GLOBAL BIFURCATIONS NEAR A DEGENERATE HETERODIMENSIONAL CYCLE

Fengjie Geng; Ting Wang; Xingbo Liu

doi:10.11948/2018.123

2018 Volume 8 Issue 1

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Fengjie Geng, Ting Wang, Xingbo Liu. GLOBAL BIFURCATIONS NEAR A DEGENERATE HETERODIMENSIONAL CYCLE[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 123-151. doi: 10.11948/2018.123

Citation:

Fengjie Geng, Ting Wang, Xingbo Liu. GLOBAL BIFURCATIONS NEAR A DEGENERATE HETERODIMENSIONAL CYCLE[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 123-151. doi: 10.11948/2018.123

GLOBAL BIFURCATIONS NEAR A DEGENERATE HETERODIMENSIONAL CYCLE

1 School of Science, China University of Geosciences(Beijing), 100083, Beijing, China;
2 School of Mathematics and Physics, Handan College, Handan 056005, China;
3 Department of Mathematics, East China Normal University, 200241, Shanghai, China

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Abstract

Abstract

This article is devoted to investigating the bifurcations of a heterodimensional cycle with orbit flip and inclination flip, which is a highly degenerate singular cycle. We show the persistence of the heterodimensional cycle and the existence of bifurcation surfaces for the homoclinic orbits or periodic orbits. It is worthy to mention that some new features produced by the degeneracies that the coexistence of heterodimensional cycles and multiple periodic orbits are presented as well, which is different from some known results in the literature. Moreover, an example is given to illustrate our results and clear up some doubts about the existence of the system which has a heterodimensional cycle with both orbit flip and inclination flip. Our strategy is based on moving frame, the fundamental solution matrix of linear variational system is chose to be an active local coordinate system along original heterodimensional cycle, which can clearly display the non-generic properties-"orbit flip" and "inclination flip" for some sufficiently large time.
- Heterodimensional cycle /
- inclination flip /
- orbit flip /
- moving frame
MSC: 34C23;34C37;37C29

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References

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