THE INSTANTANEOUS LYAPUNOV EXPONENT AND ITS APPLICATION TO CHAOTIC DYNAMICAL SYSTEMS

Abstract

Any system containing at least one positive Lyapunov exponent is defined to be chaotic and the system dynamics become unpredictable. For a mechanical system, the sum of Lyapunov exponents is negative and related to the damping, and so can be utilised to monitor any changes of the damping mechanism. However, in order to track any changes the data segment used to compute the Lyapunov exponents must be short. This leads to problems since Lyapunov exponents are calculated from a long term averaged divergence rate. To overcome this problem, it is described how the sum of Instantaneous Lyapunov exponents is related to the generalised divergence of the flow and the damping of a mechanical system. Computer simulations using differential equations are implemented to demonstrate the significance of the sum of Instantaneous Lyapunov exponents. In practice, it may be difficult to obtain accurate Instantaneous Lyapunov exponents are introduced to overcome this. The sum of short term averaged Lyapunov exponents from a time series (from experimental data), and so short term averaged Lyapunov exponents is applied to experimental data to detect evolutionary changes in damping from high to low.