Complex dynamics and chaos in the Rabinovich - Fabrikant model

Background and Objectives: In the work we consider a finitedimensional three-mode model of the nonlinear parabolic equation. It was proposed in 1979 by M. I. Rabinovich and A. L. Fabrikant. It describes the stochasticity arising from the modulation instability in a non-equilibrium dissipative medium with a spectrally narrow amplification increment. The Rabinovich - Fabrikant system presents some extremely rich dynamics die to the third-order nonlinearities presented in the equations. The considered system is universal. Despite the fact that it was developed as a physical model describing stochasticity in a non-equilibrium dissipative medium, it can simulate various physical systems, in which the three-mode interaction takes place and there is a cubic nonlinearity. Some of these systems have obvious applications, such as the Tollmien - Schlichting waves in hydrodynamic flows, wind waves on water, concentration waves during chemical reactions in a medium where diffusion occurs, Langmuir waves in plasma, etc. In addition, the Rabinovich - Fabrikant model can also simulate radio engineering systems that allow both analog simulation and implementation in a radio engineering device. Materials and Methods: The methodological apparatus of the study uses numerical methods for integrating differential equations, methods for calculating Lyapunov exponents, and the numerical bifurcation analysis using the MаtCont. Results: For the Rabinovich - Fabrikant system we present a diagram of dynamic regimes in the parameter plane, Lyapunov exponents depending on parameters, portraits of attractors and their basins of attractions. Additionally we plot numerically bifurcations lines in the parameter plane. They are plotted for stable points and period one limit cycles. It is shown that Rabinovich - Fabrikant models demonstrate transitions to chaos through the period-doubling bifurcation scenario for a limit cycle, which resulted from a direct Hopf bifurcation. The essential multistability also takes place in Rabinovich - Fabrikant models. In this case different types of attractors coexist. Conclusion: In the present work have numerically studied Rabinovich - Fabrikant models. We have shown that this model has a rich dynamics: transitions to chaos through the period-doubling bifurcation scenario; the fractal structure of attractors' basins; multistability. Depending on the parameters values, several combinations of coexisting attractors can be distinguished: a stable fixed point and a limit cycle; two limit cycles of different types; a limit cycle and a chaotic attractor; two chaotic attractors, etc. The last case is most interesting since chaotic systems with multiple attractors have received increasing attention in recent years because of their great impact on both theoretical analysis and engineering applications.