Simulation of coupled nonlinear time-delayed feedback loops using state-space representation Author: Karl Schmitt1,2 krbschmitt_at_math.umd.edu Advisors: Jim Yorke1,2,3 yorke_at_umd.edu Rajarshi Roy 4,2,3 lasynch_at_gmail.com Tom Murphy5,3 tem_at_umd.edu 1 Department of Mathematics, University of Maryland 2 Institute for Research in Energy and Applied Physics (IREAP) 3 Institute for Physics and Science Technology (IPST) 4 Department of Physics, University of Maryland 5 Department of Electrical and Computer Engineering, University of Maryland Abstract: A discrete time model for single and coupled nonlinear time-delayed Mach-Zehnder feedback loops is developed and validated. The model is used to investigate synchronization regimes related to feedback strength, time delay and optical biasing for mutually coupled loops. A comparison between simulation and experimental results is presented for synchronization regimes, and the model is used to identify parameter mismatches in the experimental system. Identification of mismatches and appropriate adjustment of synchronization regimes is discussed in context of applications to secure communications and sensor networks.Introduction The simulation of accurate models for experimental systems is vital to determining future research and validating existing research. In particular, modeling a system of coupled nonlinear time-delayed feedback loops provides a number of unique avenues for numerical investigation. Such systems are under heavy study in efforts to develop new methods of secure communication and sensing. Of particular interest to this field is the nonlinear system involving Mach-Zehnder interferometers as all of its components are readily available commercially. Published models for this system have been developed using continuous time, and model reasonably the behavior of the system. In our research however, development of the system has progressed in a manner that suggests implementing a physically discrete system. This progression motivates a change in the models for the system as well. The model for each independent loop will be a discrete time, state-space representation of the loop in Kouomuo [1] and will be tested against published results for identical systems. Coupling schemes will be initially tested on well known and previously explored systems such as the Lorenz model [2]. The final implementation will be tested against published experimental data for such a coupled system [3,4]. Since research into implementation as either a communication or sensor device has shown a significant dependence on the accuracy of synchronization which is further dependent on the quality of parameter matching, this will explored in detail. We will identify via simulation existing parameter mismatching in an experimental system. Background The nonlinear time-delayed feedback system explored in detail by Kouomuo [1] is comprised of a laser, Mach-Zehnder inferometer, filtering, delay and amplification. Through basic mathematical representations for each of these components one can form a model for the evolution of the system in terms of a time-delayed integro-differential equation as defined in Kouomuo: ∫+−=++ttoTtxdssxtxdtdtx ])([cos)(1)()(2φβθτ Here x(t) is a dimensionless variable with parameters of the normalized feedback gain β, the normalized bias offset φ, the high cut-off filter time constant τ and the low cut-off filter time constant θ.The generally established method for solving these equations would be traditional numerical methods such as RK4. However, one can examine the initial situation and formulate these equations using a completely different approach (presented below). Having established a basic nonlinear system, we can now examine more complicated behavior. It has been observed both in natural systems and mathematical models that two nonlinear systems can achieve a synchronous state when coupled in an appropriate manner. Understanding such systems may lead to better communication techniques, advanced medical procedures and a significant improvement in understanding certain biological systems [6, 9, 10]. With either formulation it is fairly easy to cast this in the form of many published pieces of work about coupling systems of nonlinear equations. What becomes interesting is examining the behavior of such coupled systems. In published work on the Lorenz system it has been demonstrated that two such coupled systems can be made to synchronize. This seems counter-intuitive to the concept of nonlinear (chaotic) systems and has therefore sparked a variety of research. Of specific relevance to this project is published experimental work which has demonstrated that given the correct setup it is possible to achieve synchronization between two Mach-Zehnder loops. Derivation of Alternative Model The approach taken by Kouomuo was to model the filters using single-pole low-pass and high-pass filters. An alternative approach is to formulate them in state-space. Then the filtering would look like: )()()()()()(tDxttytxtt+=+=CuBAuu Here x(t) represents the input to a filter, y(t) is the output from the filter and A, B, C and D are constant matrices related to the filter used. Furthermore this can be easily converted to a discrete map equation. This is highly appropriate if one is considering a discrete-time filter such as might be implemented on a digital signal processing board. Since the experimentalists related to this project have chosen to implement the system in this manner we will use the discrete versions as follows[5]: ][][][][][]1[nDxnnynxnn+=+=+CuBAuu Now we must include the concept of feedback. The simplest approach would be just a direct feedback where x[n]=y[n]. This, however, does not allow any dynamics besides the filter response to occur. Therefore we also include some function applied to the output of the filter, thus you could imagine something like: ])[(][ knyfnx−= Here is included the fact that we are using time-delayed feedback as represented by the argument [n-k]. This gives rise to a state-space representation that looks like: ])[(][][])[(][]1[knyDfnnyknyfnn−+=−+=+CuBAuu By carefully choosing our state-space to be the canonical form derived from the z-transform of the discrete time filters we are interested in modeling, we can rewrite the topequation in terms of only the state-vector u, and generate our output at a later iteration via the simplified second equation.