On The Stochastic Stability Of A Noise Driven Duffing Oscillator

The model of Duffing oscillators, as a typical nonlinear system, exists widely in Engineering, suchas gravity pendulum, flutter system and so on. With the deepening of research development, scientistsfound that many of the nature excitation must be depicted as random excitation, and the vibrationphenomenon must be depicted as random vibration, so random dynamic gradually flourished.Stochastic stability is an important part of random dynamics. Lyapunov exponent and momentLyapunov exponent is main content, especially moment Lyapunov exponent was a strong conditionfor the system, so it was widely used in the engineering. This paper will mainy research the momentLyapunov exponent.Thesis mainly is divided into two parts to study. The first part will discuss the stability of theequilibrium point, then the stability of periodic orbit. During the research mainly adopts the method ofmultiple scales and stochastic average method. The main research process above are as follows:1) first we wii research the response of Duffing oscillators under the deterministic excitation, thenconsider the stability of equilibrium under bounded noise excitation. We can get the approximateequation using the stochastic average method, then by means of linear transformation we canobtain the matrix. The moment exponent was the characteristic value of matrix. At last we willuse Monte Carlo simulation to verify the correctness of the solution of moment Lyapunovexponent, and draw the images, through which we can qualitatively draw some conclusions.2) As the previous chapter, we will firstly ananlysis the deterministic response of Duffing oscillator,and calculate the periodic orbit and get the condition of periodic orbit. Then consider the stabilityof periodic orbit of this system under the white noise excitation. We will let the orbit subtractanother, by this way we will only discuss the stability of the equilibrium. Then use the stochasticaverage method and so on to obtain the matix, and verify the correctness by meas of MonteCarlo simulation. here also need to generation into the original system to verify the correctness,through the displacement time history and phase diagram.