The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points

Deng, Bo

doi:10.1090/S0002-9947-1992-1024768-9

The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points
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Trans. Amer. Math. Soc. 331 (1992), 15-53 Request permission

Abstract:

The complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained. It is topologically conjugate to a non-Bernoulli shift called $\{ {\sum ,\sigma } \}$. We also obtain a more or less complete picture, referred to as the net weaving bifurcation, when the fixed point of such a system is undergoing the generic saddle-node bifurcation. The idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork, transcritical, periodic doubling, and Hopf bifurcations are all homotopically conjugate to our shift dynamics $\{ {\sum ,\sigma } \}$ in a neighborhood of a transverse homoclinic orbit. These bifurcations are also examined in the context of the spectral decomposition with respect to the maximal indecomposable nonwandering sets.

References

George D. Birkhoff, Dynamical systems, American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966. With an addendum by Jurgen Moser. MR 0209095

Sil’nikov problem, invariant manifolds and

lemma

Bo Deng, Homoclinic bifurcations with nonhyperbolic equilibria, SIAM J. Math. Anal. 21 (1990), no. 3, 693–720. MR 1046796, DOI 10.1137/0521037
N. K. Gavrilov and L. P. Šil′nikov, Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve. II, Mat. Sb. (N.S.) 90(132) (1973), 139–156, 167 (Russian). MR 0334280
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
Jürgen Moser, Stable and random motions in dynamical systems, Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. MR 0442980
Sheldon E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9–18. MR 339291, DOI 10.1016/0040-9383(74)90034-2

Sur le problème des trois corps et ces équations de la dynamique

13

Clark Robinson, Bifurcation to infinitely many sinks, Comm. Math. Phys. 90 (1983), no. 3, 433–459. MR 719300
Stephen Schecter, Melnikov’s method at a saddle-node and the dynamics of the forced Josephson junction, SIAM J. Math. Anal. 18 (1987), no. 6, 1699–1715. MR 911659, DOI 10.1137/0518122
Michael Shub, Global stability of dynamical systems, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR 869255, DOI 10.1007/978-1-4757-1947-5

On a Poincaré-Birkhoff problem

3

Stephen Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 63–80. MR 0182020
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1

Generic one-parameter families of vector fields

43

A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal. 72 (1987), no. 2, 209–224. MR 886811, DOI 10.1016/0022-1236(87)90086-3
Stephen Wiggins, Global bifurcations and chaos, Applied Mathematical Sciences, vol. 73, Springer-Verlag, New York, 1988. Analytical methods. MR 956468, DOI 10.1007/978-1-4612-1042-9