Limit Cycles Bifurcating of Discontinuous and Continuous Piecewise Differential Systems of Isochronous Cubic Centers with Three Zones

• Autores: Maria Elisa Anacleto, Claudio Vidal
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 4, 2024
• Idioma: inglés
• Enlaces
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• Resumen

  • We study the maximum number of limit cycles that bifurcate from the periodic annulus of the isochronous cubic centre of discontinuous and continuous piecewise differential systems with three zones formed by the discontinuity set = {(x, y) ∈ R2 : (y = 0) ∨ (x = 0 ∧ y ≥ 0)}. More precisely, we consider the following perturbed systems x˙ = −y + x2 y + pi(x, y), y˙ = x + x y2 + qi(x, y), i = 1, 2, 3, where pi and qi are polynomials of degree m. Using the averaging theory of first order, we prove that for m = 1, 2, 3, at most 3, 9 and 15 limit cycles bifurcate from the periodic annulus of the isochronous cubic centre in the discontinuous case, respectively. While for the continuous case, it can appear 1, 5 and 6 limit cycles from the periodic orbits of these centres, respectively. Furthermore, we extend our study when pi and qi are homogeneous polynomials with 1 ≤ m ≤ 3, obtaining respectively for m = 1, 2, 3 at most one, seven and at least twelve limit cycles, which bifurcate from the periodic orbits of the isochronous cubic centre.

• Referencias bibliográficas
  • 1. Algaba, A., Reyes, M.: Computing center conditions for vector fields with constant angular speed. J. Comput. Appl. Math. 154, 143–159 (2003)
  • 2. Andronov, A., Vitt, A., Khaikin, S.: Theory of Oscillations. Pergamon Press, Oxford (1966)
  • 3. Belousov, B.P.: Periodically acting reaction and its mechanism. In: Collection of Abstracts on Radiation Medicine, Moscow, pp. 145–147...
  • 4. Buzzi, C., Pessoa, C., Torregrosa, J.: Piecewise linear perturbations of a linear center. Discrete Contin. Dyn. Syst. 9, 3915–3936 (2013)
  • 5. Chavarriga, J., Sabatini, M.: A survey of isochronous centers. Qual. Theory Dyn. Syst. 1, 1–70 (1999)
  • 6. Choudhury, A.G., Guha, P.: On commuting vector fields and Darboux functions for planar differential equations. Lobachevskii J. Math. 34,...
  • 7. Conti, R.: Uniformly isochronous centers of polynomial systems in R2. Lect. Notes Pure Appl. Math. 152, 21–31 (1994)
  • 8. Coll, B., Gasull, A., Prohens, R.: Bifurcation of limit cycles from two families of centers. Dyn. Contin. Discrete Impuls. Syst. Ser. A...
  • 9. Collins, C.B.: Conditions for a center in a simples class of cubic systems. Differ. Integral Equ. 10, 333–356 (1997)
  • 10. Dias, F.S., Mello, L.F.: The center-focus problem and small amplitude limit cycles in rigid systems. Discrete Contin. Dyn. Syst. 32, 1627–1637...
  • 11. Gasull, A., Prohens, R., Torregrosa, J.: Limit cycles for rigid cubic systems. J. Math. Anal. Appl. 303, 391–404 (2005)
  • 12. Filippov, A.F.: Differential equations with discontinuous right–hand sides, translated from Russian. Mathematics and its Applications...
  • 13. Fowles, G.R., Cassidy, G.L.: Analytical Mechanics. Saunders Collegs Publishing, Philadelphia (1993)
  • 14. Han, M.A., Romanovski, V.G., Zhang, X.: Equivalence of the Melnikov function method and the averaging method. Qual. Theory Dyn. Syst....
  • 15. Hilbert, D.: Mathematische Probleme, Lecture, Second Internat.Congr. Math. (Paris, 1900), Nachr. Ges. Wiss. Göttingen Math. Phys. KL.,...
  • 16. Huang, B., Niu, W.: Limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers. J. Appl....
  • 17. Itikawa, J., Llibre, J.: Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers. J. Comp. Appl....
  • 18. Itikawa, J., Llibre, J., Mereu, A.C., Oliveira, R.: Limit cycles in uniform isochronous centers of discontinuous differential systems...
  • 19. Karlin, S., Studden, W. J.: Tchebycheff systems: With applications in analysis and statistics. In: Pure and Applied Mathematics, vol....
  • 20. Liang, H., Llibre, J., Torregrosa, J.: Limit cycles coming from some uniform isochronous centers. Adv. Nonlinear Stud. 16(2), 197–220...
  • 21. Llibre, J., Mereu, A.C., Novaes, D.D.: Averaging theory for discontinuous piecewise differential systems. J. Differ. Equ. 258, 4007–4032...
  • 22. Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241, 1826–1844 (2012)
  • 23. Poincaré, H.: Sur l’intégration des équations différentielles du premier ordre et du premier degré I and II. Rend. Circ. Mat. Palermo...
  • 24. Poincaré, H.: Sur l’intégration des équations différentielles du premier ordre et du premier degré I and II. Rend. Circ. Mat. Palermo...
  • 25. Simpson, D.J.W.: Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol. 69. World Scientific,...
  • 26. Teixeira, M.A.: Perturbation theory for non-smooth systems. In: Robert, A.M. (ed.) Mathematics of Complexity and Dynamical Systems, vol....
  • 27. Van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701–710 (1920)
  • 28. Van der Pol, B.: On relaxation-oscillations. Lond. Edinb. Dublin Philos. Mag. J. Sci. 2(7), 978–992 (1926)
  • 29. Zhabotinsky, A.M.: Periodical oxidation of malonic acidin solution (a study of the Belousov reaction kinetics). Biofizika 9, 306–311 (1964)