Stochastic Bifurcations Of Dynamical System With Fractional Derivative Under Colored Noise

Stochastic bifurcation of a class of dynamic systems with fractional order damping and fractional PID controller excited by color noise is studied.Chapter 2 discusses the stochastic averaging processing of a series of dynamics system with fractional derivative.Firstly,we propose a new method for the fractional order derivative is the approximate transformation,i.e.,the complicated mathematical expression of fractional derivative is approximated into pseudo periodic functions.Then,this new way is verified by Gauss-Laguerre formula and Runge-Kutta method.Finally,the simulation method of Gauss color noise specific Monte Carlo digital is given.Chapter 3 discusses the stochastic bifurcation problem of the generalized DVP model with fractional ? ?(0,1)under the color noise excitation.Firstly,the new method proposed at Chapter 2 and stochastic averaging method are applied to obtain the FPK equation and the stationary probability density of amplitude.After that,two critical parameters conditions of system occurs stochastic P-bifurcation which must be satisfied are obtained by using the extremum principle,singularity theory,and maple software.Then,starting from the colored noise and multiplicative noise in two aspects,the study finds that the change of noise intensity,fractional order and correlation time will lead to the stochastic bifurcation.Chapter 4 focuses on stochastic bifurcation of a generalized DVP system with higher order?(1 < ? < 2)of fractional derivative subjected to additive color noise excitation by using a similar approach to Chapter 3.At first,by using the new method in the second chapter,the fractional derivative expression of Caputo definition is replace as a pseudo periodic functions,and the FPK equation and stationary probability density for corresponding amplitude response is obtained based on the Stochastic averaging method.Analytical equations of two essential parameter conditions for stochastic P-bifurcation are derived from using catastrophe theory and extremum principle.The phenomenon of stochastic bifurcation with respect to the noise intensity,fractional order,correlation time and fractional coefficient are studied by analyzing probability density function curves of amplitude response.Chapter 5 studies stochastic bifurcation problem of a generalized DVP system under color noise excitation with the emphasis on the control of parameters in fractional order PID controllers.Firstly,the complicated mathematical expression of fractional order PID controller is approximated into pseudo periodic functions by using the properties of fractional calculus and the nature of a generalized integral.After that,the stochastic averaging method is applied to obtain the FPK equation and the stationary probability density of amplitude.And then,the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory.The stochastic bifurcation caused by the change of fractional order integral,fractional order differential and the coefficient of fractional PID controller is analyzed in detail.