Compact connected lie transformation groups on spheres with low cohomogeneity, I

Memoirs of the American Mathematical Society

Volumn , Issue 569, 1996, Pages 1-79

  Straume, Eldar ^a  

^a NONE

Author keywords

Averaging; Chaos; Discretization; Exponential smallness; Homoclinic orbit; Ordinary diffferential equations; Transversality

Indexed keywords

EID: 0029687918     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0569     Document Type: Article

References (53)

1. 1. R. Abraham & J. Robbin: Transversal Mappings and Flows. W. A. Benjamin, Inc., New York (1967).
2. 2. L. V. Ahlfors: Complex Analysis. McGraw-Hill, Tokyo (1966).
3. 3. J. C. Alexander & J. A. Yorke: Global bifurcation of periodic orbits. Amer. J. Math. 100, pp. 263-292 (1978).
4. 4. C. Amick & E. S. C. Ching & L. P. Kadanoff & V. Rom-Kedar: Beyond all orders: Singular perturbations in a mapping. J. Nonlinear Sci. 2, pp. 9-67 (1992).
5. 5. S. Angenent: A variational interpretation of Melnikov's function and exponentially small separatrix splitting. In "Symplectic Geometry", D. Salamon (ed.), London Math. Soc. Lect. Notes 192, pp. 5-35 (1993).
6. 6. V. I. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, New York (1983).
7. 7. D. G. Aronson & M.A. Chory & G. R. Hall fe R. P. McGehee: Bifurcations from an invariant circle for two-parameter familes of maps of the plane: a computer-assisted study. Comm. Math. Phys. 83, pp. 303-354 (1982).
8. 8. G. R. Belitskii: Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class. Funct. Analysis Appl. 7, pp. 268-277 (1973).
9. 9. W. J. Beyn: The effect of discretization on homoclinic orbits. In "Bifurcation: Analysis, Algorithms, Applications", T. Küpper et al. (eds.). Birkhäuser Verlag, Basel, pp. 1-8 (1987).
10. 10. W. J. Beyn: The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Analysis 9, pp. 379-405 (1990).
11. 11. H. Bohr: Almost Periodic Functions. Chelsea Publ. Co., New York (1947).
12. 12. A. Chenciner: Bifurcations de points fixes élliptiques III. Orbites périodiques de "petites" périodes et élimination résonnante des couples de courbes invariantes. Publ. Math. IHES 66, pp. 5-91 (1988).
13. 13. S.-N. Chow & B. Deng & B. Fiedler: Homoclinic bifurcation at resonant eigenvalues. To appear in J. Dyn. Diff. Eq. (1990).
14. 14. S.-N. Chow & J. K. Hale: Methods of Bifurcation Theory. Springer-Verlag, New York (1982).
15. 15. K. Deimling: Nonlinear Functional Analysis. Springer-Verlag, Berlin (1985).
16. 16. A. Delshams & M.T.M. Seara:An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. Comm. Math. Phys. 150 pp. 433-463 (1992).
17. 1-linearization, and homoclinic bifurcation. J. Diff. Eq. 79, pp. 189-231 (1989).
18. 18. P. Deuflhard: Newton Techniques for Highly Nonlinear Problems. Theory and Algorithms. In preparation (1995).
19. 19. R. Devaney: Homoclinic orbits in Hamiltonian systems. J. Diff. Eq. 21, pp. 431-438 (1976).
20. 20. E. J. Doedel & M. J. Friedinan: Numerical computation of heteroclinic orbits. J. Comp. Appl. 26, pp. 155-170 (1989).
21. 21. E. J. Doedel & M. J. Friedman: Numerical computation and continuation of invariant manifolds connecting fixed points. SIAM J. Numer. Analysis 28, pp. 789-808 (1991).
22. 22. B. Fiedler: Global pathfollowing of homoclinic orbits in two-parameter flows. Preprint (1992).
23. 23. E. Fontich: Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations. J. Nonlinear Anal. 20, pp. 733-744 (1993).
24. 24. E. Fontich & C. Simó: Splitting of separatrices for analytic diffeomorphisms. Ergod. Theor. Dyn. Sys. 10, pp. 295-318 (1990a).
