Abstract
In this work we proof analytically the existence and stability of four families of periodic orbits of the Rabinovitch-Fabrikant system that born from a Zero-Hopf Bifurcation.
Similar content being viewed by others
References
Bogoliubov, N.N., Krylov, N.: The Application of Methods of NonlinearMechanics in the Theory of Stationary Oscillations, Publ. 8 of the Ukrainian Acad. Sci: Kiev, (1934)
Bogoliubov, N.N.: On Some Statistical Methods in Mathematical Physics. Kiev, Izv. vo Akad. Nauk Ukr. SSR (1945)
Buică, A., Llibre, J.: Averaging methods for finding periodic orbits via brouwer degree. Bull. Des. Sci. Math. 128, 7–22 (2004)
Danca, M.F., Chen, G.: Bifurcation and chaos in a complex model of dissipative medium. Int. J. Bifurc. Chaos 14, 3409–3447 (2004)
Danca, M.F., Feckan, M., Kuznetsov, N., Chen, G.: Looking more closely to the Rabinovich-Fabrikant system. Int. J. Bifurc Chaos 26, 1650038 (2015)
Fatou, P.: Sur le mouvement d’un système soumis à des forces à courte période. Bull. Soc. Math, France 56, 98–139 (1928)
Guckenheimer, J.: On a codimension two bifurcation. Lect. Notes Math. 898, 99–142 (1980)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations. Springer, Dynamical Systems and Bifurcations of Vector Fields (1983)
Han, M.: Existence of periodic orbits and invariant tori in codimension two bifurcations of three dimensional systems. J. Sys. Sci Math. Sci. 18, 403–409 (1998)
Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer-Verlag, Berlin (2004)
Liu, Y., Yang, Q., Pang, G.: A hyperchaotic system from the Rabinovich system. J. Comput. Appl. Math. 234, 101 (2010)
Llibre, J.: Averaging theory and limit cycles for quadratic systems. Rad. Mat. 11(03), 215–228 (2002)
Llibre, J., Zhang, X.: Hopf bifurcation in higher dimensional di erential systems via the averaging method. Pac. J. Math. 240, 321–341 (2009)
Rabinovich, M.I., Fabrikant, A.L.: Stochastic self-modulation of waves in nonequilibrium media. J. Exp. Theor. Phys 77, 617 (1979)
Sanders, J., Verhulst, F., Murdock, J.: Averaging method in nonlinear dynamical systems, vol. 59, 2nd edn. Applied Mathematical Sciences, Springer, New York (2007)
Zhang, C.X., Yu, S.-M., Zhang, Y.: Design and realization of multi-wing chaotic attractors via switching control. Int. J. Mod. Phys. B 25, 2183 (2011)
Acknowledgements
This paper has been partially supported by Ministerio de Ciencia, Innovaci ón y Universidades, grant number PGC2018-097198-B-I00, and by Fundaci ón Séneca of Región de Murcia, grant number 20783/PI/18.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Diab, Z., Guirao, J.L.G. & Vera, J.A. On the Periodic Structure of the Rabinovitch-Fabrikant System. Qual. Theory Dyn. Syst. 20, 35 (2021). https://doi.org/10.1007/s12346-021-00474-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-021-00474-w