skip to main content
Idioma:
Tipo de recurso Mostra resultados com: Mostra resultados com: Índice

Finite-dimensionality of attractors for dynamical systems with applications: deterministic and random settings

Cunha, Arthur Cavalcante

Biblioteca Digital de Teses e Dissertações da USP; Universidade de São Paulo; Instituto de Ciências Matemáticas e de Computação 2021-01-27

Acesso online. A biblioteca também possui exemplares impressos.

  • Título:
    Finite-dimensionality of attractors for dynamical systems with applications: deterministic and random settings
  • Autor: Cunha, Arthur Cavalcante
  • Orientador: Carvalho, Alexandre Nolasco de
  • Assuntos: Atrator Global; Atrator Uniforme; Atrator Uniforme Aleatório; Dimensão Fractal; Sistemas Dinâmicos; Random Uniform Attractor; Global Attractor; Fractal Dimension; Dynamical Systems; Uniform Attractor
  • Notas: Tese (Doutorado)
  • Descrição: In this work we obtain estimates on the fractal dimension of attractors in three different settings: global attractors associated to autonomous dynamical systems, uniform attractors associated to non-autonomous dynamical systems and random uniform attractors associated to non-autonomous random dynamical systems. Firstly we give a simple proof of a result due to Mañé (Springer LNM 898, 230242, 1981) that the global attractor A (as a subset of a Banach space) for a map S is finite-dimensional if DS(x) =C(x)+L(x), where C is compact and L is a contraction (and both are linear). In particular, we show that if S is compact and differentiable then A is finite-dimensional. Using a smoothing property for the differential DS we also prove that A has finite fractal dimension and we make a comparison of this method with Mañés approach. We give applications to an abstract semilinear parabolic equation and to 2D Navier-Stokes equations. Secondly we prove using a smoothing method that uniform attractors have finite fractal dimension on Banach spaces, with bounds in terms of the dimension of the symbol space and a Kolmogorov entropy number. We also show that the smoothing property is useful to prove the finite-dimensionality of uniform attractors in more regular Banach spaces. In addition, we prove that the finite-dimensionality of the hull of a time-dependent function is fully determined by the tails of the function. We give applications to non-autonomous 2D Navier- Stokes and reaction-diffusion equations. Thirdly we prove using a smoothing and a squeezing method that random uniform attractors have finite fractal dimension. Neither of the two methods implies the other. Estimates on the dimension are given in terms of the dimension of the symbol space plus a term arising from the smoothing/squeezing property; the smoothing is applied also to more regular spaces. In this setting we give applications to a stochastic reaction-diffusion equation with scalar additive noise. In addition, a random absorbing set which absorbs itself after a deterministic period of time is constructed.
  • DOI: 10.11606/T.55.2021.tde-26032021-135356
  • Editor: Biblioteca Digital de Teses e Dissertações da USP; Universidade de São Paulo; Instituto de Ciências Matemáticas e de Computação
  • Data de criação/publicação: 2021-01-27
  • Formato: Adobe PDF
  • Idioma: Inglês
Links
Voltar para lista de resultados
Ir para página anteriorAnterior Resultado  2 AvançarIr para próxima página

  • Realização: ABCD-USP USP   Logos de Redes Sociais: Freepik

Buscando em bases de dados remotas. Favor aguardar.