Abstract
We report on the study of bifurcations of multi-circuit 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). We prove 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; (2) may have a tangency of an arbitrarily high order to subfamilies with a multicircuit homoclinic loop. These theorems show the 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.
Original language | English |
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Pages (from-to) | 409-423 |
Number of pages | 15 |
Journal | Nonlinearity |
Volume | 10 |
Issue number | 2 |
DOIs | |
State | Published - 1 Mar 1997 |
Externally published | Yes |
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy (all)
- Applied Mathematics