Homoclinic and heteroclinic bifurcations of Vector fields

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.

[1]  J. A. Rodríguez,et al.  Bifurcation to homoclinic connections of the focus-saddle type , 1986 .

[2]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[3]  C. Sparrow.  The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors , 1982 .

[4]  L. P. Šil'nikov,et al.  ON THE GENERATION OF A PERIODIC MOTION FROM TRAJECTORIES DOUBLY ASYMPTOTIC TO AN EQUILIBRIUM STATE OF SADDLE TYPE , 1968 .

[5]  Colin Sparrow,et al.  T-points: A codimension two heteroclinic bifurcation , 1986 .

[6]  J. Carr.  Applications of Centre Manifold Theory , 1981 .

[7]  L. P. Šil'nikov.  ON A POINCARÉ-BIRKHOFF PROBLEM , 1967 .

[8]  Kenneth J. Palmer,et al.  Exponential dichotomies and transversal homoclinic points , 1984 .

[9]  J. Gambaudo,et al.  Une nouvelle bifurcation de codimension 2: le collage de cycles , 1984 .

[10]  Bifurcations near homoclinic orbits with symmetry , 1984 .

[11]  S. Gils,et al.  A Note on: "Abelian Integrals and Bifurcation Theory" , 1985 .

[12]  M. Mimura,et al.  Singular perturbation approach to traveling wave solutions of the Hodgkin-Huxley equations and its application to stability problems , 1989 .

[13]  Shui-Nee Chow,et al.  Abelian integrals and bifurcation theory , 1985 .

[14]  W. A. Coppel.  Dichotomies in Stability Theory , 1978 .

[15]  Paul Glendinning,et al.  Collage de cycles et suites de Farey , 1984 .

[16]  David Terman,et al.  Propagation Phenomena in a Bistable Reaction-Diffusion System , 1982 .

[17]  H. Kokubu.  On a codimension $2$ bifurcation of heteroclinic orbits , 1987 .

[18]  Stephen Schecter,et al.  The saddle-node separatrix-loop bifurcation , 1987 .

[19]  Colin Sparrow,et al.  Local and global behavior near homoclinic orbits , 1984 .

[20]  Joseph Gruendler,et al.  The Existence of Homoclinic Orbits and the Method of Melnikov for Systems in $R^n$ , 1985 .

[21]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[22]  Pierre Gaspard,et al.  Generation of a countable set of homoclinic flows through bifurcation , 1983 .

[23]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[24]  G. Belitskii.  Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class , 1973 .

[25]  Eiji Yanagida,et al.  Branching of double pulse solutions from single pulse solutions in nerve axon equations , 1987 .

[26]  Henryk Żoł̨dek,et al.  Bifurcations of certain family of planar vector fields tangent to axes , 1987 .