Models for the Splitting of Separatrices

Abstract

In this note we show some simple examples to describe some aspects of the splitting of separatrices when we perturb a system with a high frequency periodic function of time. In this case the Melnikov method does not give reliable information because the Melnikov integral is exponentially small with respect to the frequency.

The first example is:

$$\begin{gathered} \dot x = f\left( x \right) \hfill \\ \dot y = yg\left( x \right) + h\left( x \right)sin\frac{t}{\varepsilon }, \varepsilon \ne 0 \hfill \\ \end{gathered} $$

(1)

which we consider as a perturbation of

$$\begin{gathered} \dot x = f\left( x \right) \hfill \\ \dot y = yg\left( x \right) \hfill \\ \end{gathered} $$

(2)

and we assume that f,g and h are analytic functions such that (2) has two non-degenerate hyperbolic points and an heteroclinic orbit connecting them.

We find that (1) has heteroclinic orbits and the measure of the region between the perturbed séparatrices is exponentially small with respect to ε.

Then we consider the system

$$\dot x = \frac{1}{2}\left( {1 - {x^2}} \right)$$

and we discuss the existence of heteroclinic orbits in the parameter space (β,ε). We get an infinitely flat curve β=β (ε) for which we have invariant manifolds with tangencies.

Finally, the system

$$\dot y = \left( {x + \alpha } \right) + \beta \left( {1 - {x^2}} \right) + sin\frac{t}{\varepsilon }$$

exhibits a similar behaviour.

It seems that the exponential smallness of the separatin of the perturbed separatrices only holds for conservative system and systems reducible to them, as it is (1) with a suitable x-depending time scaling.