On the computation of finite invariant sets of mappings

Gelman, Alex; Rheinboldt, Werner C.

doi:10.1090/S0025-5718-1989-0955751-0

On the computation of finite invariant sets of mappings
HTML articles powered by AMS MathViewer

by Alex Gelman and Werner C. Rheinboldt PDF

Math. Comp. 52 (1989), 545-551 Request permission

Abstract:

This paper suggests a new computational method for determining closed curves that are invariant under a given mapping. Unlike other authors, we discretize not only the curve but also the mapping itself. This allows us to avoid completely the computational difficulties connected with the numerical solution of large linear systems. The method uses simple recurrence formulas, which greatly reduce the execution times.

References

Software for Continuation and Bifurcation Problems in Ordinary Differential Equations

Chihiro Hayashi, Nonlinear oscillations in physical systems, McGraw-Hill Electrical and Electronic Engineering Series, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170071
G. Iooss, A. Arneodo, P. Coullet, and C. Tresser, Simple computation of bifurcating invariant circles for mappings, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 192–211. MR 654890
I. G. Kevrekidis, R. Aris, L. D. Schmidt, and S. Pelikan, Numerical computation of invariant circles of maps, Phys. D 16 (1985), no. 2, 243–251. MR 796271, DOI 10.1016/0167-2789(85)90061-2
Werner C. Rheinboldt and John V. Burkardt, A locally parameterized continuation process, ACM Trans. Math. Software 9 (1983), no. 2, 215–235. MR 715303, DOI 10.1145/357456.357460
M. van Veldhuizen, Convergence results for invariant curve algorithms, Math. Comp. 51 (1988), no. 184, 677–697. MR 930220, DOI 10.1090/S0025-5718-1988-0930220-1