In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris, SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the mth member of the class of algorithms ($m=0,1,...$) finds iteratively an approximation of the appropriate zero of the (m+1)st...

In this paper we summarize an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and some additional technical assumptions like boundedness or smoothing properties. We apply this result to processes (two-parameter semiflows) generated by nonautonomous semilinear parabolic evolution equations.

Using the Lyapunov-Perron method, we prove the existence of an inertial manifold for the process associated to a class of non-autonomous semilinear hyperbolic equations with finite delay, where the linear principal part is positive definite with a discrete spectrum having a sufficiently large distance between some two successive spectral points, and the Lipschitz coefficient of the nonlinear term may depend on time and belongs to some admissible function spaces.

Inertial manifold with delay (IMD) for dissipative systems of second order in time is constructed. This result is applied to the study of different asymptotic properties of solutions. Using IMD, we construct approximate inertial manifolds containing all the stationary solutions and give a new characterization of the K-invariant manifold.

We study a family of semilinear reaction-diffusion equations on spatial domains ${\Omega }_{\epsilon }$, ε > 0, in ${ℝ}^l$ lying close to a k-dimensional submanifold ℳ of ${ℝ}^l$. As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by $H{¹}_s\left(\Omega \right)$. The definition of $H{¹}_s\left(\Omega \right)$, given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable characterizations...