Related papers: Partial Inertial Manifolds for infinite-dimensiona…
These notes are devoted to the problem of finite-dimensional reduction for parabolic PDEs. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called…
Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the…
Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for…
Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of…
In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness…
We study some classes of semi-linear differential equations including both well-posed and ill-posed cases that can generate cocycles (or cocycle correspondences with generating cocycles). Under exponential dichotomy condition with other…
Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well posed, so we find an additional…
Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of…
A method for determining the dimension and state space geometry of inertial manifolds of dissipative extended dynamical systems is presented. It works by projecting vector differences between reference states and recurrent states onto local…
This work is the first attempt to treat partial differential equations with discrete (concentrated) state-dependent delay. The main idea is to approximate the discrete delay term by a sequence of distributed delay terms (all with…
This article deals with invariant manifolds for infinite dimensional random dynamical systems with different time scales. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations. Under suitable…
We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in ${\mathbb R}^{n}$ permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare--Bendixson…
We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have asymptotically finite-dimensional dynamics in the corresponding state space. This example is more natural in…
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $\mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for…
A new class of nonlinear partial differential equations with distributed in space and time state-dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state-dependent delay and discuss…
The conditions imposed in the paper ['Inertial manifolds and completeness of eigenmodes for unsteady magnetic dynamos', Physica D {\bf 194} (2004) 297-319] on the fluid velocity to guarantee the existence of inertial manifolds for the…
We obtain global and local theorems on the existence of invariant manifolds for perturbations of non autonomous linear differential equations assuming a very general form of dichotomic behavior for the linear equation. Besides some new…
We define partial differential (PD in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson…
The paper discusses linear fractional representations of parameter-dependent nonlinear systems with dynamics defined by real rational nonlinearities and a finite set of point delays. The global asymptotic stability is investigated via…