ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Kuznetsov S. P., Turukina L. V. Generalized Rabinovich–Fabrikant system: equations and its dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics, 2022, vol. 30, iss. 1, pp. 7-29. DOI: 10.18500/0869-6632-2022-30-1-7-29

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Generalized Rabinovich–Fabrikant system: equations and its dynamics

Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Turukina L. V., Saratov State University

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

Research was carried out under support of the Russian Science Foundation (project no. 21-12-00121),
  1. Kuznetsov SP. Dynamical Chaos. Moscow: Fizmatlit; 2006. 356 p. (in Russian).
  2. Ott E. Chaos in Dynamical Systems. Cambridge: Cambridge University Press; 1993. 385 p.
  3. Guckenheimer J, Holmes PJ. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag; 1983. 462 p. DOI: 10.1007/978-1-4612-1140-2.
  4. Anishchenko VS, Vadivasova TE, Astahov VV. Nonlinear Dynamics of Chaotic and Stochastic Systems. Saratov: Saratov University Publishing; 1999. 367 p. (in Russian). 
  5. Schuster HG, Just W. Deterministic Chaos: An Introduction. Weinheim: Wiley; 2005. 287 p. DOI: 10.1002/3527604804.
  6. Kuznetsov SP. Dynamical Chaos and Hyperbolic Attractors: From Mathematics to Physics. Moscow-Izhevsk: Institute for Computer Research; 2013. 488 p. (In Russian).
  7. Neimark JI, Landa PS. Stochastic and Chaotic Oscillations. Dordrecht: Springer; 1992. 500 p. DOI: 10.1007/978-94-011-2596-3.
  8. Lorenz EN. The Essence of Chaos. Seattle, WA, USA: University of Washington Press; 1995. 240 p.
  9. Alligood KT, Sauer T, Yorke J. Chaos: An Introduction to Dynamical Systems. New York: Springer-Verlag; 1996. 603 p. DOI: 10.1007/b97589.
  10. Hilborn RC. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford: Oxford University Press; 2001. 672 p.
  11. Rabinovich MI, Fabrikant AL. Stochastic self-modulation of waves in nonequilibrium media. Sov. Phys. JETP. 1979;77(2):617–629 (in Russian).
  12. Danca MF, Feckan M, Kuznetsov N, Chen G. Looking more closely to the Rabinovich–Fabrikant system. International Journal of Bifurcation and Chaos. 2016;26(2):1650038. DOI: 10.1142/S0218127416500383.
  13. Liu Y, Yang Q, Pang G. A hyperchaotic system from the Rabinovich system. Journal of Computational and Applied Mathematics. 2010;234(1):101–113. DOI: 10.1016/
  14. Agrawal SK, Srivastava M, Das S. Synchronization between fractional-order Rabinovich– Fabrikant and Lotka–Volterra systems. Nonlinear Dynamics. 2012;69(4):2277–2288. DOI: 10.1007/s11071-012-0426-y.
  15. Srivastava M, Agrawal SK, Vishal K, Das S. Chaos control of fractional order Rabinovich– Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich–Fabrikant system. Applied Mathematical Modelling. 2014;38(13):3361–3372. DOI: 10.1016/j.apm.2013.11.054.
  16. Danca MF. Hidden transient chaotic attractors of Rabinovich–Fabrikant system. Nonlinear Dynamics. 2016;86(2):1263–1270. DOI: 10.1007/s11071-016-2962-3.
  17. Danca MF, Kuznetsov N, Chen G. Unusual dynamics and hidden attractors of the Rabinovich– Fabrikant system. Nonlinear Dynamics. 2017;88(1):791–805. DOI: 10.1007/s11071-016-3276-1.
  18. Danca MF, Chen G. Bifurcation and chaos in a complex model of dissipative medium. International Journal of Bifurcation and Chaos. 2004;14(10):3409–3447. DOI: 10.1142/S0218127404011430.
  19. Luo X, Small M, Danca MF, Chen G. On a dynamical system with multiple chaotic attractors. International Journal of Bifurcation and Chaos. 2007;17(9):3235–3251. DOI: 10.1142/S0218127407018993.
  20. Kuznetsov AP, Kuznetsov SP, Turukina LV. Complex Dynamics and Chaos in the Rabinovich– Fabrikant Model. Izvestiya of Saratov University. Physics. 2019;19(1):4–18 (in Russian). DOI: 10.18500/1817-3020-2019-19-1-4-18.
  21. Hocking LM, Stewartson K. On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. R. Soc. Lond. A. 1972;326(1566):289–313. DOI: 10.1098/rspa.1972.0010.
  22. Andronov AA, Fabrikant AL. Landau damping, wind waves and whistle. In: Nonlinear Waves. Moscow: Nauka; 1979. P. 68–104 (in Russian).
  23. Kuramoto Y, Yamada T. Turbulent state in chemical reactions. Progress of Theoretical Physics. 1976;56(2):679–681. DOI: 10.1143/PTP.56.679.