Abstract
We perform a bifurcation analysis of an orbit homoclinic to a hyperbolic saddle of a vector field inR4. We give an expression of the gap between returning points in a transverse section by renormalizing system, through which we find the existence of homoclinic-doubling bifurcation in the case1+α>β>ν. Meanwhile, after reparametrizing the parameter, a periodic-doubling bifurcation appears and may be close to a saddle-node bifurcation, if the parameter is varied. These scenarios correspond to the occurrence of chaos. Based on our analysis, bifurcation diagrams of these bifurcations are depicted.
Highlights
Introduction and ProblemsHomoclinic orbits are crucial to know dynamics of differential systems in many application fields
It has an orbit homoclinic to the equilibrium (u, V, w) = 0 which corresponds to a solitary wave (u, w)(x, t) = (u, w)(ζ) of system (1)
In this paper we focus on the homoclinic-doubling problem for a kind of homoclinic flips
Summary
Homoclinic orbits are crucial to know dynamics of differential systems in many application fields. It has an orbit homoclinic to the equilibrium (u, V, w) = 0 which corresponds to a solitary wave (u, w)(x, t) = (u, w)(ζ) of system (1). For b > 2, a > 0, and shows the existence of the N-homoclinic orbit in some circumstances on two sides of the flip bifurcation. The homoclinic-doubling bifurcation, which switches a 2n−1-homoclinic orbit to a 2n-homoclinic orbit, exists extensively in systems with flips; see [3,4,5,6] and the references therein. Suppose that the system (5) has an orbit γ(t) of codimension-1 homoclinic to a saddle equilibrium p at ξ = 0. Notice that if the gap ‖γ−(T, ξ)−γ+(T, ξ)‖ = 0, it means that the homoclinic orbit is kept (see Figure 1(a)) but it may not be of codimension-1. We try to quantitate the gap size
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