Study On Two Stochastic Dynamical Systems And Their Applications In Economic System

It is very important and worthwhile to study random responses, random stabilities,random bifurcations, first-passage failures and random optimal control of stochasticnonlinear dynamical system in theory and its applications. In recent years, there aremany preliminary research results in theory about stochastic nonlinear science.However, the results of some complicated classical dynamical systems are notcompletely, and some results can not agree with others. In addition, there is sufficientwork to do as how to applying stochastic nonlinear dynamical methods to solve realproblems of economics and finance. Considering that, in the present paper, the authorhas studied the behaviors of two classical nonlinear dynamical systems, which areMathieu-Duffing system and Rayleigh oscillator, under different kinds of stochasticnoise excitations. At the same time, the author has studied a price model by usingstochastic nonlinear dynamical equations.In the first section, the dynamical behaviors of stochastic Mathieu-Duffing systemhave been studied. Firstly, the author has studied the maximal Lyapunov exponentsand almost-sure stability of Mathieu-Duffing equation under Gaussian white noiseexcitation by multiple scales method. The results show that near the parametericresonance at excitation frequencyΩ=2ω₀, the system become more unstable with theincreasing of the maximal Lyapunov exponentλ, and the maximal Lyapunovexponents reache its maximum value whenσ=0. That is to say, the stability of trivialsolution of stochastic Mathieu-Duffing equation is more stable whileσtends tozero. But for the stability of nontrivial solution, the author has got the sufficient andnecessary condition is-3βαa₀ cosη₀/2ω₀＞0.In the second section, the author has studied the onset of chaotic motion of Van der Pol-Mathieu-Duffing system under bounded noise excitation. By using randomMelnikov technique, a mean square criterion is used to detect the necessary conditionfor chaotic motion of this stochastic system. Then, the Poincare map and Lyapunovexponents have been calculated by numerical method. The results show that when thenoise is presented in the system, the shapes of Poincare maps begin to diffuse intolarger area and the threshold of bounded noise amplitude for the onset of chaos in thissystem increases as the intensity of the bounder noise increases, which is furtherverified by the maximal Lyapunov exponents of the system, i.e.,whenω₀:Ω₁:Ω₂=1:2:2 orω₀:Ω₁:Ω₂=1:1:2, the chaotic motions of thesystem can be enhanced by the bound noise.In the third section, considering the government macro-control for price system asexternal control force, a dynamically controlled nonlinear stochastic price model isproposed based on deterministic model. By using stochastic averaging method andnonlinear stochastic control strategy, the stochastic model can been stabilized. Thecontrolling aim is to enable the system becoming more stable. Through comparingthe controlled Lyapunov exponents with uncontrolled Lyapunov exponents, it isshown that the controlled strategy is effective. This is very important to ourgovernment. Then, the author has not only studied the steady-state response andfirst-passage failure but also the applications in price modeling of Rayleigh oscillatorunder external and parametric excitations. Firstly, steady-state response and the firstpassage failure of Rayleigh oscillator under external Gaussian white noise excitationhave been studied. The equation of motion of the system is first reduced to averagedIto stochastic differential equations by using the stochastic averaging method. Abackward Kolmogorov equation governing the conditional reliability function and ageneralized Pontrayagin equation governing the conditional monments offirst-passage time are established. Statistical properties of the stationary response andfirst-passage failure are analyzed by using numerical results in three groups of varying parameters. Secondly, the stochastic Hopf bifurcation of Rayleigh oscillator subject toGaussian white noise parametric excitation is studied. The equation of motion of thesystem is reduced to an averaged It(?) stochastic differential equations by using thestochastic averaging method. Then, the critical D-bifurcation parameterc_D=-D/2μis determined approximately by using lyapunov exponents of theinvariable measureδ_c is trivial, and the critical P-bifurcation parameter valuec_P=0 is determined approximately by using the theory of singular points ofdiffusion processes. Finally, the procedure has been verified with the numericalsolution of the joint probability density of the stochastic system.In the last section, the author has introduced the creations and the problems thatwill be further studied.