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

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January 2016, 21(1): 121-131. doi: 10.3934/dcdsb.2016.21.121

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

Jackson Itikawa ^1, and Jaume Llibre ^2,

1.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, Spain
2.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia

Received April 2015 Revised June 2015 Published November 2015

Abstract
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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.

Keywords: Poincaré disk, uniform isochronous center, Polynomial vector field, quartic polynomial differential system., phase portrait.

Mathematics Subject Classification: Primary: 34C05, 34C2.

Citation: Jackson Itikawa, Jaume Llibre. Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 121-131. doi: 10.3934/dcdsb.2016.21.121

References:

[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.

show all references

References:

[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.

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