Abstract In the light of the potential applications in engineering, electronics, physics, chemistry, and biology, the current work applies several techniques to achieve analytic approximate and numerical solutions of the cubic–quintic Duffing–van der Pol equation. This equation represents a second-order ordinary differential equation with quintic nonlinearity and includes two external periodic forcing terms. A classical approximate solution involves the secular terms is obtained. Unfortunately, this traditional method does not enable us to ignore these secular terms. Additionally, along with the concept of the expanded frequency, a bounded approximate solution is achieved. The Homotopy perturbation method is utilized to obtain an approximate solution with an artificial frequency of the given system. Near the equilibrium points, in the case of the autonomous system, the linearized stability is accomplished. Furthermore, in the case of the non-autonomous system, by means of the multiple time scales, the stability analysis is effectuated, together with the resonance and the non-resonance cases. Numerical computations are performed to demonstrate, graphically, the perturbed solutions as well as the stability/instability regions. Various numerical solutions to initial–boundary value problems are deduced via a three-step finite difference scheme. These are plotted and discussed to show the chaotic nature of solutions.

中文翻译：

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具有两个外部周期强迫项的三次五次 Duffing-Van der Pol 方程的解析近似解：稳定性分析

摘要 鉴于在工程、电子、物理、化学和生物学中的潜在应用，当前的工作应用了多种技术来实现三次-五次 Duffing-van der Pol 方程的解析近似和数值解。该方程表示具有五次非线性的二阶常微分方程，包括两个外部周期强迫项。一个经典的近似解涉及到长期项的获得。不幸的是，这种传统方法无法让我们忽略这些世俗术语。此外，结合扩展频率的概念，实现了有界近似解。同伦微扰方法用于获得具有给定系统的人工频率的近似解。在平衡点附近，在自治系统的情况下，实现了线性化稳定性。此外，在非自治系统的情况下，通过多个时间尺度，与共振和非共振情况一起进行稳定性分析。执行数值计算以图形方式演示扰动解以及稳定性/不稳定性区域。通过三步有限差分格式推导出初边值问题的各种数值解。这些被绘制和讨论以显示解决方案的混乱性质。执行数值计算以图形方式演示扰动解以及稳定性/不稳定性区域。通过三步有限差分格式推导出初边值问题的各种数值解。这些被绘制和讨论以显示解决方案的混乱性质。执行数值计算以图形方式演示扰动解以及稳定性/不稳定性区域。通过三步有限差分格式推导出初边值问题的各种数值解。这些被绘制和讨论以显示解决方案的混乱性质。