Non-perturbative analysis of weakly nonlinear oscillators: Stability, chaos, and bifurcation investigations

The nonlinear oscillations frequently occur in real-world challenges, such as mechanical structures and biological cycles, and can give rise to complex behaviour like bifurcations and chaos that are vital in understanding and predicting dynamical systems. Advances in analytical and numerical methods have enabled novel applications in fields including engineering, medicine, and materials science. We analyse oscillators with pronounced nonlinear features, incorporating both damping and restoring forces, using a blend of theoretical and computational approaches. Two examples are presented from diverse scientific and technical areas. The innovative methodology described significantly reduces computation time and resources when compared to conventional perturbation methods widely used in this field. Based on He’s frequency formula, the proposed non-perturbative approach transforms weakly nonlinear oscillator of ordinary differential equation into linear one, allowing us to establish a new frequency corresponding to the linearized one. The theoretical outcomes are validated via numerical simulations using Mathematica Software, with results revealing excellent regularity between the two ordinary differential equations. An inclusive investigation of system stability can be conducted with this approach, expanding the capabilities beyond those available with previous techniques. Consequently, the non-perturbative approach offers a more practical and reliable framework via numerical solutions of weakly nonlinear oscillators. Moreover, it stands out as a flexible tool for use in applied research and engineering due to its flexibility to a variety of nonlinear situations. We also examine the impact of different parameters on stability, with results approving the approach’s simplicity, efficiency, and reliability. Adjusting bifurcation parameters alters the structure of bifurcation diagrams and their associated Poincaré maps, and we further illustrate these effects by mapping the Lyapunov exponent curves.