Ghost ILDM-manifolds and their identification

One of the popular methods (Intrinsic Low-Dimensional Manifolds - ILDM)) of decomposition of multiscale systems into fast and slow sub-systems for reduction of their complexity is considered in the present paper. The method successfully locates a position of slow manifolds of considered system and as any other numerical approach has its own disadvantages. In particular, an application of the ILDM-method produces so-called ghost-manifolds that do not have any connection to the true dynamics of the system. It is shown analytically that for two-dimensional singularly perturbed system (for which the fast-slow decomposition has been already done in analytical way) the ghost-manifolds appear. The problem of discrimination/identification of the ghost-manifolds is under consideration and two numerical criteria for their identification are proposed. A number of analyzed examples demonstrate efficiency of the suggested approach.