Estimates for the fractal dimension and the number of determining modes for invariant sets of dynamical systems

1990 ◽  
Vol 49 (5) ◽  
pp. 1186-1201 ◽  
Author(s):  
O. A. Ladyzhenskaya
Keyword(s):  
Fractal Dimension ◽  
Dynamical Systems ◽  
Invariant Sets ◽  
Determining Modes

scholarly journals Characteristic exponents of dynamical systems in metric spaces

1983 ◽  
Vol 3 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Invariant Sets ◽  
Characteristic Exponents ◽  
Smooth Case

AbstractWe introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are defined also for invariant sets.

scholarly journals Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors

2019 ◽  
Vol 20 (2) ◽  
pp. 485-515
Author(s):  
Jan W. Cholewa ◽  
Radosław Czaja
Keyword(s):  
Fractal Dimension ◽  
Dynamical Systems ◽  
Global Attractor ◽  
American Mathematical Society ◽  
Reaction Diffusion Equations ◽  
Dissipative Systems ◽  
Exponential Attractor ◽  
Lattice Systems ◽  
Exponential Attractors ◽  
Lattice Dynamical Systems

Abstract In this work, we examine first-order lattice dynamical systems, which are discretized versions of reaction–diffusion equations on the real line. We prove the existence of a global attractor in $$\ell ^2$$ℓ2, and using the method by Chueshov and Lasiecka (Dynamics of quasi-stable dissipative systems, Springer, Berlin, 2015; Memoirs of the American Mathematical Society, vol 195(912), AMS, 2008), we estimate its fractal dimension. We also show that the global attractor is contained in a finite-dimensional exponential attractor. The approach relies on the interplay between the discretized diffusion and reaction, which has not been exploited as yet for the lattice systems. Of separate interest is a characterization of positive definiteness of the discretized Schrödinger operator, which refers to the well-known Arendt and Batty’s result (Differ Int Equ 6:1009–1024, 1993).

scholarly journals Invariant Sets in Quasiperiodically Forced Dynamical Systems

2020 ◽  
Vol 19 (1) ◽  
pp. 329-351
Author(s):  
Yoshihiko Susuki ◽  
Igor Mezić
Keyword(s):  
Dynamical Systems ◽  
Invariant Sets

Determining modes for dissipative random dynamical systems

1999 ◽  
Vol 66 (1-2) ◽  
pp. 1-25 ◽  
Author(s):  
Franco Flandolf ◽  
Jose A. Langa
Keyword(s):  
Dynamical Systems ◽  
Random Dynamical Systems ◽  
Determining Modes

Certain new robust properties of invariant sets and attractors of dynamical systems

1999 ◽  
Vol 33 (2) ◽  
pp. 95-105 ◽  
Author(s):  
A. S. Gorodetski ◽  
Yu. S. Ilyashenko
Keyword(s):  
Dynamical Systems ◽  
Invariant Sets
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