Euler's problem, Euler's method, and the standard map; or, the discrete charm of buckling
type: Item
instance of: scholarly article
title: Euler's problem, Euler's method, and the standard map; or, the discrete charm of buckling
zbMATH Open document ID: 0797.34041
DOI: 10.1007/BF02429861
author: Gábor Domokos
author: Philip Holmes
published in: Journal of Nonlinear Science
publication date: 1994-10-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 563607
zbMATH Keywords: Euler buckling problem
zbMATH Keywords: boundary value problem
zbMATH Keywords: bifurcation structure
zbMATH Keywords: Hamiltonian dynamical system
zbMATH Keywords: initial value problem
zbMATH Keywords: semi-implicit Euler method
zbMATH Keywords: symplectic integrator
zbMATH Keywords: two-dimensional standard map
zbMATH Keywords: discrete model
zbMATH Keywords: symmetries
reviewed by: Bodo Werner
describes a project that uses: DSTool
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 563607