Moatimid Galal M, Amer T S
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt.
Department of Mathematics, Faculty of Science, Tanta University, Tanta, 31527, Egypt.
Sci Rep. 2023 Jul 24;13(1):11942. doi: 10.1038/s41598-023-38679-5.
A remarkable example of how to quantitatively explain the nonlinear performance of many phenomena in physics and engineering is the Van der Pol oscillator. Therefore, the current paper examines the stability analysis of the dynamics of ϕ-Van der Pol oscillator (PHI6) exposed to exterior excitation in light of its motivated applications in science and engineering. The emphasis in many examinations has shifted to time-delayed technology, yet the topic of this study is still quite significant. A non-perturbative technique is employed to obtain some improvement and preparation for the system under examination. This new methodology yields an equivalent linear differential equation to the exciting nonlinear one. Applying a numerical approach, the analytical solution is validated by this approach. This novel approach seems to be impressive and promising and can be employed in various classes of nonlinear dynamical systems. In various graphs, the time histories of the obtained results, their varied zones of stability, and their polar representations are shown for a range of natural frequencies and other influencing factor values. Concerning the approximate solution, in the case of the presence/absence of time delay, the numerical approach shows excellent accuracy. It is found that as damping and natural frequency parameters increase, the solution approaches stability more quickly. Additionally, the phase plane is more positively impacted by the initial amplitude, external force, damping, and natural frequency characteristics than the other parameters. To demonstrate how the initial amplitude, natural frequency, and cubic nonlinear factors directly affect the periodicity of the resulting solution, many polar forms of the corresponding equation have been displayed. Furthermore, the stable configuration of the analogous equation is shown in the absence of the stimulated force.
范德波尔振荡器是定量解释物理和工程中许多现象非线性性能的一个显著例子。因此，鉴于其在科学和工程中的应用前景，本文研究了受外部激励的ϕ -范德波尔振荡器（PHI6）动力学的稳定性分析。许多研究的重点已转向时延技术，但本研究的主题仍然相当重要。采用一种非微扰技术对所研究的系统进行改进和准备。这种新方法产生了一个与激励非线性方程等效的线性微分方程。应用数值方法，通过该方法验证了解析解。这种新颖的方法似乎令人印象深刻且很有前景，可用于各类非线性动力系统。在各种图表中，展示了一系列固有频率和其他影响因素值下所得结果的时间历程、其不同的稳定区域及其极坐标表示。关于近似解，在存在/不存在时间延迟的情况下，数值方法显示出极高的精度。结果发现，随着阻尼和固有频率参数的增加，解更快地趋近于稳定。此外，相平面受初始振幅、外力、阻尼和固有频率特性的正向影响比其他参数更大。为了说明初始振幅、固有频率和三次非线性因子如何直接影响所得解的周期性，展示了相应方程的许多极坐标形式。此外，还展示了在没有激励力时类似方程的稳定构型。
