Approximation of long time statistical properties of large dissipative chaotic dynamical systems

It is well-known that physical laws for large chaotic systems are revealed statistically. We consider temporal and spatial approximations of stationary statistical properties of dissipative chaotic dynamical systems. We demonstrate that appropriate temporal/spatial discretization viewed as discrete dynamical system is able to capture asymptotically the stationary statistical properties of the underlying continuous dynamical system provided that appropriate Lax type criteria are satisfied.
We also show a general framework on when the long-time statistics of the system can be well-approximated by BDF2 based schemes.
Application to the infinite Prandtl number model for convection as well as the two-dimensional barotropic quasi-geostrophic equations will be discussed.