On the Periodic Structure of the Rabinovitch-Fabrikant System

• Diab, Zouhair ^[2] ; Guirao, Juan L. G. ^[1] ; Vera, Juan A. ^[1]
  1. [1] Universidad Politécnica de Cartagena

     Universidad Politécnica de Cartagena

     Cartagena, España

     
  2. [2] Larbi Tebessi University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 2, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00474-w
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • 1. Bogoliubov, N.N., Krylov, N.: The Application of Methods of NonlinearMechanics in the Theory of Stationary Oscillations, Publ. 8 of the...
  • 2. Bogoliubov, N.N.: On Some Statistical Methods in Mathematical Physics. Kiev, Izv. vo Akad. Nauk Ukr. SSR (1945)
  • 3. Buic˘a, A., Llibre, J.: Averaging methods for finding periodic orbits via brouwer degree. Bull. Des. Sci. Math. 128, 7–22 (2004)
  • 4. Danca, M.F., Chen, G.: Bifurcation and chaos in a complex model of dissipative medium. Int. J. Bifurc. Chaos 14, 3409–3447 (2004)
  • 5. Danca, M.F., Feckan, M., Kuznetsov, N., Chen, G.: Looking more closely to the Rabinovich-Fabrikant system. Int. J. Bifurc Chaos 26, 1650038...
  • 6. Fatou, P.: Sur le mouvement d’un système soumis à des forces à courte période. Bull. Soc. Math, France 56, 98–139 (1928)
  • 7. Guckenheimer, J.: On a codimension two bifurcation. Lect. Notes Math. 898, 99–142 (1980)
  • 8. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations. Springer, Dynamical Systems and Bifurcations of Vector Fields (1983)
  • 9. Han, M.: Existence of periodic orbits and invariant tori in codimension two bifurcations of three dimensional systems. J. Sys. Sci Math....
  • 10. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer-Verlag, Berlin (2004)
  • 11. Liu, Y., Yang, Q., Pang, G.: A hyperchaotic system from the Rabinovich system. J. Comput. Appl. Math. 234, 101 (2010)
  • 12. Llibre, J.: Averaging theory and limit cycles for quadratic systems. Rad. Mat. 11(03), 215–228 (2002)
  • 13. Llibre, J., Zhang, X.: Hopf bifurcation in higher dimensional di erential systems via the averaging method. Pac. J. Math. 240, 321–341...
  • 14. Rabinovich, M.I., Fabrikant, A.L.: Stochastic self-modulation of waves in nonequilibrium media. J. Exp. Theor. Phys 77, 617 (1979)
  • 15. Sanders, J., Verhulst, F., Murdock, J.: Averaging method in nonlinear dynamical systems, vol. 59, 2nd edn. Applied Mathematical Sciences,...
  • 16. Zhang, C.X., Yu, S.-M., Zhang, Y.: Design and realization of multi-wing chaotic attractors via switching control. Int. J. Mod. Phys. B...