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Complexity in the bifurcation structure of homoclinic loops to a saddle-focus. (English) Zbl 0905.34042

Summary: The authors report on bifurcations of multicircuit homoclinic loops in two-parameter families of vector fields in the neighbourhood of a main homoclinic tangency to a saddle-focus with characteristic exponents (\(-\lambda\pm i\omega\), \(\gamma\)) satisfying the Shil’nikov condition \(\lambda/\gamma< 1\) (\(\lambda\), \(\omega\), \(\gamma>0\)). It is proved that one-parameter subfamilies of vector fields transverse to the main homoclinic tangency (1) may be tangent to subfamilies with a triple-circuit homoclinic loop; and (2) may have a tangency of an arbitrary high order to subfamilies with a multicircuit homoclinic loop. These theorems show high structural instability of one-parameter subfamilies of vector fields in the neighbourhood of a homoclinic tangency to a Shil’nikov-type saddle-focus. Implications for nonlinear partial differential equations modelling waves in spatially extended systems are briefly discussed.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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