  We study the dynamics of perturbations around nonthermal fixed points
associated to universal scaling phenomena in quantum many-body systems far from
equilibrium. For an N-component scalar quantum field theory in 3+1 space-time
dimensions, we determine the stability scaling exponents using a
self-consistent large-N expansion to next-to-leading order. Our analysis
reveals the presence of both stable and unstable perturbations, the latter
leading to quasi-exponential deviations from the fixed point in the infrared.
We identify a tower of far-from-equilibrium quasi-particle states and their
dispersion relations by computing the spectral function. With the help of
linear response theory, we demonstrate that unstable dynamics arises from a
competition between elastic scattering processes among the quasi-particle
states. What ultimately renders the fixed point dynamically attractive is the
phenomenon of a scaling instability, which is the universal scaling of the
unstable regime towards the infrared due to a self-similar quasi-particle
cascade. Our results provide ab initio understanding of emergent stability
properties in self-organized scaling phenomena.
