On dimension of attractors of differential inclusions and reaction-diffussion equations
Francisco Balibrea José Valero
Discrete & Continuous Dynamical Systems - A 1999, 5(3): 515-528 doi: 10.3934/dcds.1999.5.515
In this paper we improve a general theorem of O.A. Ladyzhenskaya on the dimension of compact invariant sets in Hilbert spaces. Then we use this result to prove that the Hausdorff and fractal dimensions of global compact attractors of differential inclusions and reaction-diffusion equations are finite.
keywords: Attractor Hausdorff dimension fractal dimension.
A triangular map on $I^{2}$ whose $\omega$-limit sets are all compact intervals of $\{0\}\times I$
Francisco Balibrea J.L. García Guirao J.I. Muñoz Casado
Discrete & Continuous Dynamical Systems - A 2002, 8(4): 983-994 doi: 10.3934/dcds.2002.8.983
In this paper we construct a triangular map $F$ on $I^2$ which holds the following property. For each $[a,b]\subseteq I=[0,1]$, $a\leq b$, there exists $(p,q)\in I^2$ \ $I_0$ such that $\omega_F(p,q)=$ {0} $\times [a,b]\subset I_0$ where $I_0=${0}$\times I$. Moreover, for each $(p,q)\in I^{2}$, the set $\omega_F(p,q)$ is exactly {0} $\times J$ where $J\subset I$ is a compact interval degenerate or not. So, we describe completely the family $\mathcal W(F)=${$\omega_F(p,q):(p,q)\in I^2$} and establish $\mathcal W(F)$ as the set of all compact interval, degenerate or not, of $I_0$.
keywords: $\omega$-Limit Set. Discrete dynamical system triangular map

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