Samenvatting
In this thesis, "Network dynamics of delay coupled nonlinear oscillators for the study of lasers", we programmed software as well as theoretically, numerically and experimentally explored the effects of delays on coupled nonlinear systems.Firstly to achieve appropriate numerical calculations we proceeded by developing a computer program capable of solving large and highly connected networks. Our main intention was to create software capable of generating extremely exotic stuctures ranging from single nodes with feedback, to completely random networds. We therefore developed a user-friendly interface to construct and simulate coupled networks. We paid special attention to the efficiency of computations as well as to the optimization of the computer's working memory. Simulated data has the possibility to be stored on a hard drive. The saved file is fully compatible with Mathematica. The type of simulatons that this program can perform includes solving trajectories, bifurcation diagrams, computing auto- and cross-correlations as well as power spectrum calculations. These simulations can be set to repeat recursively in order to scan parameter spaces and delays. In this thesis all numerical calculations have been executed by this programmed numerical tool.
Secondly we explored an autoregressive model applied to a linear map in order to predict the network behavior and to understand it. This analysis shoved us that we could express the autocorrelation and cross-correlation functions of so called "circulant graphs" through the autocorrelation function of single nodes with feedback. Our objective was to numerically check our analytical results obtained for linear stochastic maps through our software. We mimited our verifications to bidirectional network structures. We therefore focused on two generic delayed differential equations, namely the Mackey-Glass equation and Stuart-Landau equation. Both of these time-evolving equations have been linked to past and ongoing research involving biological, financial and optical applications i.e. Lasers. We equally paid attention to key geometrical substructures influencing the dynamical behavior of symmetric networks. Along the way, we also took an interest in the sychronization phenomena of several networds. On several occasions we were confronted with situations where on-off intermittencey occurred.
Thirdly we thought of a realistic set-up, based on semiconductor ring lasers (SRLs), in order to observe through experimentation the effect of delays on nonlinear systems. SRLs are very intersting for several applications and have been researched extensively in the pst by many academics. Despite these efforts very little studies have looked at the behavior of delayed coupled ring lasers. To predict the dynamics of such systems we have modified an existing SRL model to include couplings and delays. Subsequently we performed numerical sumulations and compared them with our experimental results.
We finalized our work with a discussion and outlook of our results. In the appendix we introduce, describe and propose a delay model applicable to financial markets.
| Datum prijs | 6 jun 2012 |
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| Originele taal | English |
| Begeleider | Jan Danckaert (Promotor) & Guy Verschaffelt (Promotor) |