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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 (-λ±iω, γ) satisfying the Shil’nikov condition λ/γ<1 (λ, ω, γ>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:
34C37Homoclinic and heteroclinic solutions of ODE
34C23Bifurcation (ODE)
34C05Location of integral curves, singular points, limit cycles (ODE)