We study the global attractor Aε of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form g0(x, t)+ 1 ερ g1 ( x ε , t ) , x ∈Ω R2, t ∈R, 0 ρ 1. If the functions g0(x, t) and g1 (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor Aε is bounded in the space H , however, its norm ‖Aε‖H may be unb...

Many a dissipative evolution equation possesses a global attractor A with finite Hausdorff dimension d. In this paper it is shown there is an embedding X of A into IR , with N = [2d+2], such that X is the global attractor of some finite-dimensional system on IR with trivial dynamics on X. This allows the construction of a discrete dynamical system on IR which reproduces the dynamics of the time...

A crisis is a global bifurcation in which a chaotic attractor has a discontinuous change in size or suddenly disappears as a scalar parameter of the system is varied. In this Letter, we describe a global bifurcation in three dimensions which can result in a crisis. This bifurcation does not involve a tangency and cannot occur in maps of dimension smaller than 3. We present evidence of unstable ...

We consider a hydrodynamic system that models the Smectic-A liquid crystal flow. The model consists of the Navier-Stokes equation for the fluid velocity coupled with a fourth-order equation for the layer variable φ, endowed with periodic boundary conditions. We analyze the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove...

In the description of the long-time behavior of solutions to nonautonomous differential equations the notion of a pullback attractor plays a similar role as the global attractor in autonomous dynamical systems. We present the theorem on the existence of a pullback attractor if the evolution process is a family of closed operators. The abstract result is formulated in the context of the smoothin...

We prove that the weakly damped cubic Schrödinger flow in L(T) provides a dynamical system that possesses a global attractor. The proof relies on a sharp study of the behavior of the associated flow-map with respect to the weak L(T)-convergence inspired by [18]. Combining the compactness in L(T) of the attractor with the approach developed in [10], we show that the attractor is actually a compa...

We prove that the weakly damped cubic Schrödinger flow in L(T) provides a dynamical system that possesses a global attractor. The proof relies on a sharp study of the behavior of the associated flow-map with respect to the weak L(T)-convergence inspired by [18]. Combining the compactness in L(T) of the attractor with the approach developed in [10], we show that the attractor is actually a compa...

For the Kuramoto Sivashinsky equation with L-periodic boundary conditions we show that the radius of space analyticity on the global attractor is lowersemicontinuous function at the stationary solutions, and thereby deduce the existence of a neighborhood in the global attractor of the set of all stationary solutions in which the radius of analyticity is independent of the bifurcation parameter ...