On the Integrability of Liénard systems with a strong saddle

Giné, Jaume; Llibre, Jaume

doi:https://doi.org/10.1016/j.aml.2017.03.004

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2017

Author

Giné, Jaume

Llibre, Jaume

Suggested citation

Giné, Jaume; Llibre, Jaume; . (2017) . On the Integrability of Liénard systems with a strong saddle. Applied Mathematics Letters, 2017, vol. 70, p. 39-45. https://doi.org/10.1016/j.aml.2017.03.004.

Abstract

We study the local analytic integrability for real Li\'{e}nard systems, $\dot x=y-F(x),$ $\dot y= x$, with $F(0)=0$ but $F'(0)\ne0,$ which implies that it has a strong saddle at the origin. First we prove that this problem is equivalent to study the local analytic integrability of the $[p:-q]$ resonant saddles. This result implies that the local analytic integrability of a strong saddle is a hard problem and only partial results can be obtained. Nevertheless this equivalence gives a new method to compute the so-called resonant saddle quantities transforming the $[p:-q]$ resonant saddle into a strong saddle.

URI

http://hdl.handle.net/10459.1/62956

DOI

https://doi.org/10.1016/j.aml.2017.03.004

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Applied Mathematics Letters, 2017, vol. 70, p. 39-45

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Except where otherwise noted, this item's license is described as cc-by-nc-nd (c) Elsevier, 2017