Dynamics of Nonlinear Nonautonomous Systems with Delays and Applications

具有时滞的非线性非自治系统动力学及其应用

• 批准号: DDG-2015-00046
• 负责人: Idels, Lev
• 金额: $ 0.73万
• 依托单位: Vancouver Island University
• 依托单位国家: 加拿大
• 项目类别: Discovery Development Grant
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612447
• 关键词: Dynamics; Nonlinear; Nonautonomous; Systems; Delays

The global stability of systems with strong nonlinearity is a prerequisite for almost all applications, and it is very challenging to discover what can guarantee the stability of the models that arise in a series of interesting applications. Time delays which cause the poor performance and instability of dynamic systems are commonly encountered in various physical engineering and neural networks and it has a great impact on its dynamic behaviors. It remains poorly understood how the dynamical behavior depends on the network architecture since the technical difficulties in the study impose the restriction to network of either smooth functions or simple connection architectures; therefore the introduction of  non-monotonic activation functions considerably increases its memory capacity. The inclusion of explicit time lags in the models allows direct reference to experimentally measurable and/or controllable cell growth characteristics. Most results in the recent studies on the global stability of nonlinear systems were obtained under some restrictive assumptions on the kernels, and there are no general methods to study global stability for non-autonomous models with high-order nonlinearities and without symmetric structure; so even the questions about uniqueness, continuous dependence and linearization are challenging for qualitative studies. It is very important to establish explicit, necessary and sufficient conditions for the existence of unique and globally stable equilibria, the latter will guarantees that the systems are not in a chaotic state. I will construct certain classes of weakly nonlinear systems that have the global and dynamic equivalence to the highly-nonlinear systems under study, by using a specially crafted integral and differential inequalities, the properties of some special matrix norms and the properties of nonlinear Volterra operators. I will derive instability tests and design effective feedback controllers for high-order delayed equations  with chaotic dynamics. It is to be emphasized that our technique will produce explicit and easily verifiable sets of tests; and does not require a long sequence of other theorems or conditions that must be proven/cited before a proof of a new theorem.  The results obtained for the networks might be used to examine oscillatory behavior in a genetic regulatory network with delays and design sampled data controllers. I will obtain sets of necessary conditions for the stability and derive new algorithms for detecting instability to find out: How to control a motion which is harmful or dangerous? How to create a motion which is useful or needed?  This project will be ideal for training graduate and undergraduate students in applied mathematics, engineering and computer science; and will entail  cooperation with research university groups in Canada, Argentina, France, Israel and Ukraine.

强非线性系统的全局稳定性是几乎所有应用的前提条件，如何保证一系列有趣的应用中出现的模型的稳定性是一个非常具有挑战性的问题，时滞是各种物理工程和神经网络中经常遇到的导致动态系统性能差和不稳定的问题，它对系统的动态行为有很大的影响。由于研究中的技术困难限制了网络的光滑函数或简单连接结构，因此对动力学行为如何依赖于网络结构的理解仍然很少;因此，非单调激活函数的引入大大增加了其记忆容量。在模型中包含明确的时间滞后允许直接参考实验可测量和/或可控制的细胞生长特征。近年来关于非线性系统全局稳定性的研究大多是在对核函数的一些限制性假设下得到的，而对于具有高阶非线性和无对称结构的非自治模型的全局稳定性还没有通用的研究方法，因此即使是唯一性、连续依赖性和线性化等问题的定性研究也是具有挑战性的.建立全局稳定平衡点存在的显式充分必要条件是非常重要的，后者将保证系统不处于混沌状态。我将构造某些类弱非线性系统，它们与所研究的高度非线性系统具有全局和动态等价性，通过使用一个特制的积分和微分不等式，一些特殊矩阵范数的性质和非线性沃尔泰拉算子的性质。我将推导出不稳定性测试和设计有效的反馈控制器高阶延迟方程混沌动力学。需要强调的是，我们的技术将产生明确的和易于验证的测试集，并不需要一个长序列的其他定理或条件，必须证明/引用之前证明一个新的theorem. The网络获得的结果可能会被用来检查振荡行为的遗传调控网络的延迟和设计采样数据控制器。我将获得一组必要条件的稳定性和新的算法来检测不稳定性，以找出：如何控制运动是有害的或危险的？如何创建一个有用或需要的运动？ 该项目将是培训应用数学、工程和计算机科学研究生和本科生的理想项目;并将与加拿大、阿根廷、法国、以色列和乌克兰的研究型大学团体合作。