Dynamics Of The Delayed Stochastic Differential Systems

With the continuous development of natural science, people realize the nature of thereal world more and more, therefore, many mathematicians and other scientists concernedthe inevitable random and non-linear factors of the reality system, in particular, in recentyears, a large number of nonlinear stochastic systems in physics, life sciences, engineering,and many other areas of economy and finance are derived, the theory and application ofnonlinear stochastic dynamical systems are more studied. And described by the ordinarydi?erential equations to the nonlinear dynamic systems compared, the research of nonlin-ear stochastic di?erential dynamical systems has just started, and mathematical theoryand methods are far from mature , especially, in the basic concepts and applied theories ofstochastic bifurcation and stochastic chaos aspect, there exist lots of outstanding issues,the theoretical system not complete, we pending further to clarify and discuss. Germanscholar, Arnold prof. even think that the theory of stochastic bifurcation and chaos is stillin its infancy. Therefore, we further understand and study that the complexity of nonlin-ear stochastic di?erential dynamical system is not only necessary but also significance ofthe theory and application.In the paper, stochastic stability, stochastic bifurcation and stochastic chaos of thedelayed stochastic di?erential dynamical system are studied extensively. The main re-search work are as follows.In Chapter 1, we give a survey to the developments of stochastic stability, stochas-tic bifurcation and stochastic chaos for delayed stochastic di?erential dynamical system.Then we introduce the background of problem and important basics knowledge.In Chapter 2, we mainly consider stochastic stability and stochastic bifurcations of thegeneral two-dimensional stochastic di?erential systems with delay. Firstly, using reducedprinciple and first?second-order standard stochastic averaging method, we obtain thegeneral form of stochastic averaging equation for the general two-dimensional stochasticdi?erential systems with delay. Secondly, based on singular boundary theory and ergodictheory, some stability conditions for stochastic averaging equation are obtained, and werigorously prove the existence of three types stochastic bifurcation for stochastic averagingequation, some analytic criterions of stochastic stability and stochastic bifurcation for theoriginal two-dimensional stochastic di?erential systems with delay are established. Ourconclusion are applied to stochastic ocean structure model, our study shows that thetheoretical results very well consistent with the results of case study. In Chapter 3, we mainly consider the stochastic stability and stochastic bifurcation ofthe delayed non-integrable stochastic Hamilton system. By non-integrable stochastic av-eraging method, ergodic theory and singular boundary theory, etc., we obtain the stochas-tic stability, stochastic bifurcation and random limit cycles for the delay non-integrablestochastic Rayleigh-van der Pol oscillator and the delay non-integrable coupled nonlinearstochastic Rayleigh-van der Pol oscillator. We generalize and improve the correspondingresults in recent literature.In Chapter 4, we mainly consider stochastic stability and stochastic bifurcation of thedelayed integrable stochastic Hamilton system and part integrable stochastic Hamiltonsystem. Firstly, by stochastic center manifold theory, ergodic theory, integrable stochasticaveraging method, etc., some analytic criterions of stochastic stability, stochastic bifur-cation for the delayed integrable stochastic Du?ng-van der Pol oscillator are obtained.Secondly, through partly integrable stochastic averaging method, ergodic theory, reso-nance and non-resonance theory, ect., we obtain su?cient conditions of the stochasticstability and stochastic bifurcation for the resonance and non-resonant case of the de-lays four freedom degrees partly integrable stochastic oscillator. We improve and extendexisting relevant results.In Chapter 5, we mainly study the stochastic bifurcation and chaos of three typesof classical system (Kolmogorov ecosystems, Josephson systems and Lorenz systems).Firstly, through using stochastic Melnikov function, etc., we prove the existence of stochas-tic chaos for four di?erent types stochastic Kolmogorov ecosystems. Secondly, accordingto non-smooth dynamical systems theory and stochastic Melnikov function, etc., we ob-tained the related su?cient conditions of chaos for the stochastic Josephson systems.Thirdly, using generalized stochastic Hamilton system, perturbation theory and gener-alized stochastic Melnikov function, etc., we obtain su?cient conditions of stochasticbifurcation and chaos for stochastic Lorenz systems.