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.