Elsevier

Chaos, Solitons & Fractals

Volume 39, Issue 1, 15 January 2009, Pages 295-303
Chaos, Solitons & Fractals

Codimension 2 bifurcations of double homoclinic loops

https://doi.org/10.1016/j.chaos.2007.01.101Get rights and content

Abstract

In this work, the double-homoclinic-loop bifurcations in four dimensional vector fields are investigated by setting up local coordinates near the double homoclinic loops. We get the existence, uniqueness and incoexistence of the large 1-hom and large 1-per orbit, and their corresponding existence regions are located. Furthermore, the inexistence of the large 2-hom and large 2-per orbit are also demonstrated.

Section snippets

Setting of the problem

In recent years, a great number of papers have been devoted to the problem of the homoclinic or heteroclinic loop bifurcations (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and the further references therein). But few concerned the bifurcation of double homoclinic loops except [8]. In [8] the authors gave the number of limit cycles near double homoclinic loops under perturbations in planar Hamiltonian systems. In this

Preliminary results and bifurcation equations

Suppose that U is small enough, we can introduce successively three transformations such that system (1.1) has the following normal form in U:x˙=[λ1(μ)+o(1)]x+O(u)[O(y)+O(v)],y˙=[-ρ1(μ)+o(1)]y+O(v)[O(x)+O(u)],u˙=[λ2(μ)+o(1)]u+O(x)[O(y)+O(v))],v˙=[-ρ2(μ)+o(1)]v+O(y)[O(x)+O(u)].In fact, the first Cr transformation is used to straighten Ws and Wu, the second Cr-1 coordinate change is used to straighten the strong stable manifold Wss and the strong unstable manifold Wuu of O, and the last one is Cr-

Large homoclinic loop bifurcations

In this section, we study the existence, uniqueness and incoexistence of the double homoclinic, large 1-hom and large 1-per orbit. Firstly, we have the following result concerned with the uniqueness and the incoexistence.

Theorem 3.1

Suppose that (H1)-(H4) and ρ1λ1 hold. Then, for |μ| small enough, system (1.1) has at most one large loop consisting of two homoclinic orbits, one large 1-homoclinic orbit or one large 1-per orbit in a small neighborhood of Γ, and these orbits can not coexist.

Proof

Change time t to −

Inexistence of large 2-hom and large 2-per orbit

In this section, we study the inexistence of large 2-hom and large 2-per orbits under the condition ρ1λ1. Like before, we set τ3 and τ4 the times from q22S20 to q13S11 and from q14S10 to q23S21, respectively, with P10(q22)=q13,P20(q14)=q23, and q24=q20,s3=e-λ1τ3 and s4=e-λ1τ4. Then, we can easily see that there is an one to one correspondence between these large 2-hom or 2-per orbits and the solutions Y=(s1,s2,s3,s4,u11,u21,u13,u23,v10,v20,v14,v22) ofGi1=0,Gi3=0,Gi4=0,i=1,,4,with si0, i=1

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Supported by National Natural Science Foundation of PR China (Nos. 10371040, 10671069).

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