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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

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 & 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.  Google Scholar

[2]

J. Chavarriga and M. Sabatini, A survey of isochronous centers, Qualitative Theory of Dynamical Systems, 1 (1999), 1-70. doi: 10.1007/BF02969404.  Google Scholar

[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.  Google Scholar

[4]

R. Conti, Uniformly isochronous centers of polynomial systems in $\mathbbR^2$, Lecture Notes in Pure and Appl. Math., 152 (1994), 21-31.  Google Scholar

[5]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Spring-Verlag, 2006.  Google Scholar

[6]

G. R. Fowles and G. L. Cassiday, Analytical Mechanics, Thomson Brooks/Cole, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. Chavarriga and M. Sabatini, A survey of isochronous centers, Qualitative Theory of Dynamical Systems, 1 (1999), 1-70. doi: 10.1007/BF02969404.  Google Scholar

[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.  Google Scholar

[4]

R. Conti, Uniformly isochronous centers of polynomial systems in $\mathbbR^2$, Lecture Notes in Pure and Appl. Math., 152 (1994), 21-31.  Google Scholar

[5]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Spring-Verlag, 2006.  Google Scholar

[6]

G. R. Fowles and G. L. Cassiday, Analytical Mechanics, Thomson Brooks/Cole, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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