Synthesis of Lienard's Equations Having More than One Periodic Solution (Special Section on JTC-CSCC '92)

概要

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As is well known, Lienard's equation x+μf(x)x+g(x)=0 represents a wide class of oscillatory circuits as an extension of van der Pol's equation, and Lienard's theorem guarantees the existence of a unique periodic solution which is orbitally stable. However, we sometimes meet such cases in engineering applications that the symmetry of the equation is violated, for instance, by a constant bias force. While, it has been known that asymmetric Lienard's equation can have more than one periodic solution. The problem of finding the maximum number of such solutions, known as a special case of Hilbert's sixteenth problem, has recently been solved by T. Koga, one of the present authors. This paper first describes fundamental theorems due to T. Koga, and presents a solution to the synthesis problem of asymmetric Lienard's systems, which generates an arbitrarily prescribed number of limit cycles, and which is considered to be important in relation to the stability of Lienard's systems. Then, as application of this result, we give a method of determining parameters included in Lienard's systems which may produce two limit cycles depending on the parameters. We also give a Lienard's system which have three limit cycles. In addition, a new result on the parameter dependency of the number of limit cycles is presented.
一般社団法人電子情報通信学会の論文
1993-06-25