[Paper Review] An analytic technique for the solutions of nonlinear oscillators with damping using the Abel Equation

A. Ghose‐Choudhury, Partha Guha|arXiv (Cornell University)|Aug 8, 2016

Fractional Differential Equations Solutions3 citations

TL;DR

This paper presents an analytic method to solve a class of nonlinear damped oscillators by reducing the generalized Riccati equation to a first-order Abel equation and applying the Chiellini integrability condition. For the system $\ddot{x} + \alpha x^{2n+1}\dot{x} + x^{4n+3} = 0$ with $\alpha \geq 2\sqrt{2(n+1)}$, explicit closed-form solutions are derived, analytically confirming earlier numerical conjectures on periodic-to-appearance transition and isochrony in such systems.

ABSTRACT

Using the Chiellini condition for integrability we derive explicit solutions for a generalized system of Riccati equations $\ddot{x}+αx^{2n+1}\dot{x}+x^{4n+3}=0$ by reduction to the first-order Abel equation assuming the parameter $α\ge 2\sqrt{2(n+1)}$. The technique, which was proposed by Harko extit{et al}, involves use of an auxiliary system of first-order differential equations sharing a common solution with the Abel equation. In the process analytical proofs of some of the conjectures made earlier on the basis of numerical investigations in \cite{SJKB} is provided.

Motivation & Objective

• To provide analytical proofs for numerical conjectures on the transition from periodic to aperiodic motion in nonlinear damped oscillators.
• To extend the applicability of the Chiellini integrability condition to higher-order nonlinear systems, particularly generalized Riccati-type equations.
• To derive explicit closed-form solutions for a class of nonlinear oscillators with nonlinear damping and strong nonlinearity in the restoring force.
• To validate the conjecture of isochronous oscillations in the Liénard-type system $\ddot{x} + (2n+3)x^{2n+1}\dot{x} + x^{4n+3} + w_0^2x = 0$ using analytical methods.

Proposed method

• Reduction of the second-order nonlinear oscillator equation to a first-order Abel equation of the first kind via the substitution $\dot{x} = 1/v$.
• Transformation of the Abel equation into standard form $du/dx = A(x)u^2 + B(x)u^3$ using an integrating factor $\exp(-\int g_2(x)dx)$.
• Application of the Chiellini integrability condition $d/dx(B/A) = sA(x)$ with constant $s$ to ensure exact solvability by quadrature.
• Derivation of explicit solutions through integration of the reduced Abel equation under the condition $\alpha \geq 2\sqrt{2(n+1)}$.
• Use of auxiliary systems of first-order ODEs sharing common solutions with the Abel equation to facilitate integration.
• Generalization of the Chiellini condition to higher-order Liénard equations of the form $\ddot{x} + f(x)\dot{x}^{n+1} + g(x)\dot{x}^n = 0$ via variable separation and transformation techniques.

Experimental results

Research questions

• RQ1 What is the analytical mechanism behind the transition from periodic to aperiodic motion in the nonlinear oscillator $\ddot{x} + \alpha x^{2n+1}\dot{x} + x^{4n+3} = 0$?
• RQ2 Can the numerical conjecture of isochronous oscillations in the system $\ddot{x} + (2n+3)x^{2n+1}\dot{x} + x^{4n+3} + w_0^2x = 0$ be rigorously proven?
• RQ3 How does the Chiellini integrability condition extend to higher-order nonlinear damping terms in Liénard-type equations?
• RQ4 What is the role of the critical damping threshold $\alpha_c = 2\sqrt{2(n+1)}$ in determining the qualitative behavior of the system?

Key findings

• For $\alpha \geq 2\sqrt{2(n+1)}$, the system $\ddot{x} + \alpha x^{2n+1}\dot{x} + x^{4n+3} = 0$ admits explicit closed-form solutions via reduction to the Abel equation and application of the Chiellini condition.
• The critical value $\alpha_c = 2\sqrt{2(n+1)}$ marks the threshold where the discriminant of the cubic in the Abel equation vanishes, signaling a transition from periodic to aperiodic motion.
• When $\alpha = 2n+3$ and $w_0 \neq 0$, the solution is periodic and isochronous, as confirmed by satisfying Sabatini’s criterion for isochrony in Liénard systems.
• For $\alpha = 2n+3$ and $w_0 = 0$, the solutions are singular and blow up in finite time, with explicit forms $x \propto [(t_0 \mp t)^{-1}]^{1/(2n+1)}$.
• The solution for $\alpha = 2n+3$, $w_0 = 1$ can be expressed in terms of the hypergeometric function ${}_2F_1$, confirming its analytical tractability.
• The generalized Chiellini condition $\left(\frac{g}{f}\right)' = K \left(\frac{g^n}{f^{n-1}}\right)$ enables exact integration of higher-order Liénard equations with $\dot{x}^{n+1}$ and $\dot{x}^n$ damping terms.