Bifurcation of degenerate homoclinic orbits in reversible and conservative systems
- 45 Downloads
We study the bifurcation of homoclinic orbits from a degenerate homoclinic orbitγ in autonomous systems which are either conservative or reversible. More precisely we consider degenerate homoclinic orbits where along the orbit the tangent spaces to the stable and unstable manifolds of the equilibrium have an intersection which is two-dimensional. In the reversible case we demand thatγ is symmetric and we distinguish betweenγ being elementary and nonelementary. We show that the existence of elementary degenerate homoclinic orbits is a codimension-two phenomenon, while nonelementary homoclinic orbits and degenerate homoclinic orbits in conservative systems appear generically in one-parameter families. We give a complete description of the set of nearby 1-homoclinic orbits.
Key wordsBifurcation degenerate homoclinic orbits reversible systems conservative systems
AMS classifications58F 34C
Unable to display preview. Download preview PDF.
- 1.Buffoni, B., Champneys, A. R., and Toland, J. F. (1996). Bifurcation and coalescence of plethora of homoclinic orbits for a Hamiltonian system.J. Dyn. Stab. Sys. 8, 221–281.Google Scholar
- 4.Lin, X.-B. (1990). Using Melnikovs method to solve Shilnikov's problems.Proc. Roy. Soc. Edinburgh 116A, 295–325.Google Scholar
- 5.Vanderbauwhede, A. (1992). Bifurcation of degenerate homoclinics.Results Math. 21, 211–223.Google Scholar