Chaotic and quasi-periodic dynamics in fractional-order nonlinear wave systems within dispersive-dissipative media.

This investigation undertakes a detailed exploration of the nonlinear time-fractional Bogoyavlenskii-Kadomtsev-Petviashvili ([Formula: see text]) equation, emphasizing its behavioral characteristics and applications in fluid dynamics, plasma physics, and wave propagation phenomena. The [Formula: see text] framework generalizes conventional nonlinear evolution equations, providing a more nuanced representation of wave dynamics in dispersive and dissipative media. Through the complementary application of the Khater III method and an enhanced Kudryashov technique, we derive closed-form solutions and rigorously validate them via numerical implementation of He's variational iteration approach. Our analysis uncovers intricate solution behaviors, including nonlinear wave interactions and resonance dynamics within fractional-order temporal frameworks. The results substantiate the [Formula: see text] model's ability to characterize physical systems governed by fractional time evolution, thereby connecting classical wave theory with contemporary fractional calculus formulations. The integration of analytical and computational methodologies produces high-precision solutions that faithfully reproduce the system's intrinsic physical attributes. This work advances the theoretical underpinnings of fractional differential equations and their utility in modeling non-integer order wave phenomena. Additionally, we conduct an exhaustive examination of the [Formula: see text] system, clarifying its evolutionary dynamics through bifurcation analysis, characterization of chaotic/quasi-periodic regimes, and assessment of parameter sensitivity. The primary objective is to illuminate the governing mechanisms of the system's temporal evolution using advanced mathematical tools rooted in nonlinear dynamical theory.