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Research Papers

Measures of Order in Dynamic Systems

[+] Author and Article Information
Davoud Arasteh

Department of Electronic Engineering Technology,  Southern University, Baton Rouge, LA 70813davouda@engr.subr.edu

J. Comput. Nonlinear Dynam 3(3), 031002 (Apr 30, 2008) (10 pages) doi:10.1115/1.2908174 History: Received September 17, 2006; Revised October 17, 2007; Published April 30, 2008

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.

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Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

LZC analysis flowchart for sequence S of length N

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Figure 2

Exponential divergence of two nearby trajectories for a dynamical flow δX(t)=δX(0)2(tλ)

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Figure 3

Largest Lyapunov exponent as a function of control parameter a

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Figure 4

Henon bifurcation diagram versus control parameter a

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Figure 5

The LZC of strings constructed from the Henon map as a function of control parameter a

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Figure 6

Stretching and contraction along the principal axis of initial states volume element as time increases

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Figure 7

Forced-dissipative oscillator strange attractor and its Poincaré map for control parameters: Γ=3.8, k=0.5, ωd=2∕3

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Figure 8

Forced-dissipative oscillator chaotic attractor velocity distribution function; control parameters: Γ=3.8, k=0.5, ωd=2∕3

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Figure 9

Forced-dissipative-oscillator LZC measure; Control parameters: k=0.5, ωd=2∕3, 0.9≤Γ≤1.5, step size ΔΓ=0.001

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Figure 10

Forced-dissipative-oscillator bifurcation diagram; control parameters: k=0.5, ωd=2∕3, 0.9≤Γ≤1.5, step size ΔΓ=0.001

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Figure 11

Forced-dissipative-oscillator: (a) first Lyapunov exponent; (b) Lyapunov exponent spectrum. Control parameters: k=0.5, ωd=2∕3, 0.9≤Γ≤1.5, step size ΔΓ=0.001

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Figure 12

Plot of Fibonacci map LZC versus block length fn

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Figure 13

Comparison of asymptotic normalized LZC

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