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Une preuve rigoureuse de I’existence du chaos dans le double enroulement

A rigorous proof of the existence of chaos in double scroll

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Cet article présente une preuve rigoureuse de l’existence du chaos dans les montages où la trajectoire de phase a un double enroulement. Nous utilisons une représentation « linéaire par morceaux» applicable à une famille trés large d’équations, dont celles qui decrivent les circuits à double enroulement constituent un cas particulier. Les expressions analytiques de la trajectoire, permettent de calculer les sections de Poincaré et de montrer l’existence d’une trajectoire homocline de Shilnikov, donc de prouver rigoureusement l’existence du chaos. On peut aussi mettre en évidence les conditions de naissance et de disparition de la trajectoire à double enroulement.

Abstract

This paper provides a rigorous mathematical proof that the double scroll is indeed chaotic. Our approach is to derive a linearly equivalent class of piecewiselinear differential equations which includes the double scroll as a special case. The analytical expressions characterizing various half-return maps associated with the Poincaré map are used in a crucial way to prove the existence of Shilnikov-type homoclinic orbit, thereby establishing rigorously the chaotic nature of the double scroll. These analytical expressions are also fundamental in our in-depth analysis of the birth (onset of the double scroll) and death (extinction of chaos) of the double scroll.

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  1. College of Engineering, Electronics research laboratory, University of California, Berkeley, CA 94720, USA

    Léon O Chua

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  1. Léon O Chua
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Chua, L.O. Une preuve rigoureuse de I’existence du chaos dans le double enroulement. Ann. Telecommun. 42, 239–246 (1987). https://doi.org/10.1007/BF02995242

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  • DOI: https://doi.org/10.1007/BF02995242

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