Sufficient conditions for the existence of periodic solutions of the extended Duffing-Van der Pol oscillator. (English) Zbl 1348.34078
The authors consider the following first-order differential system associated to the extended Duffing-Van der Pol oscillator
\[
\begin{aligned}\dot{x}&=y,\\ \dot{y}&=-x+\rho y-\alpha x^{3}-\rho x^{2}y-\lambda x^{5}+\delta \mathrm{ cost}. \end{aligned} (1)
\]
Under some sufficient conditions on the parameters \(\rho\), \(\alpha\), \(\lambda\) and \(\delta\), they prove that the system (1) possesses one or three periodic solutions and they study the stability of some of them. For the proofs, they use an analytical approach and apply the averaging theory and some algebraic techniques.
Reviewer: Mohsen Timoumi (Monastir)
MSC:
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 37C60 | Nonautonomous smooth dynamical systems |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34C29 | Averaging method for ordinary differential equations |
Keywords:
extended Duffing-Van der Pol oscillator; periodic solutions; non-autonomous systems; averaging theoryReferences:
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