Analysis of Degenerate Fold–Hopf Bifurcation in a Three-Dimensional Differential System

• Tigan, Gheorghe ^[1]
  1. [1] Polytechnic University of Timişoara

     Polytechnic University of Timişoara

     Rumanía

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 17, Nº 2, 2018, págs. 387-402
• Idioma: inglés
• DOI: 10.1007/s12346-017-0241-4
• Enlaces
  • Texto completo
• Resumen

  • In this work we study degenerate fold–Hopf bifurcation of codimension three in a differential system of the form x˙=-x-y-z,y˙=x+ay+bxy,z˙=bx-cz.We prove that the classical theory of the fold–Hopf bifurcation cannot be applied for studying the system’s behavior. However, using tools from averaging theory we prove the existence of periodic orbits generated by this bifurcation.

• Referencias bibliográficas
  • 1. Giné, J., Grau, M., Llibre, J.: Limit cycles bifurcating from planar polynomial quasi-homogeneous centers. J. Differ. Equ. 259, 7135–7160...
  • 2. Llibre, J., Mereu, A.C., Novaes, D.D.: Averaging theory for discontinuous piecewise differential systems. J. Differ. Equ. 258, 4007–4032...
  • 3. Llibre, J., Novaes, D.D., Teixeira, M.A.: On the birth of limit cycles for non-smooth. Bull. Sci. math. 139, 229–244 (2015)
  • 4. Candido, M.R., Llibre, J.: New results on averaging theory and applications. Z. Angew. Math. Phys. 67, 106 (2016)
  • 5. Francoise, J.-P.: Poincaré–Andronov–Hopf bifurcation and the local Hilbert’s 16th problem. Qual. Theory Dyn. Syst. 11(1), 61–77 (2012)
  • 6. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1995)
  • 7. Tigan, G.: Degenerate with respect to parameters fold–Hopf bifurcations. Discrete Contin Dyn Syst Ser A 37(4), 2115–2140 (2017)
  • 8. Cid-Montiel, L., Llibre, J., Stoica, C.: Zero–Hopf bifurcation in a hyperchaotic Lorenz system. Nonlinear Dyn. 75(3), 561–566 (2014)
  • 9. Buzzi, C., Llibre, J., Medrado, J.: Hopf and zero–Hopf bifurcations in the Hindmarsh–Rose system. Nonlinear Dyn. 83(3), 1549–1556 (2016)
  • 10. Llibre, J., Zhang, X.: On the Hopf-zero bifurcation of the Michelson system. Nonlinear Anal. Real World Appl. 12(3), 1650–1653 (2011)
  • 11. Efstathiou, K., Liu, X., Broer, H.W.: The boundary-Hopf–fold bifurcation in Filippov systems. SIAM J. Appl. Dyn. Syst. 14, 914–941 (2015)
  • 12. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. 57A(5), 397–398 (1976)
  • 13. Han,M., Romanovski, V.G., Zhang, X.: Equivalence of theMelnikov function method and the averaging method. Qual. Theory Dyn. Syst. 15(2),...
  • 14. Buica, A.: On the Equivalence of the Melnikov Functions Method and the Averaging Method, Qualitative Theory of Dynamical Systems. doi:10.1007/s12346-016-0216-x...
  • 15. Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, Universitext. Springer, New York (1991)