Geometric analysis and onset of chaos for the resonant nonlinear Schrödinger system

In this paper, Jacobi stability of a resonant nonlinear Schrödinger (RNS) system is studied by the KCC theory, which is also called differential geometric method. The RNS system is transformed into an equivalent planar differential system by traveling wave transformation, then the Lyapunov stability of equilibrium points of the planar system is analyzed. By constructing geometric invariants, we analyze and discuss the Jacobi stability of three equilibrium points. The results show that the zero point is always Jacobi stable, while the Jacobi stability of the other nonzero equilibrium points are determined by the values of the parameters. In addition, the focusing tendency towards trajectories around the equilibrium points are studied by the dynamical behavior of deviation vector. Finally, numerical results show that the system presents quasi-periodic and chaotic phenomena under periodic disturbances.