Stability analysis of a time‑delayed Van der Pol-Helmholtz-Duffing oscillatorin fractal space with a non‑perturbative approach

The time-delayed fractal Van der Pol-Helmholtz-Duffing (VPHD) oscillator is the subject of this paper, which explores its mechanisms and highlights its stability analysis. While time-delayed technologies are currently garnering significant attention, the focus of this research remains crucially relevant. A non-perturbative approach is employed to refine and set the stage for the system under scrutiny. The innovative methodologies introduced yield an equivalent linear differential equation, mirroring the inherent nonlinearities of the system. Notably, the incorporation of quadratic nonlinearity into the frequency formula represents a cutting-edge advancement. The analytical solution's validity is corroborated using a numerical approach. Stability conditions are ascertained through the residual Galerkin method. Intriguingly, it is observed that the delay parameter, in the context of the fractal system, reverses its stabilizing influence, impacting both the amplitude of delayed velocity and position. The analytical solution's precision is underscored by its close alignment with numerical results. Furthermore, the study reveals that fractal characteristics emulate damping behaviors. Given its applicability across diverse nonlinear dynamical systems, this non-perturbative approach emerges as a promising avenue for future research.