Global phase portrait of a degenerate Bogdanov-Takens system with symmetry
Hebai Chen Xingwu Chen Jianhua Xie
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1273-1293 doi: 10.3934/dcdsb.2017062

In this paper we study the global phase portrait of the normal form of a degenerate Bogdanov-Takens system with symmetry, i.e., a class of van der Pol-Duffing oscillators. This normal form is two-parametric and its parameters are considered in the whole parameter space, i.e., not viewed as a perturbation of some Hamiltonian system. We discuss the existence of limit cycles and prove its uniqueness if it exists. Moreover, by constructing a distance function we not only give the necessary and sufficient condition for the existence of heteroclinic loops connecting two saddles, but also prove its monotonicity and smoothness. Finally, we obtain a complete classification on the global phase portraits in the Poincaré disc as well as the complete global bifurcation diagram in the parameter space and find more plentiful phase portraits than the case that parameters are just sufficiently small.

keywords: Bogdanov-Takens system duffing equation global phase portrait heteroclinic loop limit cycle

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