Limit cycles in Lienard equations

An analytical estimation of the existence and characteristics of limit cycles in a given planar polynomial vector field represents a significant progress towards the complete answer to the second part of Hilbert’s 16th problem. In a very recent work [1], the second author of this present paper has developed a theory to fulfil this purpose. One major conclusion of the theory is that the number of limit cycles nested around a critical point in a general planar polynomial vector field is bounded by the Hilbert number where n is the order of the vector field. It is well known that linear vector fields have no limit cycles and this, of course agrees with the conclusion. Shi [2] shows that there are maximum three limit cycles nested around a critical point in quadratic vector fields. Again, it is in an agreement with the conclusion. For cubic vector fields results from previous studies [3,4,5] are also in an agreement with the conclusion whilst the result from the work [6] is in a disagreement although there exists some doubt about the result. In this present work, a detailed study is given to the limit cycles in a fifteenth order Liénard equation by using both the theory [1] and numerical simulations to check the validity of the theory. The method of analysis is briefly given in Section 2. An application example and conclusions are presented in Section 3 and 4, respectively.