Abstract
We study a bifurcation of homoclinic and heteroclinic orbits in a two or more parameter family of autonomous ODEs, where the unperturbed system has two heteroclinic orbits joined at a common saddle point. Under some assumptions on eigenvalues of the linearized equation at equilibrium points and on a non-degeneracy condition for the system, we can show that heteroclinic orbits of new type appear besides the persistent ones of the unperturbed system. A bifurcation diagram is given for such families. Some homoclinic bifurcations are also treated including the one producing a twice-rounding homoclinic orbit.
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References
G. R. Belitskii, Functional equations and conjugacy of local diffeomorphism of a finite smoothness class. Functional Anal. Appl.7 (1973), 268–277.
H. Berestycki, B. Nicolaenko and B. Scheurer, Travelling wave solutions to reaction-diffusion systems modeling combustion. Nonlinear Partial Differential Equations, Contemporary Mathematics Vol. 17, AMS, Providence, 1983, 189–208.
J. Carr. Applications of Centre Manifold Theory. Appl. Math. Sci., Vol. 35, Springer, 1981.
J. Carr, S.-N. Chow and J. K. Hale, Abelian integrals and bifurcation theory. J. Differential Equations,59 (19850, 413–436.
S.-N. Chow, B. Deng and D. Terman, The bifurcation of a homoclinic orbit from two heteroclinic orbits—a topological approach—. Preprint.
S.-N. Chow, B. Deng and D. Terman, The bifurcation of a homoclinic and periodic orbit from two heteroclinic orbits— an analytical approach—. Preprint.
S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory. Springer, 1982.
E. A. Coddington and L. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.
W. A. Coppel, Dichotomies in Stability Theory. Lecture Notes in Math.,629, Springer, 1978.
P. Coullet, J.-M. Gambaudo et C. Tresser, Une bifurcation de codimension 2: le collages de cycles. C. R. Acad. Sci. Paris,299 (1984), 253–256.
J.-M. Gambaudo, P. Glendinning et C. Tresser, Collages de cycles et suites de Farey. C. R. Acad. Sci. Paris,299 (1984), 711–714.
P. Gaspard, Generation of a countable set of homoclinic flows through bifurcation. Phys. Lett.,97A (1983), 1–4.
S. A. van Gils, A note on “Abelian integrals and bifurcation theory.” J. Differential Equations,59 (1985), 437–441
P. Glendinning, Bifurcations near homoclinic orbits with symmetry. Phys. Lett.,103A (1984), 163–166.
P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits. J. Statist. Phys.,35 (1984), 645–696.
P. Glendinning and C. Sparrow, T-points: A codimension two heteroclinic bifurcation. J. Statist. Phys.,43 (1986), 479–488.
J. Gruendler, The existence of homoclinic orbits and the method of Melnikov for systems inR n. SIAM J. Math. Anal.,16 (1985), 907–931.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Appl. Math. Sci. Vol. 42, Springer, 1983, 2nd printing, 1986.
H. Ikeda, M. Mimura and T. Tsujikawa, Singular perturbation approach to traveling wave solutions of the Hodgkin-Huxley equations and its application to stability problems. To appear in Japan J. Appl. Math.
H. Kokubu, On a codimension 2 bifurcation of heteroclinic orbits. Proc. Japan Acad.,63, Ser. A (1987), 298–301.
E. N. Lorenz, Deterministic nonperiodic flow. J. Atom. Sci.20 (1963), 130–141.
V. K. Melnikov, On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc.,12 (1963), 1–57.
K. J. Palmer, Exponential dichotomies and transversal homoclinic points. J. Differential Equations,55 (1984), 225–256.
J. Rinzel and D. Terman, Propagation phenomena in a bistable reaction-diffusion system. SIAM J. Appl. Math.,42 (1982), 1111–1137.
J. A. Rodriguez, Bifurcations to homoclinic connections of the focus-saddle type. Arch. Rational Anal. Mech.,93 (1985), 81–90.
S. Schecter, The saddle-node separatrix-loop bifurcation. SIAM J. Math. Anal.,18 (1987), 1142–1156.
L. P. Shil’nikov, On a Poincaré-Birkhoff problem. Math. USSR-Sb.,3 (1967), 353–371.
L. P. Shil’nikov, On the generation of periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type. Math. USSR-Sb.,6 (1968), 427–438.
C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Atrractors. Appl. Math. Sci. Vol. 41, Springer, 1982.
C. Tresser, About some theorems by L. P. Shil’nikov. Ann. Inst. H. Poincaré,40 (1984), 441–461.
E. Yanagida, Branching of double pulse solutions from single pulse solutions in nerve axon equations. J. Differential Equations,66 (1987), 243–262.
H. Żolądek, Bifurcations of certain family of planar vector fields tangent to axes. J. Differential Equations,67 (1987), 1–55.
B. Deng, Shil’nikov problem, exponential expansion, strong λ-lemma,C 1-linearization and homoclinic bifurcation. To appear in J. Differential Equations.
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Kokubu, H. Homoclinic and heteroclinic bifurcations of Vector fields. Japan J. Appl. Math. 5, 455–501 (1988). https://doi.org/10.1007/BF03167912
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DOI: https://doi.org/10.1007/BF03167912