Jacobi stability analysis and restoring parameters of the damped double pendulum

Abstract

Jacobi stability for a double damped pendulum was analyzed on the basis of the Kosambi–Cartan–Chern theory making it possible to determine geometric structures and five geometric invariants of the dynamical system. Eigenvalues of the second invariant (deviation curvature tensor) provided the estimate of the Jacobi stability related to the insensitivity measure against disturbances of the system itself and the environment. Such studies are relevant in applications, where it is required to determine the system stability regions according to Lyapunov and Jacobi simultaneously. Inverse problem of restoring the system parameters from indirect information represented by the eigenvalues of the deviation curvature tensor was formulated. For a double pendulum with damping, the Jacobi stability conditions were justified in terms of its free parameters. Solution to the inverse problem of restoring the pendulum parameters was obtained using the optimization approach. When minimizing the regularized criterion function, a new hybrid global optimization algorithm was applied. Numerical example is provided.