Alexander Gorban
Constructive Methods of Invariant Manifolds in Kinetics
The Constructive Methods of Invariant Manifolds for model
reduction in physical and chemical kinetics, developed
during last two decades, are presented. The physical
problem of reduced description is studied in a most general
form as a problem of constructing the slow invariant
manifold. The invariance conditions are formulated as
the differential equation for a manifold immersed in
the phase space (the invariance equation). The equation
of motion for immersed manifolds is obtained (the film
extension of the dynamics). Invariant manifolds are
fixed points for this equation, and slow invariant
manifolds are Lyapunov stable fixed points, thus slowness
is presented as stability.
A collection of methods to derive analytically and to
compute numerically the slow invariant manifolds is
presented. Among them, iteration methods based on
incomplete linearization, relaxation method and the
method of invariant grids are developed.
In particular, the Newton method subject to incomplete
linearization is the analogue of KAM methods for
dissipative systems.