Analysis of periodic oscillations with computer algebra

Abstract

AbstractIn analyzing differential equations, computer algebra or symbolic computation is useful when we want to know the relation between the solutions and the parameters at the point near bifurcation.The object of this paper is to show the fast computation method by means of computer algebra in analyzing the bifurcation problem from the steady state to the periodic oscillatory state. The proposed method of analysis depends on Lyapunov‐Schmidt method. Instead of using trigonometric multiplication and sum formula in computation of nonlinear terms and their harmonics, both the basis vectors with polynomials in components and their inner products are utilized to reduce the computation time significantly.The example of a second‐order Van der Pol equation and the example of the higher‐order coupled system are used to give the bifurcation equation. Thus, the relation between the amplitude of oscillation and the parameter is given with the stability condition. By the proposed method with the aid of a microcomputer and symbolic computation language of interpreter type, the computation time is shorter.