Strict semistability or non-semisimplicity in the sense of representation theory manifests itself dynamically in convergence to non-hyperbolic fixed points under energy minimizing flows. In such a situation one has, quite generally, a center manifold such that coordinates on it describe degrees of freedom which do not decay exponentially fast. In appropriate settings (e.g. quiver representations) one can describe the dynamics on the center manifold combinatorially in terms of modular lattices. On certain walls in the parameter space iterated logarithms appear in the description of the asymptotics. As an application one gets a canonical weight-type filtration (labelled by reals) on any finite-dimensional representation. Conjecturally, this filtration describes asymptotics of certain geometric heat-type and Laplace-type PDEs. This is a joint project with Katzarkov, Kontsevich, and Pandit.