Saddle connections in planar systems.(English)Zbl 1090.34558

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
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