25. 25. E. Fontich & C. Simó: Invariant manifolds for near identity differentiable maps and splitting of separatrices. Ergod. Th. & Dynam. Sys. 10, pp. 319-346 (1990b).
26. 26. V. G. Gelfreich: Splitting of separatrices for the rapidly forced pendulum. Preprint (1990).
27. 27. V. G. Gelfreich & V. F. Lazutkin & M.B. Tabanov: Exponentially small splitting in Hamiltonian systems. Chaos 1, pp. 137-142 (1991).
28. 28. C. Grebogi & E. Ott & J. A. Yorke: Basin boundary metamorphoses: changes in accessible boundary orbits. Physica 24 D, pp. 243-262 (1987).
29. 29. J. Guckenheimer & P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York (1983).
30. 30. U. Habbe: Numerische Versuche zur Diskretisierung homokliner Lösungen. Diploma-thesis, Heidelberg (1991).
31. 31. E. Hairer & S. P. Nørsett & G. Wanner: Solving Ordinary Differential Equations I. Springer-Verlag, Berlin (1987).
32. 32. D. Henry: Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin (1981).
33. 33. M. W. Hirsch: Differential Topology. Springer-Verlag, New York (1976).
34. 34. M. W. Hirsch & C. C. Pugh &, M. Shub: Invariant Manifolds. Springer-Verlag, Berlin (1977).
35. 35. M. W. Hirsch & S. Smale: On algorithms for solving f(x) = 0. Comm. Pure Appl. Math. 22, pp. 281-312 (1979).
36. 36. P. J. Holmes & J. E. Marsden & J. Scheurle: Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations. Cont. Math. 81, pp. 213-244 (1988).
37. 37. H. T. Jongen & P. Jonker & F. Twilt: Gradient Newton-flows (a survey). Preprint (1987).
38. 38. T. Kato: Perturbation Theory of Linear Operators. Springer-Verlag, Berlin (1966).
39. 39. M. Kummer & J. A. Ellison & A. W. Saenz: Exponentially small phenomena in the rapidly forced pendulum. In "Asymptotics Beyond all Orders", H. Segur et al. (eds.). Plenum Press, New York, pp. 197-211 (1991).
40. 40. V. F. Lazutkin: Analytic integrals of the semistandard map, and splittings of separatrices. Leningrad Math. J. 1, pp. 427-445 (1990).
41. 41. V. F. Lazutkin & I. G. Schachmannski & M.B. Tabanov: Splitting of separatrices for standard and semistandard mappings. Physica D 40, pp. 235-248 (1989).
42. 42. J. Moser: Stable and Random Motions in Dynamical Systems. Princeton University Press, New York (1973).
43. 43. S. Newhouse: The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. IHES 50, pp. 101-151 (1979).
44. 44. J. Palis, Jr. & W. de Melo: Geometric Theory of Dynamical Systems. Springer-Verlag, New York (1982).
45. 45. J. Palis, Jr. & F. Takens: Hyperbolicity and the creation of homoclinic orbits. Annals of Math. 125, pp. 337-374 (1987).
46. 46. K. J. Palmer: Exponential dichotomies and transversal homoclinic points. J. Diff. Eq. 55, pp. 225-256 (1984).
47. 47. J. A. Sanders: Melnikov's method and averaging. Celest. Mech. 28, pp. 171-181 (1982).
48. 48. J. Scheurle: Chaos in a rapidly forced pendulum equation. Cont. Math. 97, pp. 411-419 (1989).
49. 49. J. Scheurle & J. E. Marsden & P. J. Holmes: Exponentially small estimates for separatrix splittings. In "Asymptotics Beyond all Orders", H. Segur et al. (eds.). Plenum Press, New York, pp. 187-195 (1991).
50. 50. C. Simó: Averaging under fast quasiperiodic forcing. In "Integrable and Chaotic Behaviour in Hamiltonian Systems", I. Seimenis (ed.). Plenum Press, New York, pp. 13-34 (1994).
51. 51. S. Smale: On the efficiency of algorithms of analysis. Bull. AMS (N.S.) 13, pp. 87-121 (1985).
52. 52. A. Vanderbauwhede & B. Fiedler: Homoclinic period blow-up in reversible and conservative systems. Z. angew. Math. Phys. 43, pp. 292-318 (1992).
53. 53. E. Zehnder: Homoclinic points near elliptic fixed points. Cumm. Pure Appl. Math. 26, pp. 131-182 (1973).