Advertisement

Numerical Methods for the Analysis of Dynamics and Synchronization of Stochastic Nonlinear Systems

  • How-Foo Chen
  • Jia-Ming Liu
Part of the Institute for Nonlinear Science book series (INLS)

Summary

The most important numerical tools needed in the analysis of chaotic systems performing chaos synchronization and chaotic communications are discussed in this chapter. Basic concepts, theoretical framework, and computer algorithms are reviewed. The subjects covered include the concepts and numerical simulations of stochastic nonlinear systems, the complexity of a chaotic attractor measured by Lyapunov exponents and correlation dimension, the robustness of synchronization measured by the transverse Lyapunov exponents in parameter-matched systems and parameter-mismatched systems, the quality of synchronization measured by the correlation coefficient and the synchronization error, and the treatment of channel noise for quantifying the performance of a chaotic communication system. For a dynamical system described by stochastic differential equations, the integral of a stochastic term is very different from that of a deterministic term. The difference and connection between two different stochastic integrals in the Ito and Stratonovich senses, respectively, are discussed. Numerical algorithms for the simulation of stochastic differential equations are developed. Two quantitative measures, namely, the Lyapunov exponents and the correlation dimension, for a chaotic attractor are discussed. Numerical methods for calculating these parameters are outlined. The robustness of synchronization is measured by the transverse Lyapunov exponents. Because perfect parameter matching between a transmitter and a receiver is generally not possible in a real system, a new concept of measuring the robustness of synchronization by comparing the unperturbed and perturbed receiver attractors is introduced for a system with parameter mismatch. For the examination of the quality of synchronization, the correlation coefficient and the synchronization error obtained by comparing the transmitter and the receiver outputs are used. The performance of a communication system is commonly measured by the bit-error rate as a function of signal-to-noise ratio. In addition to the noise in the transmitter and the receiver, the noise of the communication channel has to be considered in evaluating the bit-error rate and signal-to-noise ratio of the system. An approach to integrating the linear and nonlinear effects of the channel noise into the system consistently is addressed. Optically injected single-mode semiconductor lasers are used as examples to demonstrate the use of these numerical tools.

Keywords

Lyapunov Exponent Correlation Dimension Semiconductor Laser Chaotic Attractor Synchronization Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. C. Chen and K. Yao, Stochastic-calculus-based numerical evaluation and performance analysis of chaotic communication systems, IEEE Trans. Circuits Syst. I, vol. 47, pp. 1663–1672, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    T. B. Simpson and J. M. Liu, Spontaneous emission, nonlinear optical coupling, and noise in laser diodes, Opt. Commun., vol. 112, pp. 43–47, 1994.CrossRefADSGoogle Scholar
  3. 3.
    J. M. Liu, C. Chang, T. B. Simpson, Amplitude noise enhancement caused by nonlinear interaction of spontaneous emission field in laser diodes, Opt. Commun., vol. 120, pp. 282–286, 1995.CrossRefADSGoogle Scholar
  4. 4.
    R. Mannella and V. Pallesche, Fast and precise algorithm for computer simulation of stochastic differential equations, Phys. Rev. A, vol. 40, pp. 3381–3386, 1989.CrossRefADSGoogle Scholar
  5. 5.
    R. L. Honeycutt, Stochastic Runge-Kutta algorithms I. White noise, Phys. Rev. A, vol. 45, pp. 600–603, 1992.CrossRefADSGoogle Scholar
  6. 6.
    E. Helfang, Numerical integration of stochastic differential equations, The Bell Syst. Tech. J., vol.58, pp. 2289–2298, 1979.Google Scholar
  7. 7.
    R. F. Fox, I. R. Gatland, R. Roy, and G. Vemuri, Fast, accurate algorithm for numerical simulation of exponentially corrected colored noise, Phys. Rev. A, vol. 38, pp. 5938–5940, 1988.CrossRefADSGoogle Scholar
  8. 8.
    C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Nature Sciences, 2nd ed. (Springer, New York, 2001).Google Scholar
  9. 9.
    S. Cyganowski, P. Koleden, and J. Ombach, From Elementary Probability to Stochastic Differential Equations with MAPLE (Springer, New York, 2001).zbMATHGoogle Scholar
  10. 10.
    V. S. Anishchenko, V. V. Astakhov, A. B. Neiman, T. E. Vadivasova, and L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems (Springer, New York, 2001).Google Scholar
  11. 11.
    J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, Modulation bandwidth, noise, and stability of a semiconductor laser subject to strong injection locking, IEEE Photon. Techno. Lett., vol. 9, pp. 1325–1327, 1997.CrossRefADSGoogle Scholar
  12. 12.
    T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, Period-doubling route to chaos in a semiconductor laser subject to optical injection, Appl. Phys. Lett., vol. 64, pp. 3539–3541, 1994.CrossRefADSGoogle Scholar
  13. 13.
    S. K. Hwang, J. B. Gao, and J. M. Liu, Noise-induced chaos in an optically injected semiconductor laser model, Phys. Rev. E, vol. 61, pp. 5162–5170, 2000.CrossRefADSGoogle Scholar
  14. 14.
    S. Tang, H. F. Chen, S. K. Hwang and J. M. Liu, Message encoding and decoding through chaos modulation in chaotic optical communications, IEEE Trans. on Circuits Syst. I, vol. 49, pp. 163–169, 2002.CrossRefGoogle Scholar
  15. 15.
    S. Haykin, Communication Systems, 3rd ed. (John Wiley & Sons, New York, 1994).Google Scholar
  16. 16.
    H. D. I. Abarbanel, R. Brown, and M. B. Kennel, Lyapunov exponents in chaotic systems: their importance and their evaluation using observed data, Int. J. of Modern Phys., vol. 5, pp. 1347–1375, 1991.zbMATHCrossRefADSGoogle Scholar
  17. 17.
    R. Brown, P. Bryant, and H. D. I. Abarbanel, Computing the Lyapunov spectrum of a dynamical system from an observed time series, Phys. Rev. A, vol. 43, pp. 2787–2806, 1991.MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    J. B. Gao, S. K. Hwang, and J. M. Liu, Effects of intrinsic spontaneous-emission noise on the nonlinear dynamics of an optically injected semiconductor laser, Phys. Rev. A, vol. 59, pp. 1582–1585, 1999.CrossRefADSGoogle Scholar
  19. 19.
    P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett., vol. 50, pp. 346–349, 1983.MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    J. Theiler, Estimating fractal dimension, J. Opt. Soc. Am. A, vol. 7, pp. 1055–1073, 1990.MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback, Opt. Commun., vol. 165, pp. 279–292, 1999.CrossRefADSGoogle Scholar
  22. 22.
    H. F. Chen and J. M. Liu, Open-loop chaotic synchronization of injectionlocked semiconductor lasers with gigahertz range modulation, IEEE J. Quantum Electron., vol. 36, pp. 27–34, 2000.CrossRefADSGoogle Scholar
  23. 23.
    L. Kocarev and U. Parlitz, General approach for chaotic synchronization with applications to communication, Phys. Rev. Lett., vol. 74, pp. 5028–5031, 1995.CrossRefADSGoogle Scholar
  24. 24.
    L. Kocarev and U. Parlitz, Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems, Phys. Rev. Lett., vol. 76, pp. 1816–1911, 1996.CrossRefADSGoogle Scholar
  25. 25.
    L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar, and J. F. Heagy, Fundamentals of synchronization in chaotic systems, concepts, and applications, Chaos, vol. 7, pp. 520–543, 1997.zbMATHMathSciNetCrossRefADSGoogle Scholar
  26. 26.
    D. E. Knuth, Seminumerical Algorithms, vol. 2 of The Art of Computer Programming, 3rd ed. (Addison-Wesley, Reading, MA), p. 122, 1997.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • How-Foo Chen
    • 1
  • Jia-Ming Liu
    • 1
  1. 1.Electrical Engineering DepartmentUniversity of California, Los AngelesLos Angeles

Personalised recommendations