Physical insights from complex multiscale non-linear system dynamics: Identification of fast and slow variables

There are many applications governed by nonlinear dynamical systems in which the model is introduced in a standard (natural) basis of a vector space, but it is more natural to analyze this model using a different set of basis vectors, which is more suitable for the acquisition of the relevant physical insights. Here the case of multiscale dynamical systems is considered, for which the appropriate alternative set of basis vectors is the one resolving the tangent space. Simple algorithmic tools are introduced that identify the fast and slow variables, thus facilitating (i) the construction of reduced models and (ii) the design of novel pathways for controlling complex multiscale systems across diverse scientific disciplines. The usefulness of these tools is demonstrated in two well-known multiscale models: the Michaelis–Menten and the Lorenz models.