Nonlinear dynamics of the generalized unstable nonlinear Schrödinger equation: a graphical perspective

In this work, the generalized unstable nonlinear Schrödinger equation is examined, which is used to predict the temporal evolution of disturbances in marginally stable or unstable media. First, we construct Lie symmetries and then, using corresponding transformations, we reduce the governing equation to a couple of ordinary differential equations. These coupled equations are solved and establish the invariant solutions, some of which are presented through graphs. Second, the dynamical behavior of the studied model is examined from various perspectives, including bifurcation, quasi-periodic, chaotic motion, and sensitivity analysis. Bifurcation analysis of the planar dynamical system is investigated at the equilibrium points of the system using bifurcation theory. After that, an external periodic perturbation term is introduced in the dynamical system, which is called the perturbed dynamical system. The quasi-periodic and chaotic motions of the perturbed dynamical system are identified through different chaos detecting tools including 3D phase portrait visualization, Poincare map, time series analysis, multistability analysis, bifurcation diagram and Lyapunov exponents. Using these tools, we observe that a perturbed dynamical system deviates from the regular patterns and exhibits chaotic behavior. Further, the sensitivity analysis is examined at three different initial conditions, and it is noticed that the given model is highly sensitive, as it changes significantly with even small variations in the initial condition. The reported results are novel, fascinating, and theoretically useful for understanding temporal evolution of disturbances in marginally stable or unstable media. Overall, understanding the dynamical behavior of systems and processes is crucial for making predictions, designing interventions, and developing new technologies.