Dynamics of the Rabinovich–Fabrikant system and its generalized model in the case of negative values of parameters that have the meaning of dissipation coefficients

Abstract

Purpose of this work is a numerical study of the Rabinovich–Fabrikant system and its generalized model, which describe the occurrence of chaos during the parametric interaction of three modes in a nonequilibrium medium with cubic nonlinearity, in the case when the parameters that have the meaning of dissipation coefficients take negative values. These models demonstrate a rich dynamics that differs in many respects from what was observed for them, but in the case of positive values of the parameters. Methods. The study is based on the numerical solution of the differential equations, and their numerical bifurcation analysis using the MаtCont program. Results. For investigated models we present a charts of dynamic regimes in the control parameters plane, Lyapunov exponents depending on the parameters, attractors and their basins. On the parameters plane, which have the meaning of dissipation coefficients, bifurcation lines and points are numerically found. They are plotted for equilibrium point and period one limit cycle. For both models we compared dynamics observed in the case when the parameters that have the meaning of dissipation coefficients take negative values, with the one observed in the case when these parameters take positive values. And it is shown that in the first case parameter space has a simpler structure. Conclusion. The Rabinovich– Fabrikant system and its generalized model were studied in detail in the case when the parameters which have the meaning of dissipation coefficients take negative values. It is shown that there are a number of differences in comparison with the case of positive values of these parameters. For example, a new type of chaotic attractor appears, multistability that is not related to the symmetry of the system disappears, etc. The obtained results are new, since the Rabinovich–Fabrikant system and its generalized model were studied in detail for the first time in the region of negative values of parameters which have the meaning of dissipation coefficients.