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Euler’s problem, Euler’s method, and the standard map; or, the discrete charm of buckling. (English) Zbl 0797.34041

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 $\alpha \left(s\right)$ depending on the arclength $s$: $EI{\alpha }^{\text{'}\text{'}}+Psin\alpha =0$, ${\alpha }^{\text{'}}\left(0\right)=0={\alpha }^{\text{'}}\left(L\right)$. 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 ${\alpha }^{\text{'}}=-\frac{P}{EI}y$, ${y}^{\text{'}}=sin\alpha$, an area-preserving map is obtained which can be transformed into the two-dimensional standard map ${{\Theta }}_{n+1}={{\Theta }}_{n}+{I}_{n+1}$, ${I}_{n+1}={I}_{n}+Ksin{{\Theta }}_{n}$, 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.

##### MSC:
 34C23 Bifurcation (ODE) 37-99 Dynamic systems and ergodic theory (MSC2000)
DSTool
##### References:
 [1] Amick, C., Ching, E. S. C., Kadanoff, L. P., and Rom-Kedar, V. (1992) Beyond All Orders: Singular Perturbations in a Mapping.J. of Nonlinear Science 2:9–67. · Zbl 0872.58054 · doi:10.1007/BF02429851 [2] Antman, S. S., and Adler, C. L. (1987) Design of Material Properties that Yield a Prescribed Global Buckling Response.J. Applied Mech 109: 263–268. · Zbl 0613.73045 · doi:10.1115/1.3173005 [3] Arnold, V. I. (1983) Geometrical Methods in the Theory of Ordinary Differential Equations. New York, Berlin, Heidelberg: Springer. (Grundlehren der Math. Wiss.250). [4] Aubry, S. (1983) The Twist Map, the Extended Frenkel-Kontorova Model and the Devil’s Staircase.Physica D 7: 240–258. · doi:10.1016/0167-2789(83)90129-X [5] Babuška, I. (1990) The Problem of Modeling the Elastomechanics in Engineering.Computer Methods in Mechanics and Engineering 82: 155–182. · Zbl 0731.73094 · doi:10.1016/0045-7825(90)90163-G [6] Channel, P. J., and Scovel, C. (1990) Symplectic Integration of Hamiltonian Systems.Nonlinearity 3: 231–259. · Zbl 0704.65052 · doi:10.1088/0951-7715/3/2/001 [7] Chenciner, A. (1983) Bifurcations de difféomorphismes deR 2 au voinsinage d’un point fixe élliptique. Les Houches Summer School Proceedings, ed. R. Helleman, G. Iooss, North Holland. [8] Chirikov, B. V. (1979) A Universal Instability of Many-Dimensional Oscillator Systems.Phys. Reports 52:263–379. · doi:10.1016/0370-1573(79)90023-1 [9] Coxeter, H. S. M. (1969)Introduction to Geometry. New York, NY, Chichester, England: John Wiley and Sons. [10] Crandall, M. G., and Rabinowitz, P. H. (1970) Nonlinear Sturm-Liouville Eigenvalue Problems and Topological Degree.J. Math. Mech. 19:1083–1102. [11] De Vogelaére, R. (1956) Methods of Integration Which Preserve the Contact Transformation Property of the Hamiltonian Equations. Department of Mathematics, University of Notre Dame, report4. [12] Devaney, R. L. (1986)An Introduction to Chaotic Dynamical Systems. Menlo Park, CA: The Benjamin/Cummings Publishing Co., Inc. [13] Domokos G. (1991) Computer Experiments with Elastic Chains.Newsletter of the Technical University of Budapest 9(1): 14–26. [14] Domokos G. (1992) Secondary Bifurcations in the Euler Problem.Newsletter of the Technical University of Budapest 10(1): 4–11. [15] El Naschie, M. S. (1990) On the Suspectibility of Local Elastic Buckling to Chaos.ZAMM 70(12): 535–542. · Zbl 0729.73919 · doi:10.1002/zamm.19900701202 [16] Euler, L. (1744)Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Lausanne:Genf (German edition: Ostwald’s Klassiker der Exakten Wiss.75 Leipzig: W. Engelmann). [17] Fontich, E. and Simo, C. (1990) The Splitting of Separatrices for Analytic Diffeomorphisms.J. Ergod Theory and Dynamical Systems 10: 295–318. [18] Gáspár Zs., and Domokos, G. (1989) Global Investigation of Discrete Models of the Euler Buckling Problem.Acta Technica Acad. Sci. Hung. 102(3–4): 227–238. [19] Greene, J. M. (1979) Method for Determining Stochastic Transition.J. Math. Phys. 20(6): 1183–1201. · doi:10.1063/1.524170 [20] Guckenheimer, J., and Holmes, P. (1983)Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. New York, Berlin, Heidelberg (Appl. Math. Sci. 42.) [21] Guckenheimer, J., Myers, M. R., Wicklin, F. J., and Worfolk, P. A. (1991) dstool: A Dynamical System Toolkit with an Interactive Graphical Interface.Center for Applied Mathematics, Cornell University. [22] Hegedüs I. (1986) Analysis of lattice single layer cylindrical structures.J. of Space Structures 2: 87–89. [23] Holmes, P. (1982) The Dynamics of Repeated Impact with a Sinusoidally Vibrating Table.J. of Sound and Vibration 84(2): 173–189. [24] Holmes, P., and Williams, R. F. (1985) Knotted Periodic Orbits in Suspensions of Smale’s Horseshoe: Torus Knots and Bifurcation Sequences.Arch. Rat. Mech. Anal. 90(2): 115–194. · Zbl 0593.58027 · doi:10.1007/BF00250717 [25] Kirchhoff, G. (1859) Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes.J. für Math. (Crelle) 56: 285–313. [26] Lazutkin, V. F., Schachmannski, I. G., and Tabanov, M. B. (1989) Splitting of Separatrices for Standard and Semistandard Mappings.Physica D 40: 235–248. · Zbl 0825.58033 · doi:10.1016/0167-2789(89)90065-1 [27] Lichtenberg, A. J., and Lieberman, M. A. (1982)Regular and Stochastic Motion. New York, Berlin, Heidelberg: Springer. (Appl. Math. Sci.38). [28] Love, A. E. H. (1927)A Treatise on the Mathematical Theory of Elasticity. Dover Publications, N.Y. [29] Maddocks, J. H. (1984) Stability of Nonlinearly Elastic Rods.Arch. Rat. Mech. Anal. 85(4): 311–354. · Zbl 0545.73039 · doi:10.1007/BF00275737 [30] Maddocks, J. H. (1987) Stability and Folds.Arch. Rat. Mech. 99(4): 301–328. · doi:10.1007/BF00282049 [31] Marsden, J. E., O’Reilly, O., Wicklin, F. J., and Zombro, B. W. (1991) Symmetry, Stability, Geometric Phases and Mechanical Integrators.Nonlinear Sci. Today 1(1): 4–11,1(2): 14–21. [32] Melnikov, V. K. (1963) On the Stability of the Center for Time Periodic Perturbations.Trans. Moscow Math. Soc. 12: 1–57. [33] Meyer, K. (1970) Generic Bifurcation of Periodic Points.Trans. Ann. Math. Soc. 149: 95–107. · doi:10.1090/S0002-9947-1970-0259289-X [34] Meyer, K. (1971) Generic Stability Properties of Periodic Points.Trans. Ann. Math. Soc. 154: 273–277. · doi:10.1090/S0002-9947-1971-0271490-9 [35] Mielke, A., and Holmes, P. (1988) Spatially Complex Equilibria of Buckled Rods.Arch. Rat. Mech. 101(4): 319–348. · Zbl 0655.73029 · doi:10.1007/BF00251491 [36] Peitgen, H. O., Saupe, D., and Schmitt, K. (1981) Nonlinear Elliptic Boundary Value Problems Versus Finite Difference Approximations: Numerically Irrelevant Solutions.J. Reine u. Angew. Math. (Crelle) 322: 74–117. [37] Reinhall, P. G., Caughey, T. K., and Sorti, D. W. (1989) Order and Chaos in a Discrete Duffing Oscillator: Implications on Numerical Integration.J. Appl. Mech. 56(1): 162–167. · Zbl 0686.70009 · doi:10.1115/1.3176039 [38] Rózsa P. (1974)Linear Algebra and Applications. (In Hungarian:Lineáris algebra és alkalmazásai) Budapest: Müszaki Könyvkiadó. [39] Thompson, J. M. T., and Virgin, L. N. (1988) Spatial Chaos and Localization Phenomena.Physics Letters A 126(8–9): 491–496. · doi:10.1016/0375-9601(88)90045-X [40] Weinberger, H.F. (1974)Variational Methods for Eigenvalue Approximation. CBMS Conference Series 15,SIAM, Philadelphia.