Open chemical systems far from equilibrium may give rise to spatial (Turing) patterns, oscillations, and chaos. States displaying such phenomena generally coexist with other states, either stationary, or oscillatory, or chaotic, under the same boundary conditions. A local function is defined here as a sum of products of the differences between two such states of thermodynamic forces and flows that appear in an expression for entropy production. It is averaged over the volume of a system when the state under consideration is spatially nonuniform, and over time when oscillatory or chaotic states are involved. It is found that this function is always zero for systems with fixed boundary conditions and negative for other systems including continuous-flow, stirred tank reactors and continuously fed unstirred reactors. These results indicate that on average, all thermodynamic forces and flows never increase or decrease simultaneously between two states, placing some restrictions on the difference in entropy production between them becoming too large.