Value and cost functions of dynamical systems. This figure shows the value and cost functions of the Lorentz attractor used in the previous figures. These functions always exist for any global random attractor because value (negative surprise) is the log density of the eigensolution of the systems Fokker-Planck operator. This means, given any deterministic motion (flow) and the amplitude of random fluctuations (diffusion), we can compute the Fokker Planck operator and its eigensolution and thereby define value . Having defined value, cost is just the expected rate of change of value, which is given by the deterministic flow and diffusion (see (23)). In this example, we computed the eigensolution or ergodic density using a discretisation of state-space into 96 bins over the ranges: and a diffusion tensor of . The upper panels show the resulting value and (negative) cost functions for a slice through state-space at . Note how cost takes large values when the trajectory (red line) passes through large value gradients. The lower left panel shows the resulting ergodic density as a maximum intensity projection over the third state. A segment of the trajectory producing this density is shown on the lower right.
