Dynamic Stability of Stochastic Delay Systems

Abstract

This paper is aimed at examining the influence of time delays and multiplicative white noise on nonlinear dynamical systems that exhibit degenerate Hopf interactions. Our desire for this study stems from the continued interest to understand the stability bifurcations of machining systems with fluctuating regenerative excitations. The excitations introduce time delays in the restoring and damping forces, and may produce persistent modes of machining failures even at incipient time delays prior to the fully growth rates of the excitations. Generally, systems involving time delays are best described by delay differential equations, and the analysis of such equations are accompanied with complications even when nonlinearities are omitted. The consideration of multiplicative white noise adds a greater impediment to the analysis. Frequently, the assumption of small delays as compared to unity is imposed in order to reduce the infinite-dimensional character of the time delay problem to a finite- dimensional problem using conventional asymptotic techniques such as (i) the Taylor series expansion, (ii) the integral averaging method, (iii) Fourier series, (iv) multiple scale and (v) harmonic balancing. It is well known that the use of these asymptotic techniques with the assumption of small time delays usually leads itself to results, which do not reflect the long term stability bifurcations of the original time delay system. Alternatively, we focus on the use of the centre manifold theorem and classical theorem of Hopf bifurcation for the study of periodic solutions of dynamical systems, with special attention that the long term stability bifurcations of the original nonlinear stochastic delay systems is preserved, and moreover the time delays are not small. The machining system considered is modeled as single degree of freedom and the equations governing the motion contain multiplicative white noise, multiple time delays and nonlinearity. The computation of the Poincare-Lyapunov coefficients, Floquet exponents and moment Lyapunov exponents of the reduced delay systems will enable us to determine sufficient conditions for the possible Hopf interactions. The dependence of the interactions on parameter variations will be captured qualitatively.