Dimension of the attractors associated to autonomous and nonautonomous dynamical systems

• Grant number: 22/04886-2
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: August 01, 2022
• End date: March 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Alexandre Nolasco de Carvalho
• Grantee: Rafael de Oliveira Moura
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 20/14075-6 - Dynamical systems and their attractors under perturbations, AP.TEM
• Associated scholarship(s): 23/11798-5 - Dimension of the attractors associated to autonomous and nonautonomous dynamical systems, BE.EP.DR

Abstract

The main goal of this project is to study the fractal dimension and the Assouad dimension of attractors for autonomous and non-autonomous dynamical systems, considering that the study of the dimension of the attractor of a dynamical system allows us, in certain cases, to reduce the complexity of this attractor by relating it to a subset of a finite-dimensional Euclidean space.Our first objective will be finding estimates for the fractal dimension of the uniform attractor for cocycles - which arise in the study of non-autonomous dynamical systems - trying to adapt and improve the current results in order to reduce the hypothesis about the Sigma symbol space, allowing it to have infinite fractal dimension. As a second objective, we will explore the Assouad dimension of global attractors of dynamical systems, since we already know that if A is the global attractor for a semigroup and the Assouad dimension of the difference set A-A is finite, then the dynamics in the attractor resembles the dynamics described in the attractor of an ODE in a finite dimensional space. However, there is still no general method for estimating the Assouad dimension of A-A, and this is what we shall investigate.This project is linked to the thematic project Dynamical systems and their attractors under perturbation (FAPESP - number 2020/14075-6), which seeks to holistically study the attractors of autonomous and non-autonomous dynamical systems arising from semilinear and quasilinear parabolic evolution equations.