Local bifurcation behavior, chaos, and oblique travelling wave structures in the (2 + 1)-dimensional chiral Schrödinger equation with beta derivative

Qualitative analysis in mathematical modeling plays a pivotal role in understanding the dynamical behavior of the complex systems, making it a cornerstone of research in nonlinear sciences. The bifurcation method is a crucial tool to investigate dynamical properties, enabling researchers to explore and characterize structural changes in orbits within nonlinear dynamical systems. By applying the bifurcation method to the (2 + 1)-dimensional chiral nonlinear Schrödinger (NLS) equation with beta-derivative, this study investigates phase portrait bifurcations, chaotic dynamics, and examines sensitivity and multi-stability properties of the system. Initially, phase portrait bifurcations are analyzed graphically by determining the equilibrium points and assessing their stability using the Jacobian matrix. The study demonstrates how the equilibrium points evolve as different angles are varied, revealing the system's dynamic nature under changing conditions. Hereafter, by introducing a perturbation term to the planar system, chaotic patterns are identified through 2D and 3D phase portraits, time series analysis, Lyapunov exponents, and Poincaré maps. The system's sensitivity and multi-stability are further explored using the Runge-Kutta method, showing strong dependence on initial conditions. Meanwhile, the planar dynamical system approach applied to the beta-derivative chiral NLS model yields various soliton solutions, including periodic, bright, and dark solitons. All generated solutions are shown to be novel in terms of their fractionality, wave propagation behavior, and the influence of free parameters, as well as the application of the methods. Therefore, this study provides deeper insight into the dynamics of the β-derivative chiral NLS model and emphasizes its broader significance in understanding nonlinear phenomena.