The first integrals of the original oscillator system under the specified parametric circumstances were obtained using inverse transformations, and some special cases of these equations were provided correspondingly. Feng 6 was interested in the first integrals of the Duffing–Van der Pol prototype that were taken into consideration under specific parametric conditions. It was demonstrated that a single-state adaptive feed-back was adequate to guide two identical oscillators to stable synchronization and was taken into consideration when considering the synchronization for known and unknown system parameters for the PHI6. Whether multi-parameter oscillators using PPHI6 might harmonize adaptively was investigated 5. The fractal basin boundary served as an easy way to illustrate the precision of the method. In cases where the PPHI6 was a bounded or unbounded double hump, the presence of homoclinic bifurcation was investigated 4. Numerical simulations that demonstrated the fractality of the attraction basins were used to supplement the findings. The multiple times scales technique produced harmonic, subharmonic, and super-harmonic oscillatory states. In a PPHI6, the dynamics of a periodically pushed particle were considered 3. A significant amount of recent research on chaotic matching originated from various fascinating real-world applications, including those in network security, chaos-generating designs, chemical reactions, lasers, biological systems, information science, neural networks, etc. The creation of the chaotic oscillator was an interesting phenomenon that has accumulated a lot of scientific attention in recent years. Phase diagrams revealed several steady-state types, demonstrating that variability was present for a variety of external force. Numerical simulations were used to show the non-autonomous oscillator potential (PPHI6) of periodic and chaotic motions. A dynamic of PHI6 prototype response to an external excitation was studied 2. The classification of the previous physical parameters and further details about these equations were investigated 1. Where \(\alpha ,\ \beta ,\) and \(\lambda\) are parameters of the potentials. Furthermore, the stable configuration of the analogous equation is shown in the absence of the stimulated force. 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. Additionally, the phase plane is more positively impacted by the initial amplitude, external force, damping, and natural frequency characteristics than the other parameters. It is found that as damping and natural frequency parameters increase, the solution approaches stability more quickly. Concerning the approximate solution, in the case of the presence/absence of time delay, the numerical approach shows excellent accuracy. 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. This novel approach seems to be impressive and promising and can be employed in various classes of nonlinear dynamical systems. Applying a numerical approach, the analytical solution is validated by this approach. This new methodology yields an equivalent linear differential equation to the exciting nonlinear one. A non-perturbative technique is employed to obtain some improvement and preparation for the system under examination. The emphasis in many examinations has shifted to time-delayed technology, yet the topic of this study is still quite significant. Therefore, the current paper examines the stability analysis of the dynamics of ϕ 6-Van der Pol oscillator (PHI6) exposed to exterior excitation in light of its motivated applications in science and engineering. A remarkable example of how to quantitatively explain the nonlinear performance of many phenomena in physics and engineering is the Van der Pol oscillator.
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