Abstract
In this paper, external bifurcations of heterodimensional cycles connecting three saddle points with one orbit flip, in the shape of “∞”, are studied in three-dimensional vector field. We construct a poincaré return map between returning points in a transverse section by establishing a locally active coordinate system in the tubular neighborhood of unperturbed double heterodimensional cycles, through which the bifurcation equations are obtained under different conditions. Near the double heterodimensional cycles, the authors prove the preservation of “∞”-shape double heterodimensional cycles and the existence of the second and third shape heterodimensional cycle and a large 1-heteroclinic cycle connecting with P1 and P3. The coexistence of a 1-fold large 1-heteroclinic cycle and the “∞”-shape double heterodimensional cycles and the coexistence conditions are also given in the parameter space.
Highlights
In recent years, bifurcation theory has been widely concerned due to its importance in practical applications and in the study of traveling wave solutions for nonlinear partial differential equations
Near the double heterodimensional cycles, the authors prove the preservation of “∞”-shape double heterodimensional cycles and the existence of the second and third shape heterodimensional cycle and a large 1-heteroclinic cycle connecting with p1 and p3
The coexistence of a 1-fold large 1-heteroclinic cycle and the “∞”-shape double heterodimensional cycles and the coexistence conditions are given in the parameter space
Summary
Bifurcation theory has been widely concerned due to its importance in practical applications (see [1] [2] [3] [4]) and in the study of traveling wave solutions for nonlinear partial differential equations. If all the equilibrium points of the orbit have the same dimension number of the stable manifold, the heteroclinic cycle is named as an equidimensional loop, otherwise, a heterodimensional. We consider the bifurcation problem of double heteroclinic loops of ∞-type connecting three saddle points with four orbits. It’s worth noting that, in the previous studies about homoclinic and heteroclinic loop bifurcations, few scholars focused on double heterodimensional cycles bifurcations of three saddle points. Jin and Zhu [18] considered the bifurcation problem of rough heteroclinic loop connecting three saddle points in a higher-dimensional system, but the loop is not a “∞”-type. Γ3 is transversal, that is, they can be preserved even under small perturbations
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