Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: Examples and numerical study

A system of $N$ rotators is investigated with a constraint given by the condition of vanishing sum of the cosines of the rotation angles. Equations of the dynamics are formulated and results of numerical simulation for the cases $N=3$, $4$, and $5$ are presented relating to the geodesic flows on a two-dimensional, three-dimensional, and four-dimensional manifold, respectively, in a compact region (due to the periodicity of the configuration space in angular variables). It is shown that a system of three rotators demonstrates chaos, characterized by one positive Lyapunov exponent, and for systems of four and five elements there are, respectively, two and three positive exponents (“hyperchaos”). An algorithm has been implemented that allows calculating the sectional curvature of a manifold in the course of numerical simulation of the dynamics at points of a trajectory. In the case of $N=3$, curvature of the two-dimensional manifold is negative (except for a finite number of points where it is zero), and Anosov's geodesic flow is realized. For $N=4$ and $5$, the computations show that the condition of negative sectional curvature is not fulfilled. Also the methodology is explained and applied for testing hyperbolicity based on numerical analysis of the angles between the subspaces of small perturbation vectors; in the case of $N=3$, the hyperbolicity is confirmed, and for $N=4$ and $5$ the hyperbolicity does not take place.