A new method for constructing periodic orbits in nonlinear dynamicalsystems

An iterative method for constructing periodic orbits in nonlinear dynamical systems is developed. The method is a modification of the Generalized Newton-Raphson technique. Application of the method to numerically continuing natural and isoperiodic families in the restricted three-body problem was implemented on a high-speed computer and the results show broad regions of convergence, lack of sensitivity, and strong convergence properties of the method. Previous methods isolated periodic orbits by adjusting the initial conditions and the period of a solution of the dynamical system until the solution closed to a prescribed accuracy. The new technique is distinctly different in that it constructs a sequence of precisely periodic solutions of a system of linear differential equations which approximates the original dynamical system. This system of linear differential equations is modified at each step to more closely represent the dynamical system. The technique gives a way for both design of starting orbits for detailed study of orbit families with the methods of initial condition adjustment and for construction of elements of new orbit families. It is applicable to nonautonomous systems and a convergence theorem for this case is presented. Although the technique was applied with success to autonomous systems, a convergence theorem for the time-free case was not established. It is shown that the method will approach the desired solution in this case arbitrarily closely for computational purposes.