A homoclinic hierarchy

Homoclinic bifurcations in autonomous ordinaty differential equations provide useful organizing centres for the analysis of examples. There are four generic types of homoclinic bifurcation, depending on the dominant eigenvalues of the Jacobian matrix of the flow near a stationary point. A family of differential equations is presented which, for suitable choices of parameters, can exhibit each of these four homoclinic bifurcations. In one of the cases this provides the first smooth example of the bifurcation in the literature. AMS classification: 58F13; 58314 KQWOF~S: Bifurcation; Homoclinic orbit; Chaos A hom~linic orbit of an autonomous ordinary differential equation is a nontrivial solution, x,(r), which tends to a stationary point, x0, in both forward and backward time, i.e. x,(t) 4 x0 as t -+ fm and ~~(0) # x0. In typical (e.g. non-Hamiltonian) families of ordinary differentid equations the existence of a homoclinic orbit is not a s~c~lly stable situation, and typical perturbations of the system will no longer have a homoclinic orbit close to the original one. Thus, in a one-parameter family of ordinary differential equations, there may be a parameter, or, = pu say, at which the system has a homoclinic orbitandan e>Osuchthatif ~E(~u-e,~n+ E)\{ pn} there is no homoclinic orbit close to the orbit which exists at p = pn. If this is the case we say that there is a homoclinic bifurcation at p = pu. The study of homoclinic bifurcations goes back (at least) to Poincare, and later Andronov. More recent work has been stimulated by a series of papers by Shilnikov [l-3] in which it was shown that, given certain con~tions described below, there is chaotic motion in a tubular neighbourhood of the homoclinic orbit, although the net effect of the bifurcation is to create a single periodic orbit (see, e.g., Ref. [4] for a discussion). Complicated sequences of local bifurcations at parameter values near pn may also occur [5,4] as well as more complicated homoclinic bifurcations, for which the homoclinic orbit loops several times through the tubular neighbourhood of the original homoclinic orbit [6]. ’ Address from January 19961 Department of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London El 4NS. UK. These theoretical results can be a great help when investigating examples. The conditions which determine whether complicated dynamics occurs are based 0375-9601/96/$12.00 8 19% Elsevier Science B.V. All rights reserved SsDrO375-9601(95)00953-l 156 P. Gle~i~ing, &. Luing/ Physics Letters A 211 (1996) 1.55-160 on the linearized flow near the stationary point. Suppose that the stationary point is hyperbolic. Then, after a change of coordinates we may assume that it is at the origin for all values of p which are of interest and the family of differential equations can be written in the form Now, since x = 0 is hyperbolic, the eigenvalues of A can be divided into two sets, {A,], i = 1,. . . , n,, and{vJ, i= l,..., ns, n, + ffU = n, such that Re(A,) f=Ax+F(x, /.L) > 0 and Re(vi) < 0. These can be ordered so that (1) for x E lQ”, n > 2. Here F(0, p) = 0, A is a constant n X n matrix and F is smooth and contains only nonlinear terms. Assume that if ,u = 0 then the system has a homoclinic orbit, x,(r), biasymptotic to the origin, and that if p E ( E, e)\(O) there are no homocli~ic orbits close to xn (by close we mean that for 77 sufficiently small I x(t) x,(t) I < q for all t E (cQ, Ml). Re ( v,,,) G . . . GRe(Y,)GRe(v,) <O <Re(h,) <RetA,) < . . . GRe(A,“}.