DOI: 10.1007/s003320010008 J. Nonlinear Sci. Vol. 11: pp. 47–67 (2001) © 2001 Springer-Verlag New York Inc. Hidden Symmetry of Global Solutions in Twisted Elastic Rings G. Domokos 1,2 and T. Healey 2 1 Department of Strength of Materials, Technical University of Budapest, H-1521 Budapest, Hungary e-mail: domokos@ iit.bme.hu 2 Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853- 1503, USA Received March 26, 2000; accepted September 19, 2000 Online publication February 26, 2001 Communicated by Martin Golubitsky Summary. We investigate global equilibria of twisted, isotropic elastic rings. The high degree of symmetry in the problem leads to nonisolated solutions. We prove that all solutions are flip-symmetric, and from this we can globally isolate all solution branches. We derive possible other choices for boundary conditions systematically and show that other conditions necessarily lead to spurious solutions. We show global computations performed by the Parallel Simplex Algorithm. 1. Introduction In this paper we examine globally the static equilibria of twisted elastic rings. Such a ring is obtained by twisting a straight, uniform rod and connecting the two ends by bending (cf. Figure 1). In the case of circular cross section, this connection can be performed without violating continuity of the strains; the displacements (end rotations) suffer a jump. There is no external load acting on the closed ring, and the loading parameter for the boundary value problem (BVP) is either the relative angle (discontinuity) or the (constant) torque in the ring. In recent years, this problem for the Kirchhoff rod has been proposed and studied as a model for DNA [4], [17], [19], [21], [22]. One goal of the paper is to show that our problem has a rather large symmetry group, even with this displacement jump. We then prove rigorously for a relatively wide class of problems (including shearable and extensible materials) that all equilibria are invariant (modulo a group action) under a “hidden” π -flip symmetry, in addition to the almost-trivial n-fold discrete rotational symmetry C n , e.g., the case n = 3 is depicted in Figure 2. This result is a substantial generalization of a theorem in [7]. It is a rare phenomenon that all equilibria belong to the same fixed-point subspace. A partial analogy