Energy dissipation and dynamical complexity in a truncated two-dimensional Navier–Stokes dynamics

The five-mode truncated version of the planar incompressible Navier–Stokes equations developed by Franceschini and co-authors is analyzed from the standpoint of irreversible thermodynamics. The entropy production is expressed in terms of the model variables and its dependence on the control parameter (driving force) is determined. Its variability patterns as the system runs over the periodic and the chaotic attractors encountered at different parameter values are subsequently compared with various indicators of the intrinsic complexity of the dynamics, such as the local Lyapunov exponents and a local generalization of the Kolmogorov entropy. The analysis reveals some unexpected correlations. In particular, it appears that the largest span of values of entropy production occurs during stages dominated by stability, while the maximum value occurs during stages dominated by instability.