  We use Lyapunov-like functions and convex optimization to propagate
uncertainty in the initial condition of nonlinear systems governed by ordinary
differential equations. We consider the full nonlinear dynamics without
approximation, producing rigorous bounds on the expected future value of a
quantity of interest even when only limited statistics of the initial condition
(e.g., mean and variance) are known. For dynamical systems evolving in compact
sets, the best upper (lower) bound coincides with the largest (smallest)
expectation among all initial state distributions consistent with the known
statistics. For systems governed by polynomial equations and polynomial
quantities of interest, one-sided estimates on the optimal bounds can be
computed using tools from polynomial optimization and semidefinite programming.
Moreover, these numerical bounds provably converge to the optimal ones in the
compact case. We illustrate the approach on a van der Pol oscillator and on the
Lorenz system in the chaotic regime.
