Modified Lyapunov exponent,new measure of dynamics

Lyapunov exponents, defined as exponential divergent or convergent rate of initially infinitely close solution trajectories, have been widely used for diagnosing chaotic systems, as well for stability analysis of nonlinear systems. Although calculated from the evolution of disturbance vectors associated with the flow, Lyapunov exponents are not associated with any specific directions, and such evolutions are driven by the dynamics in all directions in the state space. It is desirable to explore the asymptotic behaviors of the dynamic systems along certain specific directions and the specific dynamics driving such behaviors. In this paper, the Lyapunov exponents are modified. The modified Lyapunov exponents can indicate the exponential divergent or convergent rates in certain directions, which are driven by the dynamics in the same directions. The existence and the invariance to the initial conditions of the proposed modified exponents are proven mathematically. The algorithm for calculating the modified Lyapunov exponents from mathematical models is also developed. A wide range of case studies, from classical nonlinear dynamic systems to engineering systems, are presented to demonstrate the proposed modified Lyapunov exponents, and the indications of the modified exponents are also discussed. The proposed modified Lyapunov exponents can reveal additional insights into the system dynamics to the conventional Lyapunov exponents. Such information can be instrumental for stability control design.