Limit cycle construction using Liapunov functions

This paper deals with a generalization of the method of Zubov for the construction of Liapunov functions useful in estimating the location of stability boundaries. For a system is taken as the solution of where is positive semi-definite and not identically zero on a non-trivial trajectory and exhibits the significant behavior of the system. For a second order system having (with time reversed) an unstable limit cycle analytic in a parameter ε, a suitable would be satisfying the above partial differential equation may be developed as a power series in e and the position of the limit cycle can be estimated from . As an example of the procedure, the method is applied to van der Pol\´s equation and the position of the limit cycle is estimated to order ε².