Search for chaotic output from a sigma-delta modulator: an archetypal retarded nonlinear system

One of the most common methods by which a non-linear physical system can be determined to be stable, globally unstable, or in fact chaotic is by an investigation of its Lyapunov exponents. When the system contains a delay equation the correct co-ordinate space for the system is infinite dimensional. In this case special techniques must be used to make the problem solvable by numerical methods. We have investigated a practical example of such a retarded non-linear system: the continuous Sigma-Delta modulator (CSDM). The CSDM is an adaptation of the widely used digital Sigma-Delta modulator in which, now, the input and output are continuous functions of time, the quantizer has been replaced with a sharp hyperbolic tangent function and the feedback delay time can be varied continuously. This allows the system dynamics to be modeled by delay ordinary-differential equations. We have simulated such a device for various feedback configurations and delay times and have been able to show how a full calculation of the Lyapunov exponent spectrum allows a detailed analysis of stability conditions for the CSDM. Furthermore, we discuss the implications of this approach for retarded non-linear systems in general and a range of signal processing applications in particular.