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

Gonchenko, S. V.; Turaev, D. V.; Gaspard, P.; Nicolis, G.

Nonlinearity 10, No. 2, 409-423 (1997).

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

Keywords:

bifurcations; multicircuit homoclinic loops; vector fields; saddle-focus; Shil’nikov condition