Periodic points of area - preserving mappings have been bread - and - butter to
dynamical systems theory since the end of the 19th century, when Henri Poincare first
applied them to a problem about the motion of three bodies under gravity.
In fact, the
theory of differentiable
dynamical systems — as we know and love it from the work of G.D. Birkhoff, J. Hadamard, H. PoincarĂ©, and, more recently, E.N. Lorenz, D. Ruelle, and S. Smale, among many others —
applies to autonomous
systems, in which neither the forcing nor the coefficients depend explicitly on time.