×

Numerical computation of a branch of invariant circles starting at a Hopf bifurcation point. (English) Zbl 0810.65070

Kloeden, Peter E. (ed.) et al., Chaotic numerics. An international workshop on the approximation and computation of complicated dynamical behavior, July 12-16, 1993, Deakin Univ., Geelong, Australia. Providence, RI: American Mathematical Society. Contemp. Math. 172, 169-184 (1994).
Given a one-parameter family \(P_ \lambda\) of smooth diffeomorphisms of an open set in \(\mathbb{R}^ n\), the paper aims on the numerical computation and continuation (w.r.t. \(\lambda\)) of invariant circles \(T_ \lambda\) of \(P_ \lambda\), starting at a Hopf point of maps. Assuming that an invariant circle \(T_ 0\) is available for \(\lambda = \lambda_ 0\), the “next” circle \(T_ \lambda\) is sought by introducing orthonormal coordinate systems moving around \(T_ 0\) which allows a representation of \(T_ \lambda\) by a graph of a function on \(T_ 0\) for which a 1- periodic parametrization \(u_ 0(s)\) is assumed to be given. A nonlinear operator equation \(\Psi_ \lambda(x) = 0\) is formulated for a 1-periodic function \(x(s)\) such that \(T_ \lambda = \{(s, x(s)): s\in [0,1)\}\) with respect to the coordinate system given by the reference circle \(T_ 0\).
After some discretization – a collocation method with cubic splines is proposed – one ends up with a finite-dimensional nonlinear system of equations depending on \(\lambda\) which can be attacked by any classical pseudo arclength algorithm. Special attention is paid to an appropriate choice of the collocation nodes \(s_ 0, \dots, s_ N\) with respect to the parametrization \(u_ 0(s)\) of \(T_ 0\).
Two numerical examples are presented: the delayed logistic map on \(\mathbb{R}^ 2\) and a periodically driven nonlinear electric network ordinary differential equation in the plane where the diffeos are Poincaré maps. The algorithm is not restricted to stable invariant circles, but it is reported to have problems for rational rotation number with small denominators (phase locking).
It should be noted that this method can be considered as an interesting extension of a method proposed in the paper by I. G. Kevrekides, R. Aris, L. D. Schmidt and S. Pelikan [Physica D 16, 243-251 (1985; Zbl 0581.58030)], which is not in the list of references.
For the entire collection see [Zbl 0801.00025].
Reviewer: B.Werner (Hamburg)

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
78A55 Technical applications of optics and electromagnetic theory
34A34 Nonlinear ordinary differential equations and systems
34C23 Bifurcation theory for ordinary differential equations
34C40 Ordinary differential equations and systems on manifolds
37C55 Periodic and quasi-periodic flows and diffeomorphisms

Citations:

Zbl 0581.58030

Software:

AUTO-86
PDFBibTeX XMLCite