Global phase portraits of uniform isochronous centers with quartichomogeneous polynomial nonlinearities

Jackson  Itikawa; Jaume  Llibre

doi:10.3934/dcdsb.2016.21.121

Article Contents

2016, Volume 21, Issue 1: 121-131. Doi: 10.3934/dcdsb.2016.21.121

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

Received: March 31, 2015

Revised: May 31, 2015

Published: December 2015

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Abstract

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.

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

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References

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