Author(s)

Huimiao Dong, Tiansi Zhang^*

College of Science, University of Shanghai for Science and Technology, Shanghai, China.

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.

Double Heteroclinic Loops, Orbit Flip, Heteroclinic Bifurcation, Bifurcation Theory

Dong, H. and Zhang, T. (2021) External Bifurcations of Double Heterodimensional Cycles with One Orbit Flip. Applied Mathematics, 12, 348-369. doi: 10.4236/am.2021.124025.