Differentiable dynamical systems

The statistics of random sum is studied and used to evaluate performance metrics
in wireless networks. Pertinent wireless network performance measures such as call
completion/dropping probabilities and the average number of handovers usually require
the probability distributions of the cell dwell time and call holding time; and are therefore
not easy to evaluate. The proposed performance evaluation technique requires the
moments of the cell dwell time and is given in terms of the Laplace transform function of
the call holding time. Multimedia services that have Weibull and generalized gamma
distributed call holding times are investigated. The proposed approximation method uses
the compound geometric random sum distribution and requires that the geometric
parameter be very small. For applications in which this parameter is not sufficiently
small, a result is derived that improves the accuracy (to order of the geometric parameter)
of the performance measures evaluated.

In this dissertation we present a computational approach to Conley's Decomposition
Theorem, which gives a global decomposition of dynamical systems, and
introduce an explicit numerical algorithm with computational complexity bounds
for computing global dynamical structures of a continous map including attractorrepeller
pairs and Conley's Lyapunov function. The approach is based on finite spatial
discretizations and combinatorial multivalued maps. The method is successful
in exhibiting approximations of attractor-repeller pairs, invariant sets, and Conley's
Lyapunov function. We used the C++ language to code the algorithm.

Singularly perturbed systems with or without delays commonly appear in mathematical modeling of physical and chemical processes, engineering applications, and increasingly, in mathematical biology. There has been intensive work for singularly
perturbed systems, yet most of the work so far focused on systems without
delays. In this thesis, we provide a new set of tools for the stability analysis for
singularly perturbed control systems with time delays.

A second-order Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lower-order derivatives play a key role in forcing certain types of dynamics. However, the application of these techniques requires an analytic restriction on the Lagrangian that it satisfy a twist property. In this dissertation we approach this problem from the point of view of curve shortening in an effort to remove the twist condition. In classical curve shortening a family of curves evolves with a velocity which is normal to the curve and proportional to its curvature. The evolution of curves with decreasing action is more general, and in the first part of this dissertation we develop some results for curve shortening flows which shorten lengths with respect to a Finsler metric rather than a Riemannian metric. The second part of this dissertation focuses on analytic methods to accommodate the fact that the Finsler metric for second-order Lagrangian system has singularities. We prove the existence of simple periodic solutions for a general class of systems without requiring the twist condition. Further; our results provide a frame work in which to try to further extend the topological forcing theorems to systems without the twist condition.

Despite the high-dimensional nature of the nervous system, humans produce low-dimensional cognitive and behavioral dynamics. How high-dimensional networks with complex connectivity give rise to functionally meaningful dynamics is not well understood. How does a neural network encode function? How can functional dynamics be systematically obtained from networks? There exist few frameworks in the current literature that answer these questions satisfactorily. In this dissertation I propose a general theoretical framework entitled 'Structured Flows on Manifolds' and its underlying mathematical basis. The framework is based on the principles of non-linear dynamical systems and Synergetics and can be used to understand how high-dimensional systems that exhibit multiple time-scale behavior can produce low-dimensional dynamics. Low-dimensional functional dynamics arises as a result of the timescale separation of the systems component's dynamics. The low-dimensional space in which the functi onal dynamics occurs is regarded as a manifold onto which the entire systems dynamics collapses. For the duration of the function the system will stay on the manifold and evolve along the manifold. From a network perspective the manifold is viewed as the product of the interactions of the network nodes. The subsequent flows on the manifold are a result of the asymmetries of network's interactions. A distributed functional architecture based on this perspective is presented. Within this distributed functional architecture, issues related to networks such as flexibility, redundancy and robustness of the network's dynamics are addressed. Flexibility in networks is demonstrated by showing how the same network can produce different types of dynamics as a function of the asymmetrical coupling between nodes. Redundancy can be achieved by systematically creating different networks that exhibit the same dynamics. The framework is also used to systematically probe the effects of lesion