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Singular perturbations, transversality, and Sil’nikov saddle-focus homoclinic orbits. (English) Zbl 1040.34065

Summary: We consider the singularly perturbed system \(\dot x=\varepsilon f(x,y,\varepsilon,\lambda), \dot y=g(x,y,\varepsilon,\lambda)\). We assume that for small \((\varepsilon ,\lambda)\), (0,0) is a hyperbolic equilibrium on the normally hyperbolic centre manifold \(y=0\) and that \(y_0(t)\) is a homoclinic solution of \(\dot y=g(0,y,0,0)\). Under an additional condition, we show that there is a curve in the \((\varepsilon,\lambda)\) parameter space on which the perturbed system has a homoclinic orbit also. We investigate the transversality properties of this orbit and use our results to give examples of 4-dimensional systems with Sil’nikov saddle-focus homoclinic orbits.

MSC:

34E15 Singular perturbations for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
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