Dynamics of Nonlinear Nonautonomous Systems with Delays and Applications

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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.