The asymptotic completeness of inertial manifolds

@article{Robinson1996TheAC,
  title={The asymptotic completeness of inertial manifolds},
  author={James C. Robinson},
  journal={Nonlinearity},
  year={1996},
  volume={9},
  pages={1325-1340}
}
An investigation of the asymptotic completeness property for inertial manifolds leads to the concept of `flow-normal hyperbolicity', which is more natural in this case than the traditional form of normal hyperbolicity derived from the linearized flow near the manifold. An example shows that without flow-normal hyperbolicity asymptotic completeness cannot be guaranteed. The analysis also yields a new result on the asymptotic equivalence of ordinary differential equations. 
View via Publisher
homepages.warwick.ac.uk
49 Citations
Highly Influential Citations
2
Background Citations
22
Methods Citations
6

49 Citations

Some closure results for inertial manifolds

Suppose that the family of evolution equationsdu/dt+Au+fN(u)=0 possesses inertial manifolds of the same dimension for a sequence of nonlinear termsfN withfN →f in the C0 norm. Conditions are found to…

Hyperbolic solutions and their stable and unstable manifolds

For the majority of this chapter we study the continuity under perturbation of hyperbolic global solutions and their stable and unstable manifolds, for an abstract process S( ⋅, ⋅) on a Banach space…

A DYNAMICAL APPROXIMATION FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Random invariant manifolds provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial…

Determining Asymptotic Behavior from the Dynamics on Attracting Sets

Two tracking properties for trajectories on attracting sets are studied. We prove that trajectories on the full phase space can be followed arbitrarily closely by skipping from one solution on the…

Global convergence of nonlinear cascade flows with Morse-Bott zero dynamics

Tracking properties of trajectories on random attracting Sets

The theory of random attracting sets highlights interesting properties of the asymptotic behaviour of some stochastic differential equations. In this paper some results on the relation between the…

Inertial manifolds and foliations for asymptotically compact cocycles in Banach spaces

We study asymptotically compact nonautonomous dynamical systems given by abstract cocycles in Banach spaces. Our main assumptions are given by a squeezing property in a quadratic cone field (given by…

Geometric theory of inertial manifolds for compact cocycles in Banach spaces

We present a geometric theory of inertial manifolds for compact cocycles (non-autonomous dynamical systems), which satisfy a certain squeezing property with respect to a family of quadratic Lyapunov…

Pullback Attracting Inertial Manifolds for Nonautonomous Dynamical Systems

In this paper we present an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions,…

Stochastic inertial manifolds for damped wave equations

In this paper, stochastic inertial manifold for damped wave equations subjected to additive white noise is constructed by the Lyapunov–Perron method. It is proved that when the intensity of noise…

References

SHOWING 1-10 OF 38 REFERENCES

Inertial manifolds for nonlinear evolutionary equations

Spectral barriers and inertial manifolds for dissipative partial differential equations

In recent years, the theory of inertial manifolds for dissipative partial differential equations has emerged as an active area of research. An inertial manifold is an invariant manifold that is…

Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations

In this paper, we study the long-time behavior of a class of nonlinear dissipative partial differential equations. By means of the Lyapunov-Perron method, we show that the equation has an inertial…

Smoothness of inertial manifolds

Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

Contents: Introduction.- Presentation of the Approach and of the Main Results.- The Transport of Finite Dimensional Contact Elements.- Spectral Blocking Property.- Strong Squeezing Property.- Cone…

Finite-dimensional description of convective Reaction-Diffusion equations

We are concerned with the asymptotic dynamics of a certain type of semilinear parabolic equation, namely,ut=uxx+(f(u))x+g(u)+h(x) on the interval [0,L]. Under the general condition we prove that this…

The spectrum of an invariant submanifold

Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations

Conditions are presented under which the solutions of asymptotically autonomous differential equations have the same asymptotic behavior as the solutions of the associated limit equations. An example…

Averaging and integral manifolds

An integral manifold for a system of differential equations is a manifold such that any solution of the equations which has a point on it is entirely contained on it. The method of averaging…

Normally Hyperbolic Invariant Manifolds in Dynamical Systems

In the past ten years, there has been much progress in understanding the global dynamics of systems with several degrees-of-freedom. An important tool in these studies has been the theory of normally…