Article Contents
Article Contents

# Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities

• We classify the global phase portraits in the Poincaré disc of the differential systems $\dot{x}=-y+xf(x,y),$ $\dot{y}=x+yf(x,y)$, where $f(x,y)$ is a homogeneous polynomial of degree 3. These systems have a uniform isochronous center at the origin. This paper together with the results presented in [9] completes the classification of the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.
Mathematics Subject Classification: Primary: 34C05, 34C25.

 Citation:

•  [1] A. Algaba and M. Reyes, Characterizing isochronous points and computing isochronous sections, J. Math. Anal. Appl., 355 (2009), 564-576.doi: 10.1016/j.jmaa.2009.02.007. [2] J. Chavarriga and M. Sabatini, A survey of isochronous centers, Qualitative Theory of Dynamical Systems, 1 (1999), 1-70.doi: 10.1007/BF02969404. [3] A. G. Choudhury and P. Guha, On commuting vector fields and Darboux functions for planar differential equations, Lobachevskii Journal of Mathematics, 34 (2013), 212-226.doi: 10.1134/S1995080213030049. [4] R. Conti, Uniformly isochronous centers of polynomial systems in $\mathbbR^2$, Lecture Notes in Pure and Appl. Math., 152 (1994), 21-31. [5] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Spring-Verlag, 2006. [6] G. R. Fowles and G. L. Cassiday, Analytical Mechanics, Thomson Brooks/Cole, 2005. [7] E. A. González, Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc., 143 (1969), 201-222.doi: 10.1090/S0002-9947-1969-0252788-8. [8] M. Han and V. G. Romanovski, Isochronicity and normal forms of polynomial systems of ODEs, J. Symb. Comput., 47 (2012), 1163-1174.doi: 10.1016/j.jsc.2011.12.039. [9] J. Itikawa and J. Llibre, Phase portraits of uniform isochronous quartic centers, J. Comp. and Appl. Math., 287 (2015), 98-114.doi: 10.1016/j.cam.2015.02.046. [10] W. S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Diff. Eqs., 3 (1964), 21-36. [11] D. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81.doi: 10.1090/S0002-9939-1975-0356138-6.