New Challenges in Nonautonomous Bifurcation Theory

非自治分岔理论的新挑战

• 批准号: EP/I004165/1
• 负责人: Martin Rasmussen
• 金额: $ 98.39万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2010
• 资助国家: 英国
• 起止时间: 2010 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FI004165%2F1
• 关键词: New; Challenges; Nonautonomous; Bifurcation; Theory

The technological and economical development of our society has generated the need to deal with very complex systems that require an accurate level of understanding. The present crisis of the financial markets, which is partially due to the failure of instruments to cope with volatility, and weather phenomena associated to climate change such as El Nio, are examples of dynamical processes with a deep economic impact that require sophisticated models to take nonautonomous (i.e., time-dependent or random) influences in form of changes of parameters into account.The main goal of this research project is to develop the qualitative theory of nonautonomous dynamical systems from a mathematical viewpoint in directions beyond the traditional setting which are highly relevant to the applied sciences. In particular, it aims at characterising and classifying qualitative changes of such systems and developing computational tools for their analysis. The availability of a universally applicable bifurcation theory will provide computational measurements which are able to indicate points where the systems behaviour is about to change drastically, with enormous possible applications including the prediction of (catastrophic) events, for instance, in economics (stock markets), environmental studies (climate change modelling) or health care studies (seizure prediction).

我们社会的技术和经济发展产生了处理非常复杂的系统的需要，这些系统需要准确的理解。目前的金融市场危机，部分原因是由于工具未能科普波动，以及与气候变化有关的天气现象，如厄尔尼诺现象，是具有深刻经济影响的动态过程的例子，需要复杂的模型来采取非自治（即，该研究项目的主要目标是从数学角度发展非自治动力系统的定性理论，其方向超出了传统的设置，与应用科学高度相关。特别是，它旨在描述和分类这些系统的质变，并开发用于分析的计算工具。普遍适用的分叉理论的可用性将提供计算测量，能够指示系统行为即将发生急剧变化的点，具有巨大的可能应用，包括预测（灾难性）事件，例如，在经济学（股票市场），环境研究（气候变化建模）或医疗保健研究（癫痫预测）。

项目成果

Hopf bifurcation with additive noise

• DOI: 10.1088/1361-6544/aad208
• 发表时间: 2017-10
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: T. S. Doan;Maximilian Engel;J. Lamb;M. Rasmussen

On fractional lyapunov exponent for solutions of linear fractional differential equations

• DOI: 10.2478/s13540-014-0169-1
• 发表时间: 2014-03
• 期刊: Fractional Calculus and Applied Analysis
• 影响因子: 3
• 作者: N. D. Cong;Doan Thai Son;H. T. Tuan

Nonautonomous saddle-node bifurcations: Random and deterministic forcing

非自主鞍节点分叉：随机和确定性强迫

• DOI: 10.1016/j.jde.2012.03.016
• 发表时间: 2012
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Anagnostopoulou V

On stable manifolds for planar fractional differential equations

平面分数阶微分方程的稳定流形

• DOI: 10.1016/j.amc.2013.10.010
• 发表时间: 2014
• 期刊: Applied Mathematics and Computation
• 影响因子: 4
• 作者: Cong N

Conditioned Lyapunov exponents for random dynamical systems

随机动力系统的条件 Lyapunov 指数

• DOI: 10.1090/tran/7803
• 发表时间: 2019
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Engel M