Transition Matrix Theory

• Grant number: 10/19230-8
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: March 01, 2011
• End date: March 15, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ketty Abaroa de Rezende
• Grantee: Ewerton Rocha Vieira
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 07/06896-5 - Geometry of control, dynamical and stochastic systems, AP.TEM
• Associated scholarship(s): 13/15119-3 - Homological and Homotopical methods in dynamical systems, BE.EP.DR

Abstract

In this project we use algebraic spectral sequences to study properties of flow continuation. We wish to associate to the topology of the manifold trajectories that connect critical points of a gradient flow via Conley's connection matrices with integer entries. In other words, we consider a Morse complex with differential being given by a connection matrix with integer entries. We will study connection and transition matrices obtained by the sweeping algorithm with the aim of understanding the dynamical behaviour at each stage.The final objective is to understand the bifurcations which occur at sucessive stages of the continuation which are associated to the family of transition matrices. In the continuation process as one goes through several bifurcations there are birth and death of orbits which we hope to characterize dynamically with respect to the algebraic data. That is to say certain orbits which are created at a given parameter value are stable throughout the continuation while others appear, form non-transversal connections and by disappearing serve to create new orbits at that parameter value. See the following work: 1. O. Cornea, K.A. de Rezende, M. SilveiraSpectral sequences in Conley's theory publicado emErgodic Theory and Dynamical Systems, v30 pp 1009-1054 (2010)2. K.A. de Rezende,M.P. Mello, M. SilveiraConley's Spectral Sequence via the Sweeping Algorithm publicado em Topology and its Applications, v157 pp 2111-2130 (2010)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

FRANZOSA, ROBERT; DE REZENDE, KETTY A.; VIEIRA, EWERTON R.. GENERALIZED TOPOLOGICAL TRANSITION MATRIX. TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, v. 48, n. 1, p. 183-212, SEP 2016. (12/18780-0, 10/19230-8)

FRANZOSA, ROBERT; VIEIRA, EWERTON R.. TRANSITION MATRIX THEORY. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 369, n. 11, p. 7737-7764, NOV 2017. (10/19230-8)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

VIEIRA, Ewerton Rocha. Teoria da matriz de transição. 2015. Doctoral Thesis - Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica Campinas, SP.