On a modified version of ILDM approach: asymptotic analysis based on integral manifolds

Abstract

Using the method of integral (invariant) manifolds, the intrinsic low-dimensional manifolds (ILDM) method is analysed. This is a method for identifying invariant manifolds of a system's slow dynamics and has proven to be an efficient tool in modelling of laminar and turbulent combustion. It allows treating multi-scale systems by revealing their hidden hierarchy and decomposing the system dynamics into fast and slow motions. The performed analysis shows that the original ILDM technique can be interpreted as one of the many possible realizations of the general framework, which is based on a special transformation of the original coordinates in the state space. A modification of the ILDM is proposed based on a new definition of the transformation matrix. The proposed numerical procedure is demonstrated on linear examples and highly non-linear test problems of mathematical theory of combustion and demonstrates in some cases better performance with respect to the existing one.