Generalized Rabinovich–Fabrikant system: equations and its dynamics

Abstract

The purpose of this work is to numerically study of the generalized Rabinovich–Fabrikant model. This model is obtained using the Lagrange formalism and describing the three-mode interaction in the presence of a general cubic nonlinearity. The model demonstrates very rich dynamics due to the presence of third-order nonlinearity in the equations. Methods. The study is based on the numerical solution of the obtained analytically differential equations, and their numerical bifurcation analysis using the MаtCont program. Results. For the generalized model we present a charts of dynamic regimes in the control parameter plane, Lyapunov exponents depending on parameters, portraits of attractors and their basins. On the plane of control parameters, bifurcation lines and points are numerically found. They are plotted for equilibrium point and period one limit cycle. It is shown that the dynamics of the generalized model depends on the signature of the characteristic expressions presented in the equations. A comparison with the dynamics of the Rabinovich– Fabrikant model is carried out. We indicated a region in the parameter plane in which there is a complete or partial coincidence of dynamics. Conclusion. The generalized model is new and describes the interaction of three modes, in the case when the cubic nonlinearity that determines their interaction is given in a general form. In addition, since the considered model is a certain natural extension of the well-known Rabinovich–Fabrikant model, then it is universal. And it can simulate systems of various physical nature (including radio engineering), in which there is a three-mode interaction and there is a general cubic nonlinearity.