Analysis
Department of Mathematics and Statistics
Texas Tech University
This talk is focused on the behavior near the extinction time of solutions of systems of ordinary differential equations with a sublinear dissipation term. Suppose the dissipation term is a product of a linear operator $A$ and a positively homogeneous scalar function $H$ of a negative degree $-\alpha$. Then any solution with an extinction time $T_*$ behaves like $(T_*-t)^{1/\alpha}\xi_*$ as time $t\to T_*^-$, where $\xi_*$ is an eigenvector of $A$. The proof first establishes the asymptotic behaviors of the "Dirichlet" quotient and the normalized solution. They are then combined with a perturbation technique that requires the function $H$ to satisfy some pointwise H\"older-like condition. The result allows the higher order terms to be general and the nonlinear function $H$ to take very complicated forms.
This is the continuation of the previous talk. We study the behavior near the extinction time of solutions of systems of ordinary differential equations with a sublinear dissipation term. Suppose the dissipation term is a product of a linear operator $A$ and a positively homogeneous scalar function $H$ of a negative degree $-\alpha$. Then any solution with an extinction time $T_*$ behaves like $(T_*-t)^{1/\alpha}\xi_*$ as time $t\to T_*^-$, where $\xi_*$ is an eigenvector of $A$. The proof first establishes the asymptotic behaviors of the "Dirichlet" quotient and the normalized solution. They are then combined with a perturbation technique that requires the function $H$ to satisfy some pointwise H\"older-like condition. The result allows the higher order terms to be general and the nonlinear function $H$ to take very complicated forms.
This is part 1 of a 2-part talk. The task of modelling and forecasting a dynamical system is one of the oldest problems. Broadly, this task has two subtasks- extracting the full dynamical information from a partial observation; and then explicitly learning the dynamics from this information. These two subtasks can be combined into a single mathematical framework using the language of spaces, maps, and commutations. The framework also unifies two of the most common learning paradigms- delay coordinates and reservoir computing. This framework provides a platform for two other investigations of the reconstructed system- its dynamical stability; and the growth of error under iterations. We show that these questions are deeply tied to more fundamental properties of the underlying system the behavior of matrix cocycles over the base dynamics, its non-uniform hyperbolic behavior, and properties of its Koopman operator. Co-author : Tyrus Berry, George Mason University Paper.
This is part 2 of a 2-part talk. The theory of matrix cocycles provides a foundation for random matrix theory in Ergodic theory - the measure theoretic study of deterministic dynamical systems. The foundational works in this field establish the existence of Lyapunov exponents and stable / unstable splittings, for general matrix cocycles. Matrix cocycles appear naturally in a wide variety of physical phenomenon. We shall first see how it arises in the context of learning a dynamical system. The performance of a learning technique can be modelled using a cocycle, and we demonstrate the application of this theory in establishing the limits of learning.
In this session, we finish making the connections between matrix cocycles, and learning theory -- the two independent topics that have been covered in the past talks. The theory of matrix cocycles provides a foundation for random matrix theory in Ergodic theory - the measure theoretic study of deterministic dynamical systems. The foundational works in this field establish the existence of Lyapunov exponents and stable / unstable splittings, for general matrix cocycles. Matrix cocycles appear naturally in a wide variety of physical phenomenon. The performance of a learning technique can be modelled using a cocycle, and we demonstrate the application of this theory in establishing the limits of learning.
We will discuss current status and progress in several challenging problems, which remain open for a long time. Also, I will suggest some new problems and conjectures, which sound appropriate for PhD projects of our graduate students.
Abstract pdf