The Liénard equation has at least n periodic when F(x) = ∏ (i=1...n) (x^2-i^2): Focus on functions without forced oscillation for which we identify and locate periodic solutions at even very long distances from the origin. Geometrical approach in symbiosis with the appearance of trajectories. Algorithms that make it very easy to botanize many new specimens.

Serge Reymond 1
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Abstract : The Liénard equation is a subject that has occupied mathematicians for almost a century. In spite of this, we are still dealing with the same rare examples, including that of the Van der Pol equation. Mathematicians are lacking a wide collection of cases from which to test or draw some hypotheses. We present algorithms that make it very easy to botanize many new specimens. This should energize the subject. The usual approach used for the search for periodic solutions is that of analysis. In general it deals only with the existence of periodic solutions very close to the origin for functions containing a very small forced oscillation. We deal with functions without forced oscillation and identify and locate periodic solutions at even very long distances from the origin. The tools of analysis that are usually used require the mastery of elaborate techniques and it is not easy to make the link between the analytical results and the appearance of the trajectories. We use elementary mathematical tools and a geometric approach completely in symbiosis with the appearance of trajectories. The numerical search methods for periodic solutions do not allow us to know whether we have explored sufficiently far from the origin or limit the time of calculation to the strict necessary. We give a method to determine very simply, for a very wide range of functions, the bounded space in which all the periodic solutions are located. The Grail underlying the study of the Liénard equation is that of the resolution of the 16th Hilbert problem. We have not solved it however we show that the number of periodic solutions is at least (degree F(x)-1)/2
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https://hal.archives-ouvertes.fr/hal-02264465
Contributor : Serge Reymond <>
Submitted on : Friday, August 9, 2019 - 2:47:12 PM
Last modification on : Wednesday, August 14, 2019 - 1:01:44 AM

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Serge Reymond. The Liénard equation has at least n periodic when F(x) = ∏ (i=1...n) (x^2-i^2): Focus on functions without forced oscillation for which we identify and locate periodic solutions at even very long distances from the origin. Geometrical approach in symbiosis with the appearance of trajectories. Algorithms that make it very easy to botanize many new specimens.. 2019. ⟨hal-02264465⟩

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