Lorenz’s equations determine x, y and z as functions of time and are used as a classic example of chaotic behavior.
These equations yield functions which satisfy the following equation:
This suggests that if we define and ”energies” and , we see that the total energy is conserved (with value zero) over the time evolution of the system.
This also suggests the definition of a utility function which represents the common value of and for each moment in time.
Then we have
and the utility function then is seen to determine , and .