×

Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations. (English) Zbl 0840.34045

Consider a system of the form \(x'= f_0(x)+ \mu_1 f_1(x, \mu, t)+ \mu_2 f_2(x, \mu, t)\), where \(x\in \mathbb{R}^n\), \(\mu= (\mu_1, \mu_2)\in \mathbb{R}^2\). It is assumed that for \(\mu= 0\) there exists a homoclinic trajectory. The author applies the Lyapunov-Schmidt method to construct a bifurcation equation \(H= 0\). This equation determines the bifurcation diagram for homoclinic solutions.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34D10 Perturbations of ordinary differential equations
Full Text: DOI