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arXiv · math/9811178

Interactions of Andronov-Hopf and Bogdamov-Takens bifurcations

William F. Langford·Kaijun Zhan
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Abstract

A codimension-three bifurcation, characterized by a pair of purely imaginary eigenvalues and a nonsemisimple double zero eigenvalue, arises in the study of a pair of weakly coupled nonlinear oscillators with Z_2 + Z_2 symmetry. The methodology is based on Arnold's ideas of versal deformations of matrices for the linear analysis, and Poincaré normal forms for the nonlinear analysis of the system. The stratified subvariety of primary bifurcations of codimensions one and two is identified in the parameter space. The analysis reveals different types of solutions in the state space, including equilibria, limit cycles, invariant tori and the possibility of homoclinic chaos. A mechanism is identified for energy transfer without strong resonance between two oscillation modes with widely separated frequencies.

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William F. Langford, Kaijun Zhan. 1998-11-08. Interactions of Andronov-Hopf and Bogdamov-Takens bifurcations. https://arxiv.org/abs/math/9811178

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