Tendril Perversion in Intrinsically Curved Rods

Abstract

A straight elastic rod with intrinsic curvature under varying tension can undergo an instability and bifurcate to a filament made out of two helices with opposite handedness. This inversion of handedness, known as perversion, appears in a wide range of biological and physical systems and is investigated here within the framework of thin elastic rods described by the static Kirchhoff equations. In this context, a perversion is represented by a heteroclinic orbit joining asymptotically two fixed points representing helices with opposite torsion. A center manifold reduction and a normal form transformation for a triple zero eigenvalue reduce the dynamics to a third-order reversible dynamical system. The analysis of this reduced system reveals that the heteroclinic connection representing the physical solution results from the collapse of pairs of symmetric homoclinic orbits. Results of the normal form calculation are compared with numerical solutions obtained by continuation methods. The possibility of self-contact and the elastic characteristics of the perverted rod are also studied.