Euler's problem, Euler's method, and the standard map; or, the discrete charm of buckling
The starting problem is the classical Euler buckling problem of a simply supported rod of length \(L\) and bending stiffness \(EI\) loaded by an axial force \(P\), formulated as a boundary value problem (BVP) for the slope \(\alpha(s)\) depending on the arclength \(s\): \(EI \alpha''+ P\sin \alpha= 0\), \(\alpha'(0)= 0= \alpha'(L)\). The bifurcation structure with respect to variation of \(P\) is well-known and relatively simple: For \(P> P_ n:= n^ 2\pi^ 2 EI/L^ 2\) there exist \(2n+ 1\) equilibria possessing certain maximal symmetries. This elastic equilibrium problem as a BVP is analogous to the Hamiltonian dynamical system of a mathematical pendulum as an initial value problem (IVP): equilibria correspond with periodic solutions with periods dividing \(2L\). Using the semi-implicit Euler method with time-step \(l\) as a symplectic integrator for the first order system \(\alpha'=- {P\over EI} y\), \(y'=\sin\alpha\), an area-preserving map is obtained which can be transformed into the two-dimensional standard map \(\Theta_{n+1}= \Theta_ n+ I_{n+1}\), \(I_{n+1}= I_ n+ K\sin\Theta_ n\), where \(K\) (instead of \(P\)) is the bifurcation parameter. Requiring \(I_ 0= 0= I_ N\) with \(l= L/N\), a discrete BVP of the continuous model is obtained being an exact mechanical analogue in the form of a chain of rigid links coupled by linear torsional springs of stiffness \(EI/l\). The remarkable fact of this discrete model is that it possesses an extremely richer bifurcation structure for large \(P\) (or \(K\)) than the continuous model having a lot of ``parasitic'' solutions in addition to the primary branches. This is shown by exploiting the relation between the multiplicity of periodic and chaotic solutions of the standard map as an IVP and solutions of the discrete BVP. Special attention is paid to the underlying symmetries of the problems. Methods of dynamical system theory and mechanical analysis are used.
scientific article; zbMATH DE number 563607
scientific article