Euler's problem,Euler's method,and the standard map; or,the discrete charm of buckling

Summary We explore the relation between the classical continuum model of Euler buckling and an iterated mapping which is not only a mathematical discretization of the former but also has an exact, discrete mechanical analogue. We show that the latter possesses great numbers of “parasitic” solutions in addition to the natural discretizations of classical buckling modes. We investigate this rich bifurcational structure using both mechanical analysis of the boundary value problem and dynamical studies of the initial value problem, which is the familiar standard map. We use this example to explore the links between discrete initial and boundary value problems and, more generally, to illustrate the complex relations among physical systems, continuum and discrete models and the analytical and numerical methods for their study.