Homoclinic Bifurcation Properties Near Eight-figure Homoclinic Orbit

In this paper we study the homoclinic bifurcation properties near an eight-figure homoclinic orbit of a planar dynamical system. If at a saddle equilibrium the four branches of the stable and unstable manifolds perform two single homoclinic orbits simultaneously, then it is called an eight-figure homoclinic orbit. An example of such orbit is shown in Figure 3.The eight-figure homoclinic orbit is a co-dimension 2 bifurcation phenomenon . Such orbit can lead to rich dynamics and appears in many literature, see[l]-[10] and the references therein. The eight-figure homoclinic orbit also occurs in many applications.In the paper [8], Guckcnhcimer studied a mathematical model for stirred tank reactor. This model consists of two ordinary differential equations with a polynomial nonlincarity. With the aid of numerical simulation and heuristic arguments, besides the global bifurcation properties he proved the existence of an eight-figure homoclinic orbit and he also provided a corresponding local bifurcation diagram.In a small neural network consisting of two neurons [9], Giannakopoulos and Oster also found an eight-figure homoclinic orbit. The bifurcation properties of periodic orbits nearby were well studied by numerical computation in the paper [9].In the paper [10], Giannakopoulos, Kiipper and Zou studied the global bifurcation properties of a planar system of a valve generator, which consists of an electronic valve and an oscillatory circuit. An eight-figure homoclinic orbit was found and its local bifurcation properties were studied by numerical experiments.While studying the homoclinic bifurcation properties near an eight-figure homoclinic orbit, two different types of single homoclinic orbits are often involved. In order to distinguish these two types of homoclinic orbits and also for the convenience of our statement in this paper, we name these orbits according to the relative position of all the branches of the stable and unstable manifolds. Usually, there are four branches of stable and unstable manifolds at a saddle equilibrium, and one branch of stable manifolds and one branch of unstable manifolds coincide with each other to perform a single homoclinic orbit. If the other two branches of invariant manifolds lie outside of the region created by the homoclinic orbit we call it a small homoclinic orbit, see the left picture of Figure 4. Otherwise, if the other two branches of unstable manifolds are included in the interior of the homoclinic orbit we call it a big one, see the right picture in Figure 4.It is easy to verify that there will bifurcate two families of small homoclinic orbits from the eight-figure homoclinic orbit under small perturbation. It has been observed from numerical experiments [8] that the big homoclinic orbits also emanate from the eight-figure homoclinic orbit. But. to the author's knowledge there is no proof on the existence of the big homoclinic orbit near an eight-figure homoclinic orbit in the literature.In this paper, we give a complete proof on the existence of small and big homoclinic orbits emanated from the eight-figure homoclinic orbit under small perturbation. To illustrate our results, we numerically investigate the local homoclinic bifurcation diagram near the eight-figure homoclinic of a chemical model studied by Guckenheimer in [8]. Then we compare our numerical bifurcation diagram with that in [8].