We present a closed form stability criterion for the periodic orbits of two-dimensional conservative as well as "dissipative" mappings which are analogous to the Poincaré maps of dynamical systems. Our stability criterion has a particularly simple form involving a finite, symmetric, nearly tridiagonal determinant. Its derivation is based on an extension of the stability analysis of Hill's differential equation to difference equations. We apply our criterion and derive a sufficient stability condition for a large class of periodic orbits of the widely studied "standard mapping" describing a periodically "kicked" free rotator. As another example we also obtain explicitly and in closed form the intervals of bounded (and unbounded) solutions of a discrete "Schrödinger equation" for the Kronig and Penney crystal model.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics