Spatially complex localization in twisted elastic rods constrained to an elliptic cylinder

In molecular biology, the elastic rod is wound on the elliptic cylinder can be used as a simplified model of the DNA and protein molecules, in which the elliptic cylinder is used to simulate the heterogeneity of the protein molecules. The problem can be described as a long thin weightless rod constrained, by suitable distributed forces, to lie on an elliptic cylinder while being held by end tension and twisting moment. The Cosserat director theory is used to formulate this problem. More complicated shaped are possible and special attention is given to localized configurations described by homoclinic and heteroclinic orbits of the oscillator. The analytical results of homoclinic and heteroclinic boundary conditions are obtained from Padé approximation. By using the analytical results of homoclinic and heteroclinic solution, both internal force and 3D configuration are discussed in detail. The results of analytical and numerical integration are compared to verify the effectiveness and feasibility of the analytical method.