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The degenerate Bogdanov-Takens system $\dot x = y-(a_1x+a_2x^3),~\dot y = a_3x+a_4x^3$ has two normal forms, one of which is investigated in [Disc. Cont. Dyn. Syst. B (22)2017,1273-1293] and global behavior is analyzed for general parameters. To continue this work, in this paper we study the other normal form and perform all global phase portraits on the Poincaré disc. Since the parameters are not restricted to be sufficiently small, some classic bifurcation methods for small parameters, such as the Melnikov method, are no longer valid. We find necessary and sufficient conditions for existences of limit cycles and homoclinic loops respectively by constructing a distance function among orbits on the vertical isocline curve and further give the number of limit cycles for parameters in different regions. Finally we not only give the global bifurcation diagram, where global existences and monotonicities of the homoclinic bifurcation curve and the double limit cycle bifurcation curve are proved, but also classify all global phase portraits.

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.

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