Bifurcation of an Orbit Homoclinic to a Hyperbolic Saddle of a Vector Field in R4

Abstract EN

We perform a bifurcation analysis of an orbit homoclinic to a hyperbolic saddle of a vector field in R4.

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 case 1+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.