  This paper proves an instability theorem for dynamical systems. As one adds
interactions between subystems in a complex system, structured or random, a
threshold of connectivity is reached beyond which the overall dynamics
inevitably goes unstable. The threshold occurs at the point at which flows and
interactions between subsystems (`surface' effects) overwhelm internal
stabilizing dynamics (`volume' effects). The theorem is used to identify
instability thresholds in systems that possess a core-halo or core-periphery
structure, including the gravo-thermal catastrophe -- i.e., star collapse and
explosion -- and the interbank payment network. In the core-halo model, the
same dynamical instability underlies both gravitational and financial collapse.
