A bifurcation problem to homoclinic orbits for non-autonomous systems
Given the nonlinear equation \(x'=F(x)\), \(x\in R^ 3\), with \(x=0\) as a hyperbolic fixed point and a degenerated node-saddle type homoclinic orbit \(\Gamma\), the behaviour in a neighbourhood of \(\Gamma\), of the solutions of the equation \(x'=F(x)+G(\lambda,\mu,t,x)\) is studied when G is a small non-autonomous perturbation that is periodic with respect to t and depends on two parameters. The authors prove under certain generic conditions the existence of two bifurcation curves to homoclinic orbits for a system in \(R^ 3\) (not necessarily Hamiltonian). These curves divide the space of parameters into two regions S,N such that there are homoclinic orbits in S whereas there are not any in N. In a later paper the authors prove that these bifurcation curves can be obtained as the limit of bifurcation curves to subharmonics.
degenerated node-saddle type homoclinic orbit
scientific article; zbMATH DE number 3901218