Stochastic Stability and Bifurcation of Centrifugal Governor System Subject to Color Noise

Abstract

This paper presents a detailed investigation of stochastic stability and complex dynamics of a centrifugal governor system with approximately uniform color noise. The centrifugal governor system excited by noise is transformed into Itö equation using polar coordinate transformation and stochastic average method. According to the boundary conditions of attraction and repulsion, the stochastic stability is ensured. In addition, analyses concerning the influence of parameter variation and validity are carried out by employing numerical method. The results manifest that the effects of noise intensity and correlation time on stationary probability density are opposite. The amplitudes of probability density finally tend to a limit value, and the only limit cycle appears, which shows that when the bifurcation occurs, the trivial solution of the system converges to a limit cycle with a higher probability. Finally, the two-dimensional parameter bifurcation analysis of the centrifugal governor system subject to color noise excitation is studied. An interesting distribution characteristic is found that the periodic region is organized according to the sequence of Stern–Brocot trees, and this typical characteristic is a universal characteristic of the system on the two parameter planes. Furthermore, it is concluded that based on the largest Lyapunov exponent diagram and bifurcation diagram in two-dimensional parameter plane, the effects of noise intensity and correlation time on the periodic oscillation state are opposite, but both of them can transform the quasi-periodic oscillation into periodic oscillation. It should be emphasized that with the increase of noise intensity, the coexisting oscillation behavior of the centrifugal governor system will change, which is manifested by the destruction of coexisting attractors and the generation of chaotic attractors.