Positively Invariant Sets for ODEs and Numerical Integration

We show that for an ordinary differential equation (ODE) with an exponentially stable equilibrium and any compact subset of its basin of attraction, we can find a larger compact set that is positively invariant for both the dynamics of the system and a numerical method to approximate its solution trajectories. We establish this for both one-step numerical integrators and multi-step integrators using sufficiently small time-steps. Further, we show how to localize such sets using continuously differentiable Lyapunov-like functions and numerically computed continuous, piecewise affine (CPA) Lyapunov-like functions.