On the stability of periodic orbits of two-dimensional mappings

Abstract

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.