Minnesota Mathematics of Climate Seminar

February 14, 2023

Limit cycles of small amplitude in polynomial and piecewise polynomial planar vector fields, Luiz Fernando da Silva Gouveia, São Paulo State University & University of Texas at Dallas

In this talk, we will talk about Lyapunov Constants in the classical qualitative theory of differential equations and in the theory of discontinuous systems. Moreover, we will show how to calculate the Lyapunov Constants using the parallelization process and how to use the constants calculated to obtain limit cycles.

February 21, 2023

A constructive approach to finding the Taylor expansion of the period function for Hamiltonian systems, Yagor Romano Carvalho, University of Sao Paulo at Sao Carlos & University of Texas at Dallas

In this talk, we present a constructive procedure to obtain the Taylor expansion, in terms of the energy levels, of the period function for a non-degenerated center of any planar analytic Hamiltonian system. We apply it to several examples, including the whirling pendulum and a cubic Hamiltonian system. The knowledge of this Taylor expansion of the period function for this system is one of the key points to studying the number of zeroes of an Abelian integral that controls the number of limit cycles bifurcating from the periodic orbits of a planar Hamiltonian.

February 28, 2023

Attractors of Nonsmooth and Multivalued Dynamical Systems with an Application in Oceanography, Cameron Thieme, Center for Discrete Mathematics and Theoretical Computer Science, Rutgers University

Over the past few decades, piecewise-continuous differential equations have become increasingly popular in scientific models. In particular, conceptual climate models often take this form; we will study one such model--Welander's ocean box model--as an example throughout this talk. These nonsmooth systems are typically reframed as Filippov systems or differential inclusions, a special type of multivalued dynamical system. Some qualitative properties of these inclusions have been studied over the last few decades, primarily in the context of control systems. Our interest in these systems is in understanding what behavior identified in the nonsmooth model may be continued to families of smooth differential equations which limit to the Filippov system; determining this information is particularly important in this context because the piecewise-continuous model is frequently considered to be a heuristically understandable approximation of a more realistic smooth system. In this talk we will examine how Conley index theory may be applied to the study of differential inclusions in order to address this goal. In particular, we will discuss how attractor-repeller pairs identified in a Filippov system continue to nearby smooth systems.