Exploring new dynamical patterns: Bifurcation and chaos in time-fractional (3+1)-dimensional chiral nonlinear Schrödinger equation

In this paper, we use the modified Riemann-Liouville derivative to study the dynamical behaviors of the time-fractional (3+1)-dimensional chiral nonlinear Schrödinger equation. In many scientific domains, such as quantum mechanics, nonlinear optics, and hydrodynamics, the given equation is a fundamental idea in the theory of nonlinear wave processes. To obtain analytical solutions, we apply the generalized projective Riccati equation method, which enables the construction of various exact traveling wave solutions. These solutions range from bell-shaped solitons, one soliton, and dark solitons, and demonstrate the richness of the wave structures inherent within the fractional-order system. To obtain some understanding of the qualitative nature of the resulting solutions, we conduct a careful bifurcation analysis of the governing equation. By employing phase portraits and classification of equilibrium points, we study how modifications to system parameters affect the stability and dynamical transition of the nonlinear waves. We determine threshold values of our system at which it displays qualitative changes, i.e. periodic to quasi-periodic to chaotic transitions. We then proceed to the study of chaotic and quasi-periodic oscillations in fractional-order systems, demonstrating their sensitivity to initial conditions and external perturbations. Numerical simulations, such as time series analysis, Poincaré sections, and phase space orbits, capture the complexity and unpredictability of the system under the small perturbation. These results underscore the importance of fractional-order derivatives in regulating wave evolution and nonlinear dynamic behavior. Ultimately, by revealing new analytical solutions and providing a thorough dynamical analysis of stability, bifurcations, and chaotic behaviors, our findings advance our understanding of fractional nonlinear wave equations on a global scale. Wave dynamics are a significant concern in nonlinear optics, plasma physics, and related fields, where the knowledge gained from this work may have practical applications.