scholarly journals On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence

2011 ◽  
Vol 12 (1) ◽  
pp. 281-302
Author(s):  
Tomás Caraballo ◽  
David Cheban
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence ◽  
Global Attractor ◽  
Infinite Dimensional ◽  
Autonomous Dynamical Systems

scholarly journals State space decomposition for non-autonomous dynamical systems

2011 ◽  
Vol 141 (5) ◽  
pp. 957-974 ◽  
Author(s):  
Xiaopeng Chen ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
State Space ◽  
Decomposition Theorem ◽  
Navier Stokes ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
The Lorenz System ◽  
A Chain ◽  
Autonomous Dynamical Systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

scholarly journals On the structure of the Levinson center for monotone non-autonomous dynamical systems with a first integral

2021 ◽  
Vol 38 (1) ◽  
pp. 67-94
Author(s):  
DAVID CHEBAN ◽  
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Global Attractor ◽  
Difference Equations ◽  
First Integral ◽  
Almost Periodic ◽  
Almost Automorphic ◽  
Stable Solutions ◽  
Autonomous Dynamical Systems

In this paper we give a description of the structure of compact global attractor (Levinson center) for monotone Bohr/Levitan almost periodic dynamical system $x'=f(t,x)$ (*) with the strictly monotone first integral. It is shown that Levinson center of equation (*) consists of the Bohr/Levitan almost periodic (respectively, almost automorphic, recurrent or Poisson stable) solutions. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We also give some applications of theses results to different classes of differential/difference equations.

scholarly journals Flattening, squeezing and the existence of random attractors

2006 ◽  
Vol 463 (2077) ◽  
pp. 163-181 ◽  
Author(s):  
Peter E Kloeden ◽  
José A Langa
Keyword(s):  
Dynamical Systems ◽  
Modern Theory ◽  
Qualitative Properties ◽  
Infinite Dimensional ◽  
Squeezing Property ◽  
Finite Dimensional ◽  
Bounded Absorbing Set ◽  
Skew Product Flows ◽  
The Difference ◽  
Autonomous Dynamical Systems

The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic non-autonomous dynamical systems formulated as skew-product flows.

Equations for biological evolution

1995 ◽  
Vol 125 (5) ◽  
pp. 939-958 ◽  
Author(s):  
Angel Calsina ◽  
Carles Perelló
Keyword(s):  
Dynamical Systems ◽  
Mathematical Models ◽  
Global Attractor ◽  
Biological Evolution ◽  
Stationary Solutions ◽  
Infinite Dimensional ◽  
Differential Type

In this paper we consider mathematical models inspired by the mechanisms of biological evolution. We take populations which are subject to interaction and mutation. In the cases we consider, the interaction is through competition or through a prey-predator relationship. The models consider the specific characteristics as taking values in real intervals and the equations are of the integro—differential type. In the case of competition, thanks to the fact that some of the equations have solutions which are quite explicit, we succeed in proving the existence of attracting stationary solutions. In the case of prey and predator, using techniques of dynamical systems in infinite-dimensional spaces, we succeed in showing the existence of a global attractor, which in some instances reduces to a point. Our analysis takes into account having δ distributions, corresponding to all individuals having the same characteristics, as possible populations.

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