Symbolic dynamics for the Hénon-Heiles Hamiltonian on the critical level

Journal of Differential Equations

Volumn 171, Issue 1, 2001, Pages 173-202

  Arioli, Gianni ^a   Zgliczyński, Piotr ^b,c  

^a UNIVERSITY OF EASTERN PIEDMONT   (Italy)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^c JAGIELLONIAN UNIVERSITY   (Poland)

Author keywords

Computer assisted proof; Covering relations; Periodic points; Symbolic dynamics

Indexed keywords

EID: 0035917024     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.2000.3835     Document Type: Article

References (33)

1. Bastos de Figueiredo J. C., Grotta Ragazzo C., Malta C. P. Two important numbers in the Hénon-Heiles dynamics. Phys. Lett. A. 241:1998;35-40.
2. Churchill R. C., Kummer M., Rod D. L. On averaging, reduction, and symmetry in Hamiltonian systems. J. Differential Equations. 49:1983;359-414.
3. Churchill R. C., Pecelli G., Rod D. L. A Survey of the Hénon-Heiles Hamiltonian with Applications to Related Examples. Lecture Notes in Physics. 93:1978;Springer-Verlag, New York/Berlin. p. 76-136.
4. Churchill R. C., Pecelli G., Rod D. L. Stability transitions for periodic orbits in hamiltonian systems. Arch. Rational Mech. Anal. 73:1980;313-347.
5. Churchill R. C., Rod D. L. Pathology in dynamical systems. I. General theory. J. Differential Equations. 21:1976;39-65.
6. Churchill R. C., Rod D. L. Pathology in Dynamical Systems. II. Applications. J. Differential Equations. 21:1976;66-112.
7. Churchill R. C., Rod D. L. Pathology in Dynamical Systems. III. Analytic Hamiltonians. J. Differential Equations. 37:1980;23-38.
8. Churchill R. C., Rod D. L. Homoclinic and heteroclinic orbits of reversible vectorfields under perturbation. Proc. Roy. Soc. Edinburgh Ser. A. 102:1986;345-363.
9. Easton R. Isolating blocks and symbolic dynamics. J. Differential Equations. 17:1975;96-118.
10. J. Guckenheimer, and, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York/Heidelberg/Berlin.
11. Galias Z., Zgliczyński P. Computer assisted proof of chaos in the Lorenz system. Physica D. 115:1998;165-188.
12. Grotta Ragazzo C. Nonintegrability of some Hamiltonian systems, scattering and analytic continuation. Comm. Math. Phys. 166:1994;255-277.
13. Grotta Ragazzo C. Irregular dynamics and Homoclinic orbits to Hamiltonian saddle-centers. Comm. Pure Appl. Math. 50:1997;105-147.
14. Hénon M., Heiles C. The applicability of the third integral of motion. Astron. J. 69:1964;449-457.
15. Hassard B., Zhang J., Hastings S., Troy W. A computer proof that the Lorenz equations have "chaotic" solutions. Appl. Math. Lett. 7:1994;79-83.
16. Lerman L. M. Hamiltonian systems with loops of a separatrix of a saddle-center. Sel. Math. Sov. 10:1991;297-306.
17. Lohner R. J. Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value problems. Cash J. R., Gladwell I. Computational Ordinary Differential Equations. 1992;Clarendon Press, Oxford.
18. Lunsford G. H., Ford J. On the stability of periodic orbits for nonlinear oscillator systems in regions exhibiting stochastic behavior. J. Math. Phys. 13:1972;700-705.
19. Mielke A., Holmes P., O'Reilly O. Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center. J. Dynam. Differential Equations. 4:1992;95-126.
20. Mischaikow K., Mrozek M. Isolating neighborhoods and chaos. Japan J. Indust. Appl. Math. 12:1995;205-236.
21. Mischaikow K., Mrozek M. Chaos in the Lorenz equations: A computer assisted proof. Bull. Amer. Math. Soc. 32:1995;66-72.
22. Mischaikow K., Mrozek M. Chaos in the Lorenz equations: A computer assisted proof. II: Details. Math. Comput. 67:1998;1023-1046.
23. Moser J. On the generalization of a theorem of Liapunov. Comm Pure Appl. Math. 11:1958;257-271.
24. Morales-Ruiz J. J., Simo C. Non-integrability criteria for Hamiltonians in the case of lamé normal variational equations. J. Differential Equations. 129:1996;111-135.
25. Morales-Ruiz J. J., Peris J. P. On a Galosian approach to the splitting of separatrices. Ann. Faculté Sci. Toulouse. 8:1999;125-141.
26. Neumaier A., Rage T. Rigorous chaos verification in discrete dynamical systems. Physica D. 67:1993;327-346.
27. Rage T., Neumaier A., Schlier C. Rigorous verification of chaos in a molecular model. Phys. Rev. E. 50:1994;2682-2688.
28. Rüssmann H. Über das Verhalten analitischer Hamiltonscher Differential-Gleichungen in der Nähe einer Gleichgewichtlösung. Math. Ann. 154:1964;285-300.
29. W. Tucker, Lorenz attractor exists, preprint.
30. Walters P. An Introduction to Ergodic Theory. 1982;Springer-Verlag, New York.
31. Zgliczyński P. Fixed point index for iterations, topological horseshoe and chaos. Topol. Methods Nonlinear Anal. 8:1996;169-177.
32. Zgliczyński P. Computer assisted proof of chaos in the Hénon map and in the Rössler equations. Nonlinearity. 10:1997;243-252.
33. Zgliczyński P. Sharkovskii′s Theorem for multidimensional perturbations of 1-dim maps. Ergodic Theory Dynam. Systems. 19:1999;1655-1684.