In this research, we analyze the fractional generalized reaction Duffing model (FGRDM) within the framework of Atangana’s conformable derivative, providing a novel approach to modeling memory and hereditary characteristics in nonlinear dynamical systems. To determine exact analytical solutions, we utlize a hybrid methodology combining the Riccati sub-equation method and the Riccati–Bernoulli sub-ODE method. This integrated approach successfully yields a wide variety of soliton solutions, such as dark, bright, M-shape, combo, periodic, singular, and mixed hyperbolic wave structures. Beyond the analytical construction of solutions, we present a detailed qualitative analysis of the governed model, both in its perturbed and unperturbed forms, through bifurcation and chaos investigations. To explore the nonlinear behavior and chaotic dynamics, we employ a suite of diagnostic tools, such as Poincaré sections, return maps, Lyapunov exponents, time series analysis, power spectra, strange attractors, recurrence plots, and fractal dimension estimation. These tools help uncover the rich and sensitive dependency of the system on initial conditions and parameter variations, suggesting transitions between periodic, quasi-periodic, and chaotic regimes. The importance of this work lies in its comprehensive analytical and numerical analysis of the FGRDM using fractional calculus. It offers insights into the interplay between memory effects, nonlinearity, and chaos, thereby enhancing the comprehending of complex dynamical systems. Our research have potential applications in nonlinear science, particularly in fields where fractional-order models are used to explain physical phenomena with inherent damping and memory characteristics.
Rahman et al. (Thu,) studied this question.