Jacobi Stability and Restoration of Parameters of the Nonlinear Double Pendulum

Abstract

The Jacobi stability analysis of the nonlinear dynamical system on base of Kosambi–Cartan–Chern theory is considered. Geometric description of time evolution of the system is introduced, that makes it possible to determine five geometric invariants. Eigenvalues of the second invariant (the deviation curvature tensor) give an estimate of Jacobi stability of the system. This approach is relevant in applications where it is required to identify the areas of Lyapunov and Jacobi stability simultaneously. For the nonlinear system – the double pendulum – the dependence of the Jacobi stability on initial conditions is investigated. The components of the deviation curvature tensor corresponding to the initial conditions and the eigenvalues of the tensor are defined explicitly. The boundary of the deterministic system transition from regular motion to chaotic one determined by the initial conditions has been found. The formulation of the inverse eigenvalue problem for the deviation curvature tensor associated with the restoration of significant parameters of the system is proposed. The solution of the formulated inverse problem has been obtained with the use of optimization approach. Numerical examples of restoring the system parameters for cases of its regular and chaotic behavior are given.