时滞随机微分系统的动力学性质研究
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摘要
随着自然科学的不断发展,人们对现实世界的认识越来越贴近本质,因此,现实系统中不可避免的随机和非线性因素已成为众多数学家和其他领域科学家关注的焦点,特别是,近年来物理学、生命科学、工程技术、经济金融等诸多领域中推导得出大量的非线性随机模型,进一步促使了人们对非线性随机微分动力系统的理论和应用的深入研究.与确定常微分方程所描述的非线性微分动力系统比较,非线性随机微分动力系统的研究刚刚起步,目前的数学理论和方法远未成熟,特别是在随机分岔、随机极限环和随机混沌等基本概念、理论和应用方面存在大量悬而未决的科学理论问题,其理论体系很不完备,有待进一步澄清和探讨.以至于德国学者Arnold教授认为对该领域的研究还处于“Infancy”阶段.因此,进一步深入研究非线性随机微分动力系统的复杂动力学性质不仅是必要的,而且具有重要的理论意义和应用价值.
本论文主要研究具有时滞的非线性随机微分系统的随机稳定性、随机分岔和随机混沌等问题,具体的研究工作如下:
第一章为绪论,主要简述了时滞随机微分动力系统发展概况及随机稳定性、随机分岔和随机混沌的研究进展,并介绍了本文问题产生的背景和一些相关的预备基本知识.
第二章,主要考虑具有时滞的二维随机微分系统的随机稳定性和随机分岔问题.首先利用时滞退化原理、一阶或二阶标准随机平均法分别给出具有时滞的一般二维随机微分系统所对应的随机平均方程的具体表达式.其次,将所得随机平均方程的相应表达式与奇异边界值和遍历性中的相关理论有机结合,获得了随机平均方程的随机稳定性条件,严格地证明了随机平均方程的随机分岔的存在性,进一步建立了具有时滞的一般二维随机微分系统的随机稳定性与随机分岔的解析判据.然后,将所得到的理论结果应用到随机海洋结构模型.
第三章,主要考虑具有两个不同时滞的随机拟不可积Hamilton系统的动力学性质.运用随机拟不可积平均法、遍历性和奇异边界值中的相关理论,获得了时滞随机拟不可积Rayleigh-van der Pol振子和耦合Rayleigh-van der Pol振子的随机稳定性、随机分岔与随机极限环存在性的判据.所用方法与以往不同,所得的结果推广和改进了最近文献中的相关结果.
第四章,主要考虑时滞随机拟可积Hamilton系统和拟部分可积Hamilton系统的动力学行为.首先,基于随机中心流形理论、遍历性、随机拟可积平均法,获得了时滞随机拟可积Du?ng-van der Pol振子的随机稳定性、随机分岔与随机极限环存在性的解析判据.其次,运用随机拟部分可积平均法、遍历性、共振和非共振相关理论,得到具有四个自由度的时滞随机拟部分可积振子在共振和非共振情形下的随机稳定性和随机分岔的一些判据.改进和推广了已有的相关结果.
第五章,主要研究具有随机扰动的三类经典系统(包括二维Kolmogorov生态系统、Josephson系统和Lorenz系统)的随机分岔与混沌的复杂动力学性质.首先,应用随机Melnikov函数,获得四种不同类型随机Kolmogorov生态系统存在随机混沌的一些解析判据.其次,应用随机Melnikov函数和非光滑微分系统相关理论,给出随机Josephson系统存在随机混沌的判据.最后,基于广义随机Hamilton系统相关理论、摄动法和广义随机Melnikov函数,获得了随机Lorenz系统发生音叉分岔和存在随机混沌的定理.
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.
本论文主要研究具有时滞的非线性随机微分系统的随机稳定性、随机分岔和随机混沌等问题,具体的研究工作如下:
第一章为绪论,主要简述了时滞随机微分动力系统发展概况及随机稳定性、随机分岔和随机混沌的研究进展,并介绍了本文问题产生的背景和一些相关的预备基本知识.
第二章,主要考虑具有时滞的二维随机微分系统的随机稳定性和随机分岔问题.首先利用时滞退化原理、一阶或二阶标准随机平均法分别给出具有时滞的一般二维随机微分系统所对应的随机平均方程的具体表达式.其次,将所得随机平均方程的相应表达式与奇异边界值和遍历性中的相关理论有机结合,获得了随机平均方程的随机稳定性条件,严格地证明了随机平均方程的随机分岔的存在性,进一步建立了具有时滞的一般二维随机微分系统的随机稳定性与随机分岔的解析判据.然后,将所得到的理论结果应用到随机海洋结构模型.
第三章,主要考虑具有两个不同时滞的随机拟不可积Hamilton系统的动力学性质.运用随机拟不可积平均法、遍历性和奇异边界值中的相关理论,获得了时滞随机拟不可积Rayleigh-van der Pol振子和耦合Rayleigh-van der Pol振子的随机稳定性、随机分岔与随机极限环存在性的判据.所用方法与以往不同,所得的结果推广和改进了最近文献中的相关结果.
第四章,主要考虑时滞随机拟可积Hamilton系统和拟部分可积Hamilton系统的动力学行为.首先,基于随机中心流形理论、遍历性、随机拟可积平均法,获得了时滞随机拟可积Du?ng-van der Pol振子的随机稳定性、随机分岔与随机极限环存在性的解析判据.其次,运用随机拟部分可积平均法、遍历性、共振和非共振相关理论,得到具有四个自由度的时滞随机拟部分可积振子在共振和非共振情形下的随机稳定性和随机分岔的一些判据.改进和推广了已有的相关结果.
第五章,主要研究具有随机扰动的三类经典系统(包括二维Kolmogorov生态系统、Josephson系统和Lorenz系统)的随机分岔与混沌的复杂动力学性质.首先,应用随机Melnikov函数,获得四种不同类型随机Kolmogorov生态系统存在随机混沌的一些解析判据.其次,应用随机Melnikov函数和非光滑微分系统相关理论,给出随机Josephson系统存在随机混沌的判据.最后,基于广义随机Hamilton系统相关理论、摄动法和广义随机Melnikov函数,获得了随机Lorenz系统发生音叉分岔和存在随机混沌的定理.
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.
引文
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