  In this paper, we provide sufficient conditions for dissipativity and local
asymptotic stability of discrete-time dynamical systems parametrized by deep
neural networks. We leverage the representation of neural networks as pointwise
affine maps, thus exposing their local linear operators and making them
accessible to classical system analytic and design methods. This allows us to
"crack open the black box" of the neural dynamical system's behavior by
evaluating their dissipativity, and estimating their stationary points and
state-space partitioning. We relate the norms of these local linear operators
to the energy stored in the dissipative system with supply rates represented by
their aggregate bias terms. Empirically, we analyze the variance in dynamical
behavior and eigenvalue spectra of these local linear operators with varying
weight factorizations, activation functions, bias terms, and depths.
