Tracing of stable and unstable steady state periodic solutions of autonomous systems: algorithm and bifurcation analysis

Abstract

This paper describes a numerical algorithm and its computer implementation for the tracing of stable and unstable steady state periodic solutions of autonomous systems of ordinary differential equations. The problem is posed as an initial value problem. The autonomous system considered is a function of n state variables. The period is unknown for autonomous systems. The total number of unknowns to be determined at each step is n+1, i.e., n state variables plus the time period T. Since autonomous systems admit an infinite number of periodic solutions each one differing from the others by a translation in time, to have a unique solution, an appropriate value for one of this n+1 variables is assumed. The recasted system of n nonlinear algebraic equations in n unknowns is solved iteratively using Newton-Raphson method. This will give one periodic solution and its period. To have a continuum of solutions, a locally parameterised continuation procedure is adopted. Stability of periodic solutions along the continuous branch of solutions is determined by computing characteristic multipliers. The effectiveness of the algorithm is demonstrated by conducting bifurcation analysis on a three-node power system.