Measures of Order in Dynamic Systems

The usefulness of the Lempel-Ziv complexity and the Lyapanov exponent as two metrics to characterize the dynamic patterns is studied. System output signal is mapped to a binary string and the complexity measure of the time-sequence is computed. Along with the complexity we use the Lyapunov exponent to evaluate the order and disorder in the nonlinear systems. Results from the Lempel-Ziv complexity are compared with the results from the Lyapunov exponent computation. Using these two metrics, we can distinguish the noise from chaos and order. In addition, using same metrics we study the complexity measure of the Fibonacci map as a quasiperiodic system. Our analytical and numerical results prove that for a system like Fibonacci map, complexity grows logarithmically with the evolutionary length of the data block. We conclude that the normalized Lempel-Ziv complexity measure can be used as a system classifier. This quantity turns out to be 1 for random sequences and non-zero value less than 1 for chaotic sequences. While for periodic and quasiperiodic responses as data string grows, their normalized complexity approaches zero. However, higher deceasing rate is observed for periodic responses.