The focus of this paper is to examine the motion of a novel double pendulum (DP) system with two degrees of freedom (DOF). This system operates under specific constraints to follow a Lissajous curve, with its pivot point moving along this path in a plane. The nonlinear differential equations governing this system are derived using Lagrange's equations. Their analytical solutions (AS) are subsequently calculated using the multiple-scales method (MSM), which provides higher-order approximations. These solutions are considered new, as the traditional MSM has been applied to this novel system for the first time. Additionally, the accuracy of these solutions is validated through numerical results obtained using the fourth-order Runge–Kutta method. The solvability conditions and characteristic exponents are determined based on resonance cases. The Routh–Hurwitz criteria (RHC) are employed to assess the stability of the fixed points corresponding to the steady-state solutions. They are also used to demonstrate the frequency response curves. The nonlinear stability analysis is performed by examining the stability and instability ranges. Resonance curves and time history plots are presented to analyze the behavior of the system for specific parameter values. The investigation delves into a comprehensive analysis of bifurcation diagrams (BDs) and Lyapunov exponent spectra (LEs), aiming to uncover the various types of motion present within the system. Systematic examination of these charts reveals critical insights into transitions between stable, quasi-stable, and chaotic dynamical behaviors. This work has practical applications in various fields, such as robotics, pump compressors, rotor dynamics, and transportation devices. It can be used to study the vibrational motion of these systems.

Purpose: Under certain conditions, the governing equation of motion of magnetic spherical pendulum results in a cubic-quintic Duffing equation. The current work aims to achieve an analytical bounded procedure of this equation. Methods: This may be accomplished by grouping nonlinear expanded frequency, Homotopy perturbation method (HPM), and Laplace transforms. Therefore, this technique helps disregard the appearance of the source of secular terms. Results: To validate the obtained explanation, based on the method of Runge–Kutta of the fourth order (RK4), the numerical calculation is performed. On the other hand, the linearized stability analysis is carried out to explore stability neighbouring the fixed points. Moreover, the time history of the attained solution and the corresponding phase plane plots are obtained to expose the influence of the affecting factors in the behavior of motion. Conclusions: A comparison between both solutions gives a good matching between them, which explores the worthy accuracy of the approach in question. Several phase portraits are planned toward illustrating the different types of stability and instability near the equilibrium points, where the relation between the expanded and the cyclotron frequency (that are generated by the magnetic field) is characterized for diverse standards of the azimuthal angular velocity.

Purpose: A group of classical oscillators of high nonlinearity, which cannot be completely analyzed, is addressed by introducing a novel technique. The main objective of the current investigation is to utilize the generalized He’s frequency formula (HFF) in studying the analytical explanations of specific types of extremely nonlinear oscillators. This interest arises from the growing fascination in the realm of nonlinear oscillators. Regarding several engineering and scientific fields, together with three particular situations, a generic example is presented. Methods: Compared to prior perturbation approaches utilized in this field, the new strategy is straightforward and requires less processing and timing. This ground-breaking tactic, which converts the nonlinear ordinary differential equation (ODE) into a linear one, is referred to as the non-perturbative approach (NPA), as an innovative approach. A new frequency that is comparable to a linear ODE, like in a simple harmonic situation, is produced in the procedure. When evaluating the physiologically significant specialized instances, the outcome from this straightforward approach not only exhibits a strong agreement with the numerical findings but also demonstrates that it is more accurate than the outcomes from other well-known approximate methodologies. An extensive description of the NPA is presented to ensure the maximum benefits. Results: The theoretical findings are confirmed by conducting a numerical analysis with the aid of Mathematica Software (MS). The numerical solution (NS) and the theoretical responses demonstrated remarkable congruity. Conventional perturbation techniques typically use Taylor expansion to enlarge restoring forces, thereby reducing problem complexity. However, this weakness disappears with the NPA. Additionally, stability analysis of the problem alongside the NPA becomes feasible, unlike with prior conventional methodologies. Conclusion: The NPA emerges as a more responsible resource when examining the NS for oscillators with significant nonlinearity. Its exceptional versatility in addressing various nonlinear problems underscores the NPA as a valuable benefit in the fields of engineering and applied science.

A new numerical scheme based on the tau spectral method for solving the linear hyperbolic telegraph type equation is presented and implemented. The derivation of this scheme is based on utilizing certain modified shifted Chebyshev polynomials of the sixth-kind as basis functions. For this purpose, some new formulas concerned with the modified shifted Chebyshev polynomials of the sixth-kind have been stated and proved, and after that, they serve to study the hyperbolic telegraph type equation with our proposed scheme. One advantage of using this scheme is that it reduces the problem into a system of algebraic equations that can be simplified using the Kronecker algebra analysis. The convergence and error estimate of the proposed technique are analyzed in detail. In the end, some numerical tests are presented to demonstrate the efficiency and high accuracy of the proposed scheme.

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.

The present article examines the nonlinear stability of two viscoelastic electrified cylindrical fluids immersed in permeable media. The current structure consists of two endless vertical cylinders containing two electrified fluids. An axial unchanged electric field (EF) is applied to the entire construction; additionally, the impact of the surface tension is reflected. The main driving force for understanding this challenge has increasing significance in atmospheric and oceanic dynamics. The viscous potential theory is employed to ease the mathematical processes. The fundamental hydrodynamic equations are combined with Maxwell's equations in the quasi-static approximation to set the boundary-value problem. The appropriate boundary conditions (BCs) are expressed in a nonlinear form; this nonlinearity is achieved by addressing the linearized controlling equations of the motion. The viscoelastic impacts are considered to illustrate how the BCs produce their contributions. Consequently, the equations of motion are tackled without the effects of viscoelasticity parameters. The interface displacement consequently interacts vertically along with the cylindrical axis. The Rayleigh Helmholtz–Duffing oscillator describes the propagation of the interface between the two fluids. The non-perturbative approach (NPA), based on the He's frequency formula, transforms the typical nonlinear differential equation (NDE) into a linear one. The non-dimensional analysis reveals a lot of dimensionless physical numerals. These non-dimensional physical characteristics can be utilized to study the fundamental character of the liquid movement. They are also used to reduce the quantity of variables that are needed to comprehend the framework. A quick explanation of NPA is also presented. The stability study reveals the real/complex coefficients of the NDE. The numerical simulations show that there is a consistent solution and that the increases in the axial EF, as well as axial wavenumber, stabilize the system. The obtained findings help to understand and explain diverse nonlinear progressions that have taken place in fluid mechanics. To show the impact of the different factors and the efficiency of the stability approach, diverse PolarPlot diagrams are graphed for both actual and hypothetical portions.