The high-dimensional property of delay differential equations makes them useful for various purposes. The applications of systems modelled with delay differential equations demand different degrees of complexity. One solution to tune this property is to make the dynamics of the current state dependent on more delayed states. How the system responds to more delayed states depends on the system under study, as both decreases and increases in the complexity were observed in different nonlinear systems. However, it is also known that when there is an infinite number of delays that follow a continuous distribution, simpler dynamics usually expected due to the averaging over previous states that the delay kernel provides. The present thesis investigates the role of multiple delays in nonlinear time delay systems, as well as methods for evaluating their complexity. Through the use of pseudospectral differentiation, we first compute the Lyapunov exponents of such multi-delay systems. In systems with a large number of delays, chaos is found to be less likely to occur. However, in systems with oscillatory feedback functions, the entropy can increase just by adding a few delays. Our study also demonstrates that the transition to simpler dynamics in nonlinear delay systems can be either monotonous or abrupt. This is particularly true in first-order nonlinear systems, where increasing the width of the distribution of delays results in complexity collapse, even in the presence of a few discrete delays. The roots of the characteristic equation around a fixed point can be used to approximate the degree of complexity of the dynamics of such time-delay systems, as they can be linked to other dynamical invariants such as the Kolmogorov-Sinai entropy.
The phenomenon of complexity collapse uncovered in our work was further studied in an 80/20 ratio excitatory-inhibitory neural network. We found that the smaller the time delay, the higher the likelihood of chaotic dynamics, and this also promotes asynchronous spiking activity. But for larger values of the delay, the neurons show synchronized oscillatory spiking activity. A global inhibition at a longer delay results in a novel dynamical pattern of randomly occurring epochs of synchrony within the chaotic dynamics.
The final part of the thesis examines the behavior of time delay reservoir computing when there are multiple time delays. It is shown that the choice of spacing between time delays is crucial, and depends on the task at hand. The system was studied for a prediction task with one chaotic input as well as for a mixture of two chaotic inputs. It was found that, similar to the single delay case, there is a resonance when the difference between delays is equal to the clock cycle. Together, our research provides valuable insights into the dynamics and complexity of nonlinear multi-delay systems and the importance of the spacing between delays.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/45569 |
| Date | 23 October 2023 |
| Creators | Tavakoli, Kamyar |
| Contributors | Longtin, André |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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