Purpose

The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these solutions.

Design/methodology/approach

The double-hump Duffing equation is presented as a Hamiltonian system and a phase portrait of this system has been found. On the ground of analytical calculations performed using Hamiltonian-based technique, the periodic solutions of this system are represented by Jacobi elliptic functions sn, cn and dn.

Findings

Expressions for the periodic solutions and their periods of the double-hump Duffing equation have been found. An expression for the solution, in the time domain, corresponding to the heteroclinic trajectory has also been found. An important element in various applications is the relationship obtained between constant Hamiltonian levels and the elliptic modulus of the elliptic functions.

Originality/value

The results obtained in this paper represent a generalization and improvement of the existing ones. They can find various applications, such as analysis of limit cycles in perturbed Duffing equation, analysis of damped and forced Duffing equation, analysis of nonlinear resonance and analysis of coupled Duffing equations.

中文翻译：

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Duffing 方程的解析解

目的

本文的目的是找到双峰杜芬方程周期解的精确解析表达式以及这些解的周期表达式。

设计/方法/方法

双峰杜芬方程表示为哈密顿系统，并已找到该系统的相图。在使用基于哈密顿量的技术进行分析计算的基础上，该系统的周期解由雅可比椭圆函数 sn、cn 和 dn 表示。

发现

已经找到了双峰杜芬方程的周期解及其周期的表达式。还找到了对应于异宿轨迹的时域解的表达式。各种应用中的一个重要元素是在恒定哈密顿量级与椭圆函数的椭圆模量之间获得的关系。

原创性/价值

本文得到的结果代表了对现有结果的概括和改进。他们可以找到各种应用，例如微扰 Duffing 方程中的极限环分析、阻尼和受迫 Duffing 方程分析、非线性共振分析和耦合 Duffing 方程分析。