We present a modular technique for simulation-based bounded verification for nonlinear dynamical systems. We introduce the notion of input-to-state discrepancy of each subsystem Ai in a larger nonlinear dynamical system A which bounds the distance between two (possibly diverging) trajectories of Ai in terms of their initial states and inputs. Using the IS discrepancy functions, we construct a low dimensional deterministic dynamical system M (δ). For any two trajectories of A starting δ distance

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... apart, we show that one of them bloated by a factor determined by the trajectory of M contains the other. Further, by choosing appropriately small δ's the overapproximations computed by the above method can be made arbitrarily precise. Using the above results we develop a sound and relatively complete algorithm for bounded safety verification of nonlinear ODEs. Our preliminary experiments with a prototype implementation of the algorithm show that the approach can be effective for verification of nonlinear models.