Parametric excitation and chaos in a nonlinear forced Mathieu system: A comprehensive analysis

The primary goal of this work is to examine the dynamic movement and chaos of a governed system by the nonlinear forced Mathieu equation (NFME). The multiple-time-scales approach (MTSA) is applied at the third-order level of approximation to provide the approximate solutions of this equation. Concurrently, the modulation equations (MEs) are created and the resonance case is investigated according to the solvability restrictions. Consequently, the solutions are verified at steady-state’s scenario. The criteria of Routh-Hurwitz (CRH) are applied to investigate and appraise the stability/instability zones in line with steady-state solutions. The amplitudes and phases throughout a designated period have been graphed to depict the motion at a particular instant. Furthermore, graphed representations of the acquired results, resonance responses, and stability regions are supplied to assess the impacts of different applied parameter values on the system's conduct. Three first-order equations are created from a non-autonomous second-order differential equation (DE) for solving computationally. Utilizing the Runge-Kutta algorithm, various motion types of the system are shown according to a bifurcation diagram, temporal history, Poincaré map, Lyapunov exponent (LE), and phase portrait. The outcomes also unequivocally show that the studied model possesses rich and sophisticated nonlinear dynamic phenomena, including periodic, period-doubling, period-quadrupling, and chaotic vibrations. Overall, the pendulum model described by the NFME has diverse applications ranging from mechanical engineering and structural dynamics to seismology, and robotics. It offers a mathematical foundation for comprehending the complex dynamics of pendulum-based systems under external forcing and nonlinear effects.