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Title: Lyapunov exponents and extensivity in multiplex networks
Author: Araujo, Maria Angélica
