An iteration scheme for determining equations of dynamical systems

This paper considers the problem of determining solutions for the Eq. [d]=0 where F:R[d] and has the form: [d] This situation arises naturally when one attempts to value the determining equations when searching for periodic solution of ordinary differential equations possessing a first integral. Under appropriate differentiability conditions, (1) will generate a family of solutions where dF9u* will be singular at any solution u*. Thus, in order to establish a convergent iteration scheme to determine solutions of (1), a modified form of Newton's method is employed. Provided certain natural and appropriate conditions are met, this procedure establishes a contraction map on an invariant (n-1)-dimensional hyperplane of Rn provided the initial estimate is close enough to the solution curve