A Contribution to Liapunov’s Second Method: Nonlinear Autonomous Systems

Abstract

The stability of nonlinear autonomous systems with nonlinearity representable in polynomial form is investigated. For the case of locally stable systems the following theorem is presented: A sufficient condition for local stability of the system ẋ = X (x ) is the existence of a definite function v = φ(x ) such that dv/dt = θ(x )g[ξ(x )], where θ(x ) is a semidefinite function not identically equal to zero on a solution of ẋ = X (x ), g(x ) is such that g(0) = 0 and sign g(u) ≠ sign g(−u), and ξ(x ) = 0 is a closed surface. A procedure for constructing Liapunov functions based upon the use of a generating v-function is developed. Such a generating v-function may have the form: v(x) = x′ A (x)x where A(x ) = {aij (xi , xj )}, and aij = aji . The coefficients aij (xi , xj ) can be computed in order to obtain dv/dt of the wanted form. Particular emphasis is given to the case of systems with limit cycles and, as an example, the limit cycle of the van der Pol equation is identified with good approximation. It is also analytically proved that outside a closed algebraic curve, circumscribing the limit cycle, the system is asymptotically stable.