Synthesis of Linard's Equations Having More than One Periodic Solution

Tosiro KOGA  Masaharu SHINAGAWA  Satoshi HASAKO  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E76-A   No.6   pp.848-857
Publication Date: 1993/06/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Papers Selected from 1992 Joint Technical Conference on Circuits/Systems,Computers and Communications (JTC-CSCC'92))
Category: Nonlinear Circuits and Neural Nets
Keyword: 
asymmetric Linard's equation,  limit cycles,  

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Summary: 
As is well known, Linard's equation f (χ)+g(χ)=0 represents a wide class of oscillatory circuits as an extension of van der Pol's equation, and Linard'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 Linard'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 Linard's systems, which generates an arbitrarily prescribed number of limit cycles, and which is considered to be important in relation to the stability of Linard's systems. Then, as application of this result, we give a method of determining parameters included in Linard's systems which may produce two limit cycles depending on the parameters. We also give a Linard's system which have three limit cycles. In addition, a new result on the parameter dependency of the number of limit cycles is presented.