Bifurcation of degenerate homoclinics
Copyright policy )Bifurcation of degenerate homoclinics
The continuation and bifurcation of homoclinic orbits near a given degenerate homoclinic orbit is analyzed. It is shown that the existence of such degenerate homoclinic orbits is a codimension three phenomenon and that generically the set of parameter values at which a homoclinic solution exists forms a codimension one surface which shows a singularity of Whitney umbrella type at the critical parameter value. The line of self-intersecting points of that surface corresponds to systems with two nearby homoclinics, which collide at the critical parameter value. The results are derived by investigating the singularities of the equation for the homoclinic orbits. The construction method for the bifurcation equations might be more important than the conclusion of the theorem. It should lead to some understanding of the very complicated dynamics close to the center manifold, involving coexistent homoclinics and periodic solutions, as it is mentioned in the paper.
- Ghent University Belgium
Bifurcation theory for ordinary differential equations, continuation and bifurcation of homoclinic orbits, degenerate homoclinic orbit, codimension three phenomenon, singularity of Whitney umbrella type, Theory of singularities and catastrophe theory, Homoclinic and heteroclinic solutions to ordinary differential equations, center manifold, Differentiable maps on manifolds, Local and nonlocal bifurcation theory for dynamical systems, Periodic solutions to ordinary differential equations
Bifurcation theory for ordinary differential equations, continuation and bifurcation of homoclinic orbits, degenerate homoclinic orbit, codimension three phenomenon, singularity of Whitney umbrella type, Theory of singularities and catastrophe theory, Homoclinic and heteroclinic solutions to ordinary differential equations, center manifold, Differentiable maps on manifolds, Local and nonlocal bifurcation theory for dynamical systems, Periodic solutions to ordinary differential equations
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