Limit cycle construction using Liapunov functions

Abstract
This paper deals with a generalization of the method of Zubov for the construction of Liapunov functionsV(x)useful in estimating the location of stability boundaries. For a system\dot{x}=f(x), V(x)is taken as the solution of(\nablaV)' f(x)=-h(x)g(V)whereh(x)is positive semi-definite and not identically zero on a non-trivial trajectory andg(V)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 suitableg(V)would beg(V) = V(1-V)\dotVsatisfying the above partial differential equation may be developed as a power series in e and the position of the limit cycle can be estimated fromV = 1. 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.