A Lyapunov-Type Theorem for Dissipative Numerical Integrators with Adaptive Time-Stepping

The asymptotic behavior of continuous dissipative systems and dissipative numerical integrators with fixed time-stepping can be fully investigated by Lyapunov-type theorem on continuous and discrete dynamical systems, respectively. However, once adaptive time-stepping is involved, such theories cease to work, and usually the dynamics should be investigated in a backward way, such as in terms of pullback attractors. In this paper, we present a different approach—we stick to a forward definition of limit sets, and show that still we can establish a Lyapunov-type theorem, which reveals the precise asymptotic behavior of adaptive time-stepping integrators in the presence of a discrete Lyapunov functional.