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.
Highlights
The purpose of this work is to numerically study of the generalized Rabinovich – Fabrikant model
The study is based on the numerical solution of the obtained analytically differential equations, and their numerical bifurcation analysis using the MаtCont program
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
Summary
Что модель Рабиновича–Фабриканта можно трактовать как результат применения метода медленных амплитуд к системе трех осцилляторов, описываемой функцией Лагранжа. Имея в виду наложенное резонансное условие 2ω0 ≈ ω1 + ω2, где ωi = √︀ki/mi, в предположении малой нелинейности и диссипации, можно полагать, что амплитуды мод за характерный интервал времени меняются незначительно, и применить к системе (6) метод медленных амплитуд. Далее следуя подходу работы [11], считаем, что амплитуда основной компоненты a0 существенно больше амплитуд сателлитов a1,2 и что комплексные амплитуды сателлитов одинаковы, так как из-за близости частот коэффициенты в амплитудных уравнениях для сателлитов 1 и 2 практически совпадают. Первая представлена параметрами γ и ν, которые идентичны параметрам модели Рабиновича–Фабриканта и имеют смысл коэффициентов диссипации — положительной для сателлитов и отрицательной для основной моды. Характеризующие нелинейное взаимодействие между осцилляторами в системе
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