Bifurcations of double homoclinic flip orbits with resonant eigenvalues |
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| Authors: | Zhang Tian-si Zhu De-ming |
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| Affiliation: | College of Science, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China;Department of Mathematics, East China Normal University, Shanghai 200062, P. R. China |
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| Abstract: | Concerns double homoclinic loops with orbit flips and two resonant eigenvalues in a four-dimensional system. We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium, and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops, then compose them to get the important Poincaré map. A simple calculation gives explicitly an expression of the associated successor function. By a delicate analysis of the bifurcation equation, we obtain the condition that the original double homoclinic loops are kept, and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface, 2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions. We also locate the large periodic orbits and large homoclinic orbits and their number. Project supported by the National Natural Science Foundation of China (No. 10371040) |
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| Keywords: | double homoclinic orbits orbit flip periodic orbit resonance |
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