derive properties analogous to those known for smooth flows. The structure of the phase-locking regions for a Fitzhugh–Nagumo system in the singular limit is also analyzed. A singular perturbation argument shows that some of the general results persist for the nonsingularly-perturbed system, and some numerical results on phase-locking in the forced Fitzhugh–Nagumo equations illustrate this fact. The results explain much of the phase-locking behavior seen experimentally and numerically in forced excitable systems, including the existence of threshold stimuli for phase-locking. The results are compared with known results for forced oscillatory systems.