Solutions of an extended Duffing-van der Pol equation with variable coefficients

TL;DR

This work derives exact solutions for an extended, unforced Duffing-van der Pol oscillator with time-dependent coefficients by intertwining two analytical frameworks: a factorization approach augmented by the Field Method and Painlevé analysis. Both routes impose the same coefficient constraints, transforming the DVDP equation into an integrable form related to Painlevé transcendents, specifically PXXIV, and yielding equivalent explicit solutions for x(t) along with the corresponding Lagrangian formulations via the Jacobi Last Multiplier. The study also presents three detailed examples to illustrate the method’s power, revealing diverse dynamical regimes including kink-type, bounded, and decaying behaviors, all supported by explicit expressions and Lagrangians. The results demonstrate a coherent bridge between algebraic factorization, Painlevé integrability, and variational structure, highlighting potential applications to nonlinear oscillators with variable coefficients in physics and engineering.

Abstract

In this work, exact solutions of the nonlinear cubic-quintic Duffing-van der Pol oscillator with variable coefficients are obtained. Two approaches have been applied, one based on the factorization method combined with the Field Method, and a second one relying on Painlevé analysis. Both procedures allow us to find the same exact solutions to the problem. The Lagrangian formalism for this system is also derived. Moreover, some examples for particular choices of the time-dependent coefficients, and their corresponding general and particular exact solutions are presented.

Figures (4)

• Figure 1: General solution $x_{+}(t)$ from (\ref{['ex3']}) for $A=1$, $C=1$, $k_2=1$, and different values of $k_1$ (upper figure). A similar behavior is obtained by changing the value of $k_2$ and fixing the remaining parameter values. General solution $x_{+}(t)$ from (\ref{['ex4']}) for $A=1$, $k_2=1$, and different values of $k_1$ as shown in the graphics (middle figure). Particular solution $x_{+}(t)$ in (\ref{['ex5']}) for $A=1$, $C=1$ and different values of $k_3$ (lower figure).
• Figure 2: General solution $x_{+}(t)$ from (\ref{['ex8']}) for $a_1=1$, $a_2=0$, $b_1=0.8$, $k_2=20$, and different values of $k_1$ (upper figure). General solution $x_{+}(t)$ from (\ref{['ex8']}) for $a_1=1$, $a_2=0$, $b_1=0.8$, $k_1=1$, and different values of $k_2$ (middle figure). Particular solution (\ref{['ex9']}) for $a_1=10$, $a_2=1$, $b_1=-1$ and different values of $k_3$.
• Figure 3: General solution $x_{+}(t)$ from (\ref{['eq23b']}) for different choices of the parameters: $f=1$, $k_2=1$, and different values of $k_1>0$ (upper left figure); $f=1$, $k_2=10$, and different values of $k_1<0$ (upper right figure); $f=-6$, $k_2=10$, varying $k_1<0$ (lower left figure); $f=-10$, $k_1=5$, and increasing values of $k_2>0$ (lower right figure).
• Figure 4: Particular solution (\ref{['eq23c']}) for $f=-1$ when increasing $k_3$ (upper figure), and for $k_3=1$ while varying $f$ (lower figure).