Research On The Stochastic Dynamics Of Typical Fractional-order Systems

In the past few decades,fractional calculus has attracted considerable attention,partly due to the fact that the fractional-order mathematical models can be used to make a more accurate description than the integer-order models.We put forward different analytical methods for typical stochastic fractional-order systems in which its effectiveness is verified by the numerical results.The contents and results are as follows:In Chapter 1,we introduce the history of fractional calculus and review the development of stochastic fractional-order systems,stochastic vibro-impact systems,and stochastic vibration energy harvesters.In Chapter 2,the stochastic responses of the Van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation are studied.First,the fractional Van der Pol oscillator is replaced by an equivalent Van der Pol oscillator without fractional derivative terms by using the generalized harmonic balance technique.Then,the stochastic averaging method is applied to the equivalent Van der Pol oscillator to obtain the analytical solution.Finally,the analytical solutions are validated by numerical results from the Monte Carlo simulation of the original fractional Van der Pol oscillator.In Chapter 3,stochastic bifurcations of a vibro-impact oscillator with two kinds of fractional derivative elements driven by Gaussian white noise excitation are explored in this paper.We can obtain the analytical approximate solutions with the help of non-smooth transformation and stochastic averaging method.The numerical results from Monte Carlo simulation of the original system are regarded as the benchmark to verify the accuracy of the developed method.The results demonstrate that the proposed method has a satisfactory level of accuracy.The important and interesting result we can conclude in this paper is that the effect of the first fractional derivative order on the system is totally contrary to that of the second fractional derivative order.In Chapter 4,the stochastic bifurcations in the nonlinear vibro-impact system with fractional derivative element under random excitation are discussed.Firstly,the original stochastic vibro-impact system with fractional derivative element is transformed into equivalent stochastic vibro-impact system without fractional derivative element.Then,the non-smooth transformation and stochastic averaging method are used to obtain the analytical solutions of the equivalent stochastic system.At last,in order to verify the effectiveness of the above mentioned approach,the Van der Pol vibro-impact system with fractional derivative element is worked out in detail.A very satisfactory agreement can be found between the analytical results and the numerical results.An interesting phenomenon we found in this paper is that the fractional derivative order and fractional derivative coefficient of the stochastic Van der Pol vibro-impact system can induce the occurrence of stochastic P-bifurcation.In Chapter 5,we investigate the stationary response of a class of quasi-linear systems with fractional derivative excited by Poisson white noise.The equivalent stochastic system of the original stochastic system is obtained.Then,approximate stationary solutions are obtained with the help of the perturbation method.Finally,two typical examples are discussed in detail to demonstrate the effectiveness of the proposed method.In Chapter 6,we investigate the stochastic response of monostable vibration energy harvesters with fractional derivative damping under Gaussian white noise excitation.First,we obtain the fractional-order vibration energy harvesters by introducing the fractional derivative damping into the integer-order models.We can get the equivalent stochastic system with the help of variable transformation.Then,the approximately analytical solutions of the equivalent stochastic system can be obtained by the stochastic averaging method.Third,the numerical results are considered as the benchmark to prove the effectiveness of the proposed method.The results indicate that the proposed method has a satisfactory level of accuracy.