Singular Perturbations, Transversality, and Sil'nikov Saddle-Focus Homoclinic Orbits |
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Authors: | Flaviano Battelli Kenneth J. Palmer |
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Affiliation: | 1. Dipartimento di Scienze Matematiche, Facoltà di Ingegneria, Università Politecnica delle Marche, Via Brecce Bianche 1, 60100, Ancona, Italy 2. Department of Mathematics, National Taiwan University, Taipei, 106, Taiwan
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Abstract: | We consider the singularly perturbed system $dot x$ =εf(x,y,ε,λ), $dot y$ =g(x,y,ε,λ). We assume that for small (ε,λ), (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 (ε,λ) 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. |
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