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Mathematical and numerical analysis for some nonlinear second-order ordinary differential equations of Duffing type

Yusuke Kunimoto, Ikki Fukuda

TL;DR

The paper analyzes the large-time behavior of nonlinear second-order Duffing-type ODEs $x''(t)+\mu x'(t)+\alpha x(t)^p=0$ with odd $p\ge3$ and damping $\mu\ge0$. It combines energy methods to establish decay rates $|x(t)|\le C t^{-1/(p-1)}$ and $E(t)\le C t^{-2/(p-1)}$, and develops a structure-preserving numerical scheme that preserves a discrete energy and dissipation consistent with the continuous model. It proves global well-posedness for all admissible parameters, derives a differential- inequality for an augmented energy, and demonstrates numerically the predicted energy decay and damping effects across parameter regimes. The results provide rigorous insight into the asymptotic dynamics of Duffing-type systems and offer robust energy-preserving numerics for long-time simulations in nonlinear mechanics.

Abstract

In this paper, we consider the initial value problem for some nonlinear second-order ODEs of Duffing type. We study the large time behavior of the solutions to this problem, from both the perspectives of mathematical and numerical analysis. First, we derive the decay estimate of the solutions, by using the energy method. Moreover, we numerically investigate the large time behavior of the energy function related to this problem, by using a structure-preserving difference method.

Mathematical and numerical analysis for some nonlinear second-order ordinary differential equations of Duffing type

TL;DR

The paper analyzes the large-time behavior of nonlinear second-order Duffing-type ODEs with odd and damping . It combines energy methods to establish decay rates and , and develops a structure-preserving numerical scheme that preserves a discrete energy and dissipation consistent with the continuous model. It proves global well-posedness for all admissible parameters, derives a differential- inequality for an augmented energy, and demonstrates numerically the predicted energy decay and damping effects across parameter regimes. The results provide rigorous insight into the asymptotic dynamics of Duffing-type systems and offer robust energy-preserving numerics for long-time simulations in nonlinear mechanics.

Abstract

In this paper, we consider the initial value problem for some nonlinear second-order ODEs of Duffing type. We study the large time behavior of the solutions to this problem, from both the perspectives of mathematical and numerical analysis. First, we derive the decay estimate of the solutions, by using the energy method. Moreover, we numerically investigate the large time behavior of the energy function related to this problem, by using a structure-preserving difference method.

Paper Structure

This paper contains 3 sections, 3 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical Analysis
  3. Numerical Analysis

Key Result

Theorem 1

Let $p\ge3$ be an odd number, $\mu\ge0$ and $\alpha>0$. Then, for any initial data $(x_{0}, x_{1})\in \mathbb{R}^{2}$, there exists a unique global solution $x\in C^{2}([0, \infty))$ to Duffing. Moreover, if $\mu>0$, there exists a positive constant $C_{*}>0$ such that the solution $x(t)$ satisfies

Figures (1)

  • Figure 1: Numerical results for $E(t)$ by using a structure-preserving numerical schemes.

Theorems & Definitions (5)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof