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Zhang, W.P. (2007) Bifurcation of Double Homoclinic Loops in Four Dimensional Systems and Problems of Periodic Solutions in Population Dynamics. Ph.D. Thesis, East China Normal University, Shanghai.

has been cited by the following article:

• TITLE: External Bifurcations of Double Heterodimensional Cycles with One Orbit Flip

  AUTHORS: Huimiao Dong, Tiansi Zhang

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

  JOURNAL NAME: Applied Mathematics, Vol.12 No.4, April 30, 2021

  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.