The existence of transverse homoclinic solutions for higher order equations (Q1815649)

Parametrized differential equations of the form \(\dot x(t)=f(x(t),\mu,t)\), where \(x\in \mathbb{R}^n\), \(\mu\in \mathbb{R}^N\), are considered. It is assumed that \(f\) is of \(C^3\)-class, \(f(x,0,t)\) is independent of \(t\), for all sufficiently small \(|\mu|\), \(x=0\) is a hyperbolic equilibrium, \(f\) is periodic in \(t\), and there exists a homoclinic solution (with the limits at the origin) when \(\mu=0\). The main result of the paper establishes a sufficient condition on the existence of a transverse homoclinic point (hence also chaos) for the equation \(\dot x=f(x,s^2\mu_0,t)\), where \(0\neq|s|\) is small enough. Results of the paper of \textit{K. Palmer} [J. Differ. Equations 55, 225-256 (1984; Zbl 0539.58028)] are used in the proof.