Abstract
Interaction of homoclinic bifurcation and bifurcation on the center manifold is studied. We show that the occurrence of different types of solutions near the homoclinic orbit is determined asymptotically by a reduced system on the center manifold. The method is applied to cases where the center manifold is one- or two-dimensional. When the center manifold is one-dimensional, we can obtain all the solutions near the homoclinic orbit. When a Hopf bifurcation occurs on a two-dimensional center manifold, the system can have infinitely many periodic and aperiodic solutions. These solutions disappear in a manner predicted by the reduced system when the perturbation term is increased. We prove that certain periodic and aperiodic solutions disappear through inverse period doubling or saddle-node bifurcation.
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References
Carr, J. [1981], “Applications of Center Manifold Theory”, Applied Mathematical Sciences, 35, Springer-Verlag, New York.
Chow, S. -N., Deng, B. and Fiedler, B. [ 1990 ], “Homoclinic bifurcation at resonant eigenvalues”, J. Dynamics and Diff. Eqns, 2, 177–244.
Chow, S. -N. and Hale, J. K. [ 1982 ], “Methods of bifurcation theory”, Grund. der Math. Wissen, 251, Springer-Verlag, New York.
Chow, S. -N. and Lin, X. B. [ 1990 ], “Bifurcation of a homoclinic orbit with a saddle-node equilibrium”, Differential and Integral Equations, 3, 435–466.
Chow, S. -N. and Lu, K. [ 1988 ], “Ck center-unstable manifolds”, Royal Soc. Edinburgh, 108A, 303–320.
Chow, S.-N. and Mallet-Paret, J., [ 1977 ], “Integral averaging and bifurcation,” J. Diff Eqns, 26, 112–159.
Crandall, M. G. and Rabinowitz, P. H. [ 1971 ], “Bifurcation from simple eigenvalue”, J. Funct. Anal, 8, 321–340.
Deng, B. [ 1989 ], ‘The Silnikov problem, exponential expansion, strong A-lemma, cl-linearization and homoclinic bifurcation”, J. Differential Equations, 79, 189–231.
Deng, B. [ 1990 ], “Homoclinic bifurcation with nonhyperbolic equilibria”, SIAM J. Math. Anal, 21, 693–720.
Deng, B. and Sakamoto, K. [ 1995 ], “Silnikov-Hopf bifurcations”, J. Differential Equations, 119, 1–23.
Fenichel, N. [ 1974 ], “Asymptotic stability with rate conditions”, Indiana Univ. Math. J, 23, 1109–1137.
Glendinning, P. and Sparrow, C. [ 1984 ], “Local and global behavior near homoclinic orbits”, J. Stat. Phys, 35, 645–696.
Guckenheimer, J. and Holmes, P. [ 1983 ], “Nonlinear oscillations, dynamical systems, and bifurcations of vector fields”, Springer-Verlag, New York.
Guckenheimer, J., Moser, J. and Newhouse, S.E. [ 1980 ], “Dynamical systems”, Birkäuser, Boston.
Hale, J.K. [ 1980 ], “Ordinary differential equations”, 2nd Edition, Krieger Malabar, Florida
Hale, J. and Lin, X. -B., [ 1986 ], “Heteroclinic Orbits for Retarded Functional Differential Equations”, J. Differential Equations, 65, 175–202.
Hirsch, M. W., Pugh, C. C. and Shub, M. [ 1977 ], “Invariant manifolds”, Lecture Notes in Math, 583, Springer-Verlag, New York.
Hirschberg, P. and Knobloch, E. [ 1992 ], “Silnikov-Hopf Bifurcation”, Preprint.
Lin, X. B. [ 1990 ], “Using Melnikov’s method to solve Silnikov’s problems”, Proc. Roy. Soc. Edinburgh, 166A, 295–325.
Lukyanov, V. I. [ 1982 ], “Bifurcation of dynamical systems with a saddle point separatrix loop”, Differential Equations, 18, 1049–1059.
Moser, J. [ 1973 ], “Stable and Random Motions in Dynamical Systems”, Annals of Mathematics Studies, Princeton Univ. Press.
Newhouse, S. E. [ 1974 ], “Diffeomorphisms with infinitely many sinks”, Topology, 13, 9–18.
Robinson, C. [ 1983 ], “Bifurcation to infinitely many sinks”, Commun. Math. Phys, 90, 433–459.
Schecter, S. [ 1987a ], “The saddle-node separatrix-loop bifurcation”, SIAM J. Math. Anal, 18, 1142–1156.
Schecter, S. [ 1987b ], “Melnikov’s method at a saddle-node and the dynamics of the forced Josephson junction”, SIAM J. Math. Anal, 18, 1699–1715.
Schwartz, J. T. [ 1971 ], “Nonlinear functional analysis”, Academic Press, New York and London.
Silnikov, L. P. [ 1968 ], “On the generation of a periodic motion from trajectories double asymptotic to an equilibrium state of saddle type”, Math. USSR Sbornik, 6, 427–437.
Silnikov, L. P. [ 1970 ], “A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type”, Math. USSR Sbornik, 10, 92–102.
Vanderbauwhede, A. and Van Gils, S.A. [ 1987 ], “Center manifolds and contraction on a scale of Banach spaces”, J. Func. Anal, 72, 209–224.
Wiggins, S. [ 1988 ], “Global bifurcations and chaos”, Appl. Math. Science, 73, Springer-Verlag, New York.
Yorke, J. A. and Alhgood, Kathleen T. [ 1985 ], “Periodic Doubling Cascades of Attractors: A Prerequisite for Horseshoes”, Commun. Math. Phys, 101, 305–321.
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Lin, XB. (1996). Homoclinic Bifurcations with Weakly Expanding Center Manifolds. In: Jones, C.K.R.T., Kirchgraber, U., Walther, HO. (eds) Dynamics Reported. Dynamics Reported. New Series, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-79931-0_3
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DOI: https://doi.org/10.1007/978-3-642-79931-0_3
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