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

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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.
    Mathematics Subject Classification: Primary: 34C05, 34C25.

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  • [1]

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    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.

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    W. S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Diff. Eqs., 3 (1964), 21-36.

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