On the existence of chaotic behaviour of diffeomorphisms. (English) Zbl 0789.58056

The existence of chaotic behaviour is shown for a number of mappings by slowing the existence of transversal homoclinic orbits. Results are shown analytically. The method, based on Lyapunov-Schmidt reduction is applied to ordinary differential equations with impulsive effects and two singularly perturbed mappings.
Reviewer: B.F.Gray (Sydney)


37G99 Local and nonlocal bifurcation theory for dynamical systems
34C23 Bifurcation theory for ordinary differential equations
Full Text: EuDML


[1] K. R. Meyer & C. R. Sell: Melnikov transforms, Bernoulli bundles, and almost periodic perturbations. Trans. Amer. Math. Soc. 314 (1) (1989), 63-105. · Zbl 0707.34041
[2] K. J. Palmer: Exponential dichotomies, the shadowing lemma and transversal homoclinic points. Dynamics Reported 1 (1988), 265-306. · Zbl 0676.58025
[3] K. J. Palmer: Exponential dichotomies and transversal homoclinic points. J. Diff. Equations 55 (1984), 225-256. · Zbl 0508.58035
[4] S. N. Chow, J . K. Hale & J. Mallet-Paret: An example of bifurcation to homoclinic orbits. J. Diff. Equations 37 (1980), 351-373. · Zbl 0439.34035
[5] M. Fečkan: Bifurcations of heteroclinic orbits for diffeomorfisms. Aplikace Matematiky 36 (1991), 355-367. · Zbl 0748.58022
[6] S. Smale: Diffeomorphisms with infinitely many periodic points. in Differential and Combinatorical Topology, Princeton Univ. Press, New Jersey, 1963, pp. 63-80.
[7] C. Pugh M. Shub & M. W. Hirsch: Invariant Manifolds. Lec. Not. Math. 583, Springer- -Verlag, New York, 1977. · Zbl 0355.58009
[8] S. Wiggins: Global Bifurcations and Chaos. Appl. Math. Sci. 73, Springer- Verlag, New York, 1988. · Zbl 0661.58001
[9] D. Henry: Geometric Theory of Semilinear Parabolic Equations. Lec. Not. Math. 840, Springer- Verlag, New York, 1981. · Zbl 0456.35001
[10] M. W. Hirsh & S. Smale: Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, New York, 1974.
[11] M. L. Glasser V. G. Papageoriou & T. C. Bountis: Melnikov’s function for two-dimensional mappings. SIAM J. Appl. Math. 49 (1989), 692-703. · Zbl 0687.58023
[12] M. Medved’: Dynamical Systems. Veda, Bratislava, 1988.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.