Uniformly Isochronous Quartic Systems

• Alwash Mohamad A M ^[1]
  1. [1] West Los Angeles College

     West Los Angeles College

     Estados Unidos

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00339-8
• Enlaces
  • Texto completo
• Resumen

  • We show that the order of a fine focus of a uniformly isochronous quartic system is at most 8. Moreover, the system has a center if and only if it satisfies the composition condition. In particular, we prove a recent conjecture about these systems.

• Referencias bibliográficas
  • 1. Algaba, A., Reyes, M.: Centers with degenerate infinity and their commuters. J. Math. Anal. Appl. 278, 109–124 (2003)
  • 2. Algaba, A., Reyes, M., Bravo, A.: Geometry of the uniformly isochronous centers with polynomial commutator. Differ. Equ. Dyn. Syst. 10,...
  • 3. Algaba, A., Reyes, M., Bravo, A.: Uniformly isochronous quintic planar vector fields. In: International Conference on Differential Equations,...
  • 4. Algaba, A., Reyes, M.: Computing center conditions for vector fields with constant speed. J. Comput. Appl. Math. 154, 143–159 (2003)
  • 5. Alwash, M.A.M.: Computing the Poincare–Liapunov constants. Differ. Equ. Dyn. Syst. 6, 349–361 (1998)
  • 6. Alwash, M.A.M.: On the center conditions of certain cubic systems. Proc. Am. Math. Soc. 126, 3335– 3336 (1998)
  • 7. Alwash, M.A.M., Lloyd, N.G.: Periodic solutions of a quartic non-autonomous equation. Nonlinear Anal. 11, 809–820 (1987)
  • 8. Chavarriga, J., Garcia, A, Giné, j: On integrability of differential equations defined by the sum of homogeneous vector fields with degenerate...
  • 9. Collins, C.B.: Conditions for a center in a simple class of cubic systems. Differ. Integral Equ. 10, 333–356 (1997)
  • 10. Conti, R.: Uniformly isochronous centers of polynomial systems in R2. Lect. Notes Pure Appl. Math. 152, 21–32 (1993)
  • 11. Giné, J., De Prada, P.: The non-degenerate center problem in certain families of planar differential systems. Int. J. Bifurcat. Chaos...
  • 12. Giné, J., Santallusia, X.: On the Poincare–Lyapunov constants and Poincare series. Appl. Math. 28, 17–30 (2001)
  • 13. Giné, J., Grau, M., Llibre, J.: Universal centers and composition conditions. Proc. Lond. Math. Soc. 3(106), 481–507 (2013)
  • 14. Ivanov, V.V., Volokitin, E.P.: Uniformly isochronous polynomial centers. Electron. J. Differ. Equ. 133, 1–10 (2005)
  • 15. Llibre, J., Rabanal, R.: Center conditions for a class of planar rigid polynomial differential systems. Discrete Contin. Dyn. Syst. 35,...
  • 16. Pakovich, F.: Weak and strong composition conditions for the Abel differential equation. Bull. Sci. Math. 138, 993–998 (2014)
  • 17. Volokitin, E.P.: Centering conditions for planar septic systems. Electron. J. Differ. Equ. 34, 1–7 (2002)
  • 18. Zhou, Z., Romanovski, V.G.: The center problem and composition condition for a family of a quartic differential systems, Electron. J....