Periodic solutions of nonlinear ordinary differential equations computed by a boundary shape function method and a generalized derivative-free Newton method

Abstract

In the paper, the period of an n-dimensional nonlinear dynamical system is computed by a formula derived in an (n+1)-dimensional augmented state space. The periodic conditions and nonlinear first-order ordinary differential equations constitute a specific periodic boundary value problem within a time interval, whose length is an unknown finite constant. Two periodic problems are considered: (I) boundary values are given and (II) boundary values are unknown. A boundary shape function method (BSFM), using the derived shape functions, is devised to an initial value problem with the initial values of new variables given, whereas the terminal values and period are determined by iterative algorithms. The periodic solutions obtained by the BSFM satisfy the periodic conditions automatically. For the sake of comparison, the iterative algorithms based on the shooting method are developed, directly implementing the Poincaré map with the fictitious time integration method to determine the periodic solutions, where the periodic conditions are transformed to a mathematically equivalent scalar equation. Owing to the implicit, non-differentiable and nonlinear property of the scalar equation, we develop a generalized derivative-free Newton method (GDFNM) to solve the periodic problem of case (I), which can pick up very accurate period through a few iterations. In numerical examples the computed order of convergence displays the merit of the proposed iterative algorithms. The BSFM and GDFNM are better than the shooting method from the aspects of convergence speed, accuracy and stability. A conventional Poincaré mapping method is introduced to solve the periodic problems with the same parameters. The BSFM converges faster and more accurate than the Poincaré mapping method and is less sensitive to the initial guesses of initial values and period.