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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.
34C37Homoclinic and heteroclinic solutions of ODE
34C23Bifurcation (ODE)
34C05Location of integral curves, singular points, limit cycles (ODE)