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| Nongeneric Bifurcations Near a Nontransversal Heterodimensional Cycle |
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Citation： |
Xingbo LIU,Xiaofei WANG,Ting WANG.Nongeneric Bifurcations Near a Nontransversal Heterodimensional Cycle[J].Chinese Annals of Mathematics B,2018,39(1):111~128 |
| Page view： 1875
Net amount： 1043 |
Authors： |
Xingbo LIU; Xiaofei WANG;Ting WANG |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (No.11371140) and the Shanghai Key Laboratory
of PMMP. |
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| 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. |
Keywords： |
Local moving frame, Nontransversal heterodimensional cycle, Orbit flip, Poincar'{e} return map |
Classification： |
34C23, 34C37, 37C29 |
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