Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

This paper deals with forced vibrations of two-DOF systems with more than one equilibrium positions. Such systems may be obtained by digitization of elastic post-buckling systems. A vibration mode, which is periodic at small force amplitudes and becomes chaotic as the force amplitudes are slowly increased, is selected. It is possible to formulate and solve the problem of stability of a periodic or chaotic vibration mode in a space with greater dimension using the classical Lyapunov stability definition and some calculating procedures. Instability of phase trajectories is used as a criterion of the chaotic behavior in dynamical systems. Trajectories with very close initial values are compared. Use of the Lyapunov stability definition shows mutual stability/instability of the trajectories. Calculations permit to observe an appearance and enlargement of the chaotic behavior regions. Specific results are obtained for the nonautonomous Duffing equation and pendulum system.