Symmetric homoclinic tangles in reversible systems

Ergodic Theory and Dynamical Systems

Volumn 26, Issue 6, 2006, Pages 1769-1789

  Homburg, Ale Jan ^a   Lamb, Jeroen S W ^b  

^a UNIVERSITY OF AMSTERDAM   (Netherlands)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33751068656     PISSN: 01433857     EISSN: 14694417     Source Type: Journal    
DOI: 10.1017/S0143385706000472     Document Type: Article

References (28)

1. [Arn84] V. I. Arnol'd. Reversible systems. Nonlinear and Turbulent Processes in Physics. Vol. 3. Harwood, New York, 1984.
2. [Cha98] A. R. Champneys. Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Phys. D 112 (1998), 158-186.
3. [CH00] A. R. Champneys and J. Härterich. Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems. Dyn. Stab. Syst. 15 (2000), 231-252.
4. 1-linearization, and homoclinic bifurcation. J. Differential Equations 79 (1989), 189-231.
5. [Dev76a] R. L. Devaney. Reversible diffeomorphisms and flows. Trans. Amer. Math. Soc. 218 (1976), 89-113.
6. [Dev76b] R. L. Devaney. Homoclinic orbits in Hamiltonian systems. J. Differential Equations 21 (1976), 431-438.
7. [Dev77] R. L. Devaney. Blue sky catastrophes in reversible and Hamiltonian systems. Indiana Univ. Math. J. 26 (1977), 247-263.
8. [DeV58] R. DeVogelaere. On the structure of symmetric periodic solutions of conservative systems, with applications. Contributions to the Theory of Nonlinear Oscillations. Vol. IV (Annals of Mathematics Studies, 41). Princeton University Press, Princeton, NJ, 1958.
9. [Dia95] L. J. Díaz. Robust nonhyperbolic dynamics and heterodimensional cycles. Ergod. Th. & Dynam. Sys. 15 (1995), 291-315.
10. [GilVan87] S. A. van Gils and A. Vanderbauwhede. Center manifolds and contractions on a scale of Banach spaces. J. Funct. Anal. 72 (1987), 209-224.
11. [GS95] V. S. Gonchenko and L. P. Shil'nikov. On geometrical properties of two-dimensional diffeomorphisms with homoclinic tangencies. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), 819-829.
12. [GI00] A. S. Gorodetskiǐ and Yu. S. Il'yashenko. Some properties of skew products over a horseshoe and a solenoid. Proc. Steklov Inst. Math. 231 (2000), 90-112.
13. [Har98] J. Härterich. Cascades of reversible homoclinic orbits to a saddle-focus equilibrium. Phys. D 112 (1998), 187-200.
14. [HPS77] M. W. Hirsch, C. C. Pugh and M. Shub. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.
15. [Hom96] A. J. Homburg. Global aspects of homoclinic bifurcations of vector fields. Memoirs Amer. Math. Soc. 578 (1996) 128 pp.
16. [HK06] A. J. Homburg and J. Knobloch. Multiple homoclinic orbits in conservative and reversible systems. Trans. Amer. Math. Soc. 358 (2006), 1715-1740.
17. [HVS03] A. J. Homburg, R. de Vilder and D. Sands. Computing invariant sets. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 497-504.
18. [Ioo98] G. Iooss. Travelling water-waves, as a paradigm for bifurcations in reversible infinite-dimensional 'dynamical' systems. Proc. Int. Conf. Mathematicians (Berlin, 1998). Vol. III. Doc. Math. (1998), Extra Vol. III, 611-622.
19. [Irw80a] M. C. Irwin. Smooth Dynamical Systems. Academic Press, New York, 1980.
20. [Irw80b] M. C. Irwin. A new proof of the pseudo-stable manifold theorem. J. London Math. Soc. 21 (1980), 557-566.
21. [LR98] J. S. W. Lamb and J. A. G. Roberts. Time-reversal symmetry in dynamical systems: a survey. Phys. D 112 (1998), 1-39.
22. [New04] S. E. Newhouse. New phenomena associated with homoclinic tangencies. Ergod. Th. & Dynam. Sys. 24 (2004), 1725-1738.
23. [Pal00] J. Palis. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque 261 (2000), 335-347.
24. [PT93] J. Palis and F. Takens. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Fractal Dimensions and Infinitely Many Attractors. Cambridge University Press, Cambridge, 1993.
25. [Per29] O. Perron. Über Stabilität und asymptotisches Verhalten der Lösungen eines Systems endlicher Differentialgleichungen. J. reine angew. Math. 161 (1929), 41-64.
26. [RL95] J. A. G. Roberts and J. S. W. Lamb. Self-similarity of period-doubling branching in 3-D reversible mappings. Phys. D 82 (1995), 317-332.
27. [Sev86] M. B. Sevryuk. Reversible Systems (Lecture Notes in Mathematics, 1211). Springer, Berlin, 1986.
28. [Sma65] S. Smale. Diffeomorphisms with many periodic points. Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse). Princeton University Press, Princeton, NJ, 1965.