# 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.

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