Jacobi Stability Analysis and the Onset of Chaos in a Two-Degree-of-Freedom Mechanical System

In this paper, the Jacobi stability of a two-degree-of-freedom mechanical system is studied by the innovative application of KCC-theory, namely differential geometric methods. We discuss the Jacobi stability of two equilibria and a periodic orbit by constructing geometric invariants. Both the regions of Jacobi stability and Lyapunov stability are presented to show the difference. We draw the phase portraits of the deviation vector near two equilibria under specific parameter values and initial conditions, and point out the sensitivity of deviation vector to initial conditions. In addition, the corresponding instability exponent and curvature are applicable for predicting the onset of chaos, which help us to detect chaotic behaviors quantitatively.

Automated summary

In this paper, the Jacobi stability of a two-degree-of-freedom mechanical system is investigated using differential geometric methods. The construction of geometric invariants allows for a discussion on the stability of two equilibria and a periodic orbit. The phase portraits of deviation vectors near equilibria, along with the sensitivity of these vectors to initial conditions, are illustrated to predict chaos onset quantitatively.