随机激励下多稳态系统的临界过渡识别及Basin Stability分析

The study of multi-steady-state dynamics is one of the most active scientific frontiers today. So far, most of the achievements have been made on studies of deterministic system or under small-intensity Gaussian white noise excitation. In the practical problems, there are many uncertainties in the interaction between the system and the external environment, and there are many types of noise. Different types of noise may greatly affect the transitions of the particles or the critical transitions of the system. At present, the research is still rare and theoretically very immature. It is urgent to carry out this research. This project intends to take stochastic multi-stability system as the research object. According to different critical transitional behaviors induced by different noise types, corresponding identification or warning indicators are established. At the same time, Basin Stability formula is established to quantify the stability of the system state under random conditions, and the related theoretical framework is gradually perfected. Based on this, the Basin Stability analysis of multi-potential well system induced by different noise types is carried out and compared with the corresponding deterministic situation, the random environment which is most favorable for the system to be in a perfect state is also analyzed. Finally, the research results are applied to related fields such as finance, ecology and disease to determine the stability of the current state of the system and to achieve early warning of the critical transitional behaviors.

多稳态动力学的研究是当今最活跃的科学前沿之一。迄今为止，绝大部分的成果都属于确定性或微扰高斯白噪声激励的研究。而实际问题中，系统与外界环境的相互作用力有许多不确定性，噪声刻画类型也较多，不同的噪声类型会对粒子的跃迁或系统的临界过渡产生极大的不同影响，目前这方面的研究甚少，理论上也很不成熟，亟待开展这一领域的研究。本项目拟以随机多稳态系统为研究对象，针对不同噪声类型诱导的不同临界过渡行为，建立相应的识别或预警指标。同时，建立随机情形下量化系统状态稳定性的Basin Stability计算公式，逐渐完善相关理论框架，在此基础上，对不同噪声类型诱导的多势阱系统进行Basin Stability分析，并与相应的确定情形比较，分析最有利于系统处于理想态的随机环境。最终将研究结果应用到金融、生态和疾病等相关领域中，确定系统当前所处状态的稳定性，实现对临界过渡行为的早期预警。

通过四年的努力和系统研究,按照申请书的研究思路和技术路线，顺利完成了预定的研究计划。通过对随机多稳态系统的动力学分析，取得了一系列成果，主要在如下几个方面取得进展;建立了高斯白噪声激励下周期性双稳系统中临界过渡的预警指标、获得了肿瘤细胞生长系统逃逸的最可能轨迹和最可能灭绝时间，发展了对称列维噪声激励下的λ噬菌体的基因转录调控系统的basin stability分析理论，在生态系统的临界过渡识别、首次穿越时和basin stability的研究、以及微机电系统的随机跳跃和吸引盆的研究等方面取得创新性突破。初步建立一套随机激励下典型多稳态系统的动力学分析理论和数值方法。按计划完成项目研究内容。.主办国际研讨会（网络）3个，主办国内会议5个，参加各类专业学术会80人次，参加项目的人员10人，包括博士生7人。本项目在Chaos、Nonlinear . Dynamics、Physical A、International Journal Non-Linear Mechanics、International Journal of Bifurcation and Chaos等本领域国际著名杂志上发表论文22篇.

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