Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum

Continuation is an efficient algorithm for finding solutions of systems of nonlinear algebraic equations where the solutions form a one-dimensional continuum. Such systems arise naturally when investigating equilibrium points and periodic solutions of ordinary differential equations with one parameter. Continuation of isolated periodic solutions of dissipative systems is a well-established technique. Less attention has been devoted to continuation of periodic solutions of conservative systems, where periodic solutions typically form a one-parameter family. To specify a single periodic solution, additional condition must be considered. However, this gives an over-determined system, which has no solution when working with approximate numerical values. We propose a simple algorithm which solves this difficulty by using singular value decomposition of the Jacobian matrix. This algorithm is applied to the conservative model of elastic pendulum. A branch of periodic solutions with constant energy is found which is born by the period doubling bifurcation of vertical oscillations.