Bifocal homoclinic orbits in four dimensions. (English) Zbl 0741.34017
From authors’ abstract: “We study the bifurcations which occur as we perturb four-dimensional systems of ordinary differential equations having homoclinic orbits that are bi-asymptotic to a fixed point with a double-focus structure. We give several methods of understanding the geometry of the invariant set that exists close to the homoclinic orbit and introduce a multi-valued one-dimensional map which can be used to predict the behaviour and bifurcation patterns which may occur. We argue that, although local strange behaviour is likely to occur, in a global sense (i.e for large enough perturbations) the whole sequence of bifurcations produces a single period orbit, just as in the three- dimensional saddle-focus case.”.
Reviewer: A.Klíč (Praha)
MSC:
34C23 | Bifurcation theory for ordinary differential equations |
34A34 | Nonlinear ordinary differential equations and systems |
37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
34A26 | Geometric methods in ordinary differential equations |