A perturbation-incremental (PI) method for strongly non-linear oscillators and systems of delay differential equations

Abstract

A perturbation-incremental (PI) method is presented for the computation, continuation and bifurcation analysis of periodic solutions of strongly autonomous/ non-autonomous nonlinear oscillators and systems of delay differential equations. Explicit form of a limit cycle with arbitrary parameter values can be obtained, which enables the phase portraits to be constructed. Periodic solutions can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. As the parameter varies, bifurcations such as period-doubling, saddle-node, Hopf and torus bifurcations can be identified. In particular, branch switching at a period-doubling bifurcation is made simple by the present scheme as neither the tangent of the new branch nor the second derivatives need to be calculated. Subsequent continuation of an emanating branch is also discussed. Besides, the complex responses of a nonlinear system due to time delay can be observed and analysed since the delay is used as the bifurcation parameter. The advantage of the PI method lies in its simplicity and ease of implementation.