Purpose: The objective of the present study is to analyze a damped Mathieu–cubic quintic Duffing oscillator as a parametric nonlinear oscillatory dynamical system. This equation has multiple applications in diverse fields, including optics, quantum physics, and general relativity. There are multiple concerns related to periodic motion and the analysis of boundary-value problems with elliptic symmetries. The current effort aims to determine the frequency amplitude of parametric nonlinear issues. Method: The non-perturbative approach (NPA) is employed to transform the nonlinear ordinary differential equation (ODE) into a linear equation. The derivation of the approximate solutions is achieved without relying on typical perturbation approaches, separate from the series expansion. Hence, the objective of this study is to depart from traditional perturbation methods and acquire approximated solutions for minor amplitude parametric components without imposing any limitations. Furthermore, the technique is extended to ascertain optimal solutions for the nonlinear large amplitude of fluctuation. Results: The current approach allows for rapid estimation of the frequency-amplitude relationship in order to attain successive approximations of the solutions for parametric nonlinear fluctuations. A validation is obtained for the derived parametric equation, demonstrating a high level of agreement with the original equation. An analysis of stability behavior is conducted in multiple scenarios. In addition, the Floquet theory is used to examine the transition curves. Conclusion: The current technique is characterized by its clear principles, making it practical, user-friendly, and capable of achieving exceptionally high numerical precision. The current approach is highly beneficial for addressing nonlinear parametric problems due to its ability to minimize algebraic complexity during implementation.

Purpose: The aim of the current study is to comparehend an analytical description of different classes of nonlinear Mathieu oscillators. The van der Pol-Duffing-Mathieu oscillator, and the generalized van der Pol-Duffing-Mathieu oscillator are examined. Additionally, the hybrid Rayleigh-van der Pol-Duffing-Mathieu oscillators, as well as the nonlinear Mathieu oscillator, are scrutinized. Method: The non-perturbative approach (NPA) is utilized to convert the nonlinear ordinary differential equations (ODEs), of the illustrated dynamical systems, into linear ones. The approximate solutions are derived independently in the series expansion and without the use of conventional perturbation techniques. Therefore, the goal is to deviate from conventional perturbation techniques and get approximations of small amplitude parametric components without imposing any restrictions. The method is also expanded to determine the best solutions for the nonlinear immense amplitude of fluctuation. Results: The current method offers successive approximations of the solutions of parametric nonlinear fluctuations may be obtained by quickly estimating the frequency-amplitude relationship. The resulting parametric equations are validated, showing high degree of agreement with the original equation. Stability behavior is analyzed under various circumstances. The transition curves, bifurcation diagram, Poincaré map, and phase portrait are also examined using the Floquet theory. Conclusion: The stability regions are found to be diminishing with the rise of the natural frequency, and the excited frequency. Moreover, the achieved regions are found to be growing with the rise of the damping coefficient and the excitation amplitude. The stability settings have been examined by considering the effects of various factors in both the damped and un-damped phases for each situation. In the un-damped state, PolarPlots are examined of the transition curves of the two corresponding solutions, namely Cos- and Sin-oscillations. The conclusions of the acquired results suggested that the approach presented here is highly efficient, robust, founded on solid premises, and remarkably intuitive.

The nonlinear stability examination of two revolving magnetized liquid columns connecting two completely submerged fluids in a permeable region is the aim of the existing paper. Two endless vertical cylinders occupied with two magnetic fluids make up the present structure. Significantly, the disturbance at the border displays an azimuthal behavior. The entire structure is activated by an azimuthal unchanging magnetic field (MF). The increasing interest in the atmospheric and oceanic dynamics is the primary motivation in exploring this problem. To relax the complication of the mathematical processes, the viscous potential theory (VPT) is established. The motion is assessed using three basic coexistent field formulations: Maxwell's formula, Brinkman's formula, and the continuity condition, in the construction of the Coriolis force and centrifugal implications. The explanations of the linearized formula of motion produce a nonlinear categorizing diffusion structure because of the implications of the nonlinear boundary conditions (BCs). The non-perturbative approach (NPA) based on the He's frequency formulation (HFF) is employed to transform the nonlinear characteristic ordinary differential equation (ODE) into a linear one. A short description of the NPA is also presented. The nonlinear ODE with real and imaginary coefficients is exposed by the stability analysis. The stability requirements are implemented using only a nonlinear analysis. As demonstrated, as an unusual state, it is exposed that ignoring the Weber number removes all complex items of the nonlinear formulation. Physically, this means the absence of the angular velocities from the physical model. For both the real and complex situations of the original equation, the stability remains unchanged. It is found that the azimuthal MF, rotating parameter, and Darcy’s numeral have a maintenance impact. On the other hand, the azimuthal wave numeral has a destabilizing one. Several polar designs are drawn to agreement the stability situations.

The present study is concerned with an incompressible nano-liquid movement obeying the non-Newtonian Sisko model (NSM). The flow occupies a cylindrical arterial tube with mild stenosis through permeable media, in accordance with the modified Darcy's law. The motile gyrotactic microorganisms are involved in the flow. The originality of the current work emerges from the need to realize the effects of fluctuating viruses and microbes in addition to nanoparticles through arterial stenosis. The buoyancy-driven flow (BA) is considered, where density is a function of heat and particles' concentration. Furthermore, the nanoparticles dispersal is illustrated along with an activated energy chemical reaction. The mathematical construction is implemented by a group of nonlinear partial differential equations (NPDEs) concerning momentum, energy, nanoparticles' volume fraction, and microorganism concentration. Appropriate time-varying tapered stenosis BCs are utilized to comprehend the present boundary-value problem (BVP). The RK-4 analysis, in conjunction with the shooting approach, is performed on the prior system. Subsequently, a significant graphical construction of the assessed data is designed to understand the controls of the generated physical features. Biological infections by harmful microbes represent reasons for blood thinner diseases. According to the current study, the presence of microorganisms improves the flow and increases its speed. Therefore, the rapidity of the flow can be controlled by regulating the characterized parameters of the microbe's divergence. Moreover, most of the heat-related parameters in this discussion are observed as raising features of heat fluctuation, which is a vital characteristic in diagnosing narrowing vessels therapy.