Spatial chaos of buckled elastica by the Kirchhoff analogy of a gyrostat
The Kirchhoff analogy between the equilibrium of a 3D originally straight uniform rod and the dynamics of a spinning top is well known and used extensively. The rod is described by spatial coordinate and the spinning top is described by time coordinate. The extended Kirchhoff analogy between the spatial equilibrium of a 3D force-free buckled elastica and the temporal dynamics of the torque-free gyrostat is less well known and hardly used. The extended Kirchhoff analogy and the Melnikov integral are used to determine analytically the conditions for the possible onset of spatial chaos in the elastica by exploring the Hamiltonian structure of the rotational motion of a perturbed gyrostat. The analytical results are cross-checked by the seventheighth order RungeKutta algorithm to numerically integrate the governing equations of the 3D equilibrium of the elastica. Interesting spatial buckling patterns are depicted the first time. The elastica appears at different scales from microscopic chains of super coiling DNA structures to macroscopic rods/ropes/cables/satellite tethers. Apprehension of the complex deformations of the elastica under different load conditions is of both theoretical and practical interest. The simulation results show that there exist homoclinic/heteroclinic bifurcations to chaos in the equilibrium of the elastica under the appropriate load conditions, equivalently, boundary conditions.
chaos buckled elastica melnikovs integral homoclinic/heteroclinic bifurcations