Calculus of variations for functionals containing compositions

Discrete time dynamical systems generated by the iteration of nonlinear maps provide simple and interesting examples of chaotic systems. But what is the physical principle behind the emergence of these maps? In this note we present an approach to this problem by extremizing a Hamiltonian functional defined on spaces of chaotic functions and their invariant measures. We derive a generalized Euler–Lagrange equation that contains a new term involving inverse images of the extremizing map. A number of examples are presented.  2004 Elsevier Inc. All rights reserved