复杂时滞系统的多稳定性研究及其应用

项目名称： 复杂时滞系统的多稳定性研究及其应用

项目编号： No.10871019

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 彭名书

作者单位： 北京交通大学

项目金额： 24万元

中文摘要： 本项目研究复杂时滞系统的非线性扩散、多级分岔、混沌等问题, 探索和发现具有实际背景意义的理论模型，如神经网络模型以及寡头经济模型等, 结合次贷危机分析了经济增长过程中出现的泡沫现象, 探讨了多稳定性的经济学含义及从（数学）动力系统的角度揭示强共振情形下的复杂动力学和多个混沌吸引子的存在性. 运用分岔理论，揭示多种耦合效应（线性或非线性耦合）下非线性时滞系统的多稳定性(multistability)或多模态(oscillation patterns)现象，揭示系统的复杂机理、时空行为和通向混沌的不同道路，发现了时滞所导致的高余维现象，建立了多稳定（Multistability）状态下一种控制方法，从而能实现多种状态（运动模式）的相互转化,并实现了从单个子系统到大系统（多个子系相互耦合）的研究跨越. 部分成果已发表在Chaos, nonlinear dynamics, 中国科学（数学）等数学和应用数学类顶级杂志上,SCI已收录6篇, EI收录3篇. 人才培养方面, 现已形成一个富有进取精神的年轻研究团队, 并同国内外专家学者、研究机构保持紧密联系.

中文关键词： 复杂系统;多稳定性;分岔;混沌;时滞

英文摘要： In this program, the focus is on its complex dynamics in delayed systems, including stability, bifurcation, mutistability and chaotic behavior. Both continuous-time systems and discrete-time syetms are concerned.Some interesting neural models are proposed, rich dynamics are explored, which has been published by Chaos, nonlinear dynamics, International Journal of bifurcation and chaos, Science in China etc. Main Results: 1.There has been an increasing interest in nonlinear coupled systems mainly because of their possible modeling the long-time evolution dynamics among neurons (such as Hopfield/Cohen-Grossberg neuron networks) or oligopolists (such as Cournot duopoly models). We propose a generalized type of nonlinear discrete-time delayed neuron networks without self-connections, which can model the interaction among groups (subsystems) rather than single neurons. Asymptotic behavior and all possible bifurcations are discussed, which extend and generalize those obtained in recent papers. 2. A multistability phenomenon is discovered and observed in two-parameter families of nonlinear discrete-time neuron networks. A specific strategy of controlling multistability (multiple oscillation patterns) to unimode oscillation pattern (synchronization) by linear asymmetric diffusion coupling is proposed and the numerical simulation provides solid evidence for its effectiveness. 3. We give a detailed study of rich dynamics in two-parameter families of twodimensional generalized delayed discrete Cournot duopoly models. Multistability, such as the coexistence of period-2/quasiperiodic (limit-cycle), chaotic/regular motions or synchronized/asynchronized solutions are discussed. Complexity caused by delay, including the change of local stability regions and the occurrence of higher-codimension bifurcations, is to be discovered. 4.It is proved that for any homeomorphism on a compact metric space, if it has the shadowing property and it is expansible on chain recurrent set, then it is chain topologically stable.

英文关键词： Complex system; multistability; bifurcation; chaos; delay