Bifocal homoclinic orbits in four dimensions

Abshad. We study the bifurcations which (I C E U ~ 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 pertuibationsj the whoie sequence oi biiurcatians produces a singie periodic orbit, just as in the three-dimensional saddle-focus case. 1. Introduction Many features of chaotic behaviour in ordinary differential equations can be understood by the analysis of homoclinic bifurcations associated with homoclinic orbits which are bi-asymptotic (as f-fm) to a fixed point of the flow. Gaspard (1984) in extending Shii'nikov's pioneering work jiY65, iY7iJj on systems with a homoclinic orbit to a saddle-focus. The idea of such an analysis is that one approximares a Poincart map for a flow by approximating trajectories which are sufficiently close to the homoclinic orbit, and that one then studies this map to deduce facts about the dynamics of the flow. To be specific, consider the differential equation. i =f(x, p), x E Iw" (1.1) where p is a parameter. We assume that when p = 0, there is a homoclinic orbit r = (xo(t),-m < f < m, xo(0) # 0, xo(l)+ 0 as f + im} biasymptotic to a fixed point at x = 0. For p small we choose a 'box' B : 1x1 S c, where c << 1, and take one 'face' of it, S, as the Poincark surface. Suppose that r leaves B through a face S' at a point