On the computation of finite invariant sets of mappings

Authors:
Alex Gelman and Werner C. Rheinboldt

Journal:
Math. Comp. **52** (1989), 545-551

MSC:
Primary 58F25; Secondary 39B10, 58F08, 65H10

DOI:
https://doi.org/10.1090/S0025-5718-1989-0955751-0

MathSciNet review:
955751

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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.

**[1]**E. Doedel,*Software for Continuation and Bifurcation Problems in Ordinary Differential Equations*, California Institute of Technology, Applied Mathematics Report 1986.**[2]**C. Hayashi,*Nonlinear Oscillations in Physical Systems*, Princeton Univ. Press, Princeton, N. J., 1964. MR**0170071 (30:312)****[3]**G. Iooss, A. Arneodo, P. Coullet & C. Tresser,*Simple Computation of Bifurcating Invariant Circles for Mappings*, Lecture Notes in Math., Vol. 898, Springer-Verlag, Berlin and New York, 1981, pp. 192-211. MR**654890 (83i:58067)****[4]**I. G. Kevrekidis, R. Aris, L. D. Schmidt & S. Pelikan, "Numerical computation of invariant circles of maps,"*Phys. D*, v. 16, 1985, pp. 243-251. MR**796271 (86j:58113)****[5]**W. C. Rheinboldt & J. V. Burkardt, "A locally parametrized continuation process,"*ACM Trans. Math. Software*, v. 9, 1983, pp. 215-235. MR**715303 (85f:65052)****[6]**M. Van Veldhuizen, "Convergence results for invariant curve algorithms,"*Math. Comp.*, v. 51, 1988, pp. 677-697. MR**930220 (89b:65201)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0955751-0

Article copyright:
© Copyright 1989
American Mathematical Society