Nongeneric Bifurcations Near a Nontransversal Heterodimensional Cycle


Xingbo LIU,Xiaofei WANG,Ting WANG.Nongeneric Bifurcations Near a Nontransversal Heterodimensional Cycle[J].Chinese Annals of Mathematics B,2018,39(1):111~128
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Xingbo LIU; Xiaofei WANG;Ting WANG


This work was supported by the National Natural Science Foundation of China (No.11371140) and the Shanghai Key Laboratory of PMMP.
Abstract: In this paper bifurcations of heterodimensional cycles with highly degenerate conditions are studied in three dimensional vector fields, where a nontransversal intersection between the two-dimensional manifolds of the saddle equilibria occurs. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, the authors construct a Poincar\'{e} return map under the nongeneric conditions and further obtain the bifurcation equations. By means of the bifurcation equations, the authors show that different bifurcation surfaces exhibit variety and complexity of the bifurcation of degenerate heterodimensional cycles. Moreover, an example is given to show the existence of a nontransversal heterodimensional cycle with one orbit flip in three dimensional system.


Local moving frame, Nontransversal heterodimensional cycle, Orbit flip, Poincar'{e} return map


34C23, 34C37, 37C29
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