Saddle connections in planar systems. (English) Zbl 1090.34558

Lichardová, Hana

Arch. Math., Brno 36, Suppl., 507-512 (2000).

Consider a system of the form \[ \dot x=f(x)+\varepsilon g(x,\alpha ), \quad x\in \mathbb R^2, \quad \varepsilon , \alpha \in \mathbb R, \] where \(f\), \(g\) are \(C^r\), \(r\geq 2\) and bounded on bounded sets, \(\varepsilon \) being a small parameter. Supposing that the unperturbed system with \(\varepsilon =0\) possesses a saddle connection (i.e.a trajectory connecting two saddles), the author studies a question whether there are values of a parameter \(\alpha \) for which the perturbed system possesses a saddle connection. It is shown that under a convenient assumption on the existence of a suitable \(\alpha _0\) there exists \(\alpha (\varepsilon )=\alpha _0+O(\varepsilon )\) for each \(\varepsilon \) sufficiently small such that the perturbed system \[ \dot x=f(x)+\varepsilon g(x,\alpha (\varepsilon )) \] possesses a saddle connection, which is \(C^r\)-close to the saddle connection of the unperturbed system. The result is illustrated by an example of a planar Hamiltonian system (planar pendulum equation).

Reviewer: Josef Kalas (Brno)

MSC:

34C37

Homoclinic and heteroclinic solutions to ordinary differential equations

Keywords:

saddle connection; stable and unstable manifolds; small perturbation; Hamiltonian system