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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 ($-\lambda ±i\omega$, $\gamma$) 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 of ODE 34C23 Bifurcation (ODE) 34C05 Location of integral curves, singular points, limit cycles (ODE)