A minimum principle for chaotic dynamical systems

Discrete time dynamical systems generated by the iteration of nonlinear maps, such as the logistic map or the tent map, provide interesting examples of chaotic systems. But what is the physical principle behind the emergence of these maps? In the continuous time settings, differential equations of mechanics arise from the minimization of the energy function (Hamiltonian). However, there is no general physical principle for the discrete time analogue of differential equations, namely, maps. In this note, we present an approach to this problem. Using a natural definition of energy for chaotic systems, we minimize energy subject to the constraint that the observed dynamical system has a known entropy. We consider the case where the natural invariant measure is Lebesgue. Invoking the Euler–Lagrange equation, we derive a nonlinear second order differential equation whose solution is the chaotic map that minimizes energy.