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On the Takens-Bogdanov Bifurcation in the Chua's Equation

By Antonio Algaba, Emilio Freire, Estanislao Gamero, Alejandro J. Rodrígues-Luis and Ro J. Rodr


this paper, we have carry out the analysis of the Takens-Bogdanov bifurcation of the equilibrium at the origin in the Chua's equation with a cubic nonlinearity. Deriving the corresponding normal form, we put in evidence the presence of degenerate cases. Then, we obtain theoretically local and global bifurcations, that provide information about periodic behaviours and homoclinic and heteroclinic motions. The completion of the bifurcation set requires numerical methods. These allow us to detect the presence of a cusp of saddle-node bifurcation of periodic orbits, and also of a beak-tobeak singularity. Moreover, our analysis explains the presence of several codimension-two bifurcations detected numerically in Khibnik et al. [7]. Namely, a degenerate Hopf bifurcation of the origin, a degenerate homoclinic and a cusp of saddle-node of periodic orbits (see Fig. 10 of [7])

Year: 1999
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