Bautin bifurcation of a modified generalized Van der Pol-Mathieu equation. (Q2798084)

This work deals with the modified generalized Van der Pol-Mathieu equation NEWLINE\[NEWLINE\frac{d^2x}{dt^2} \, - \, \varepsilon \left( \alpha_0 + \beta_{01}x^2 + \beta_{02}x^4- \beta_{03} x^{2n} \right) \frac{dx}{dt} \, + \, \omega_0^2 \left( 1+\varepsilon h_0 \cos \gamma t\right) x \, = \, 0,NEWLINE\]NEWLINE where \(t\) is the real independent variable, \(x\) is a real dependent variable, \(n \in \mathbb{N}\), \(n>2\), \(\gamma = 2 \omega_0 + 2 d_0 \varepsilon\), \(\alpha_0, \beta_{01}, \beta_{02}, \beta_{03}, d_0 \in \mathbb{R}\), \(h_0>0\), \(\omega_0 >0\) and \(\varepsilon>0\).NEWLINENEWLINE The author first states several classical results on averaging theory, normal form theory, generalized Hopf bifurcation and Dulac criteria which allow to study the bifurcation of limit cycles and, in particular, the Bautin bifurcation. Then, the author considers the averaged system related to the non-autonomous differential system corresponding to the modified generalized Van der Pol-Mathieu equation. After renaming the parameters so as to get an autonomous system with only essential parameters and under some assumptions on them, the author proves that the obtained system exhibits Bautin bifurcation at the equilibrium \((0,0)\). In order to prove his result, he computes the first and second Lyapunov quantities associated to the system. Conditions on the parameters in order to have zero, one or two limit cycles are provided. The author also gives numerical computations and figures to illustrate the existence of the prescribed number of limit cycles.