Dynamics of Nonlinear Nonautonomous Systems with Delays and Applications

具有强非线性的系统的全局稳定性是几乎所有应用的先决条件，而在一系列有趣的应用中，发现什么可以保证模型的稳定性是非常具有挑战性的。时间延迟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，它对动态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平滑功能或简单连接结构的网络，因此对动态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，可以直接参考实验可测量和/或可控的细胞生长特性。近年来关于非线性系统全局稳定性的研究大多是在对核的一些限制性假设下得到的，对于具有高阶非线性且无对称结构的非自治模型，没有研究全局稳定性的一般方法；因此，即使是关于唯一性、连续依赖性和线性化的问题，也对定性研究具有挑战性。建立全局唯一稳定平衡点存在的明确的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。我将利用一个特别设计的积分和微分不等式、一些特殊矩阵范数的性质和非线性Volterra算子的性质，构造若干类与所研究的高度非线性系统具有全局和动态等价的弱非线性系统。我将推导不稳定性测试和设计有效的反馈控制器的高阶延迟方程与混沌动力学。需要强调的是，我们的技术将产生明确的、易于验证的测试集；并且不需要在证明一个新定理之前必须证明/引用一长串其他定理或条件。所得结果可用于研究具有延迟的遗传调控网络的振荡行为和设计采样数据控制器。我将得到一组稳定的必要条件，并推导出检测不稳定的新算法，以找出：如何控制有害或危险的运动？如何创建一个有用或需要的运动？该项目将是培养应用数学、工程和计算机科学方面的研究生和本科生的理想项目；并将需要与加拿大、阿根廷、法国、以色列和乌克兰的研究型大学团体合作。

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.