Dimensional reduction of nonlinear time delay systems

摘要

Whenever there is a time delay in a dynamical system, the studyof stability becomes an infinite-dimensional problem. The centre manifold theorem, together with the classical Hopf bifurcation, is the most valuable approach for simplifying theinfinite-dimensional problem without the assumption of small time delay. This dimensional reduction is illustrated in this paper with the delay versions of the Duffing and van der Pol equations. For both nonlinear delay equations, transcendental characteristic equations of linearized stability are examined through Hopf bifurcation. The infinite-dimensional nonlinear solutions of the delay equations are decomposed into stable and centre subspaces, whose respective dimensions are determined by the linearized stability of the transcendental equations. Linear semigroups, infinitesimal generators, and their adjoint forms with bilinear pairings are the additional candidates for the infinite-dimensional reduction.