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Analysis of a procedure for finding numerical trajectories close to chaotic saddle hyperbolic sets†

Published online by Cambridge University Press:  19 September 2008

Helena E. Nusse and
James A. Yorke
Helena E. Nusse
Affiliation:
University of Maryland, College Park, Maryland 20742, USA
James A. Yorke
Affiliation:
University of Maryland, College Park, Maryland 20742, USA
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Abstract

In dynamical systems examples are common in which there are regions containing chaotic sets that are not attractors, e.g. systems with horseshoes have such regions. In such dynamical systems one will observe chaotic transients. An important problem is the ‘Dynamical Restraint Problem’: given a region that contains a chaotic set but contains no attractor, find a chaotic trajectory numerically that remains in the region for an arbitrarily long period of time.

We present two procedures (‘PIM triple procedures’) for finding trajectories which stay extremely close to such chaotic sets for arbitrarily long periods of time.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 1 , March 1991, pp. 189 - 208
Copyright
Copyright © Cambridge University Press 1991

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References

REFERENCES

[AY]Alligood, K. T. & Yorke, J. A.. Accessible saddles on fractal basin boundaries. Preprint 1989.Google Scholar
[B]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics 470, Springer Verlag: Berlin, 1975.Google Scholar
[BR]Bowen, R. & Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181–202.CrossRefGoogle Scholar
[DN]Devaney, R. & Nitecki, Z.. Shift automorphisms in the Henon mapping. Commun. Math. Phys. 61 (1979), 137–146.Google Scholar
[GOY]Grebogi, C., Ott, E. & Yorke, J. A.. Basin boundary metamorphoses: changes in accessible boundary orbits. Physica 24D (1987), 243–262.Google Scholar
[GNOY]Grebogi, C., Nusse, H. E., Ott, E. & Yorke, J. A.. Basic sets: sets determine the dimension of basin boundaries. In: Dynamical Systems, ed. Alexander, J. C.. Proceedings of the University of Maryland 1986–87. Lecture Notes in Math. 1342, pp. 220–250. Springer-Verlag: Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988.Google Scholar
[GH]Guckenheimer, J. & Holmes, P.. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences 42, Springer Verlag: New York, 1983.Google Scholar
[M]Melo, W. de. Structural stability of diffeomorphisms on two-manifolds. Invent. Math. 21 (1973), 233–246.Google Scholar
[NP]Newhouse, S. & Palis, J.. Hyperbolic nonwandering sets on two-dimensional manifolds. In: Dynamical Systems, pp. 293–301, ed. Peixoto, M. M.. Academic Press: New York and London, 1973.Google Scholar
[Ne]Newhouse, S. E.. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. I.H.E.S. 50 (1979), 101–151.Google Scholar
[Ni]Nitecki, Z.. Differentiate Dynamics, MIT Press, Cambridge, 1971.Google Scholar
[Nu]Nusse, H. E.. Asymptotically periodic behaviour in the dynamics of chaotic mappings. SIAM J. Appl. Math. 47 (1987), 498–515.Google Scholar
[NY]Nusse, H. E. & Yorke, J. A.. A procedure for finding numerical trajectories on chaotic saddles. Physica D36 (1989), 137–156.Google Scholar
[PT]Palis, J. & Takens, F.. Homoclinic bifurcations and hyperbolic dynamics. 16° Colóquio Brasileiro Matemática, IMPA, 1987.Google Scholar
[S]Smale, S.. Differentiate dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[Y]Yorke, J. A.. DYNAMICS. A Program for IBM PC Clones. 1987, 1988.Google Scholar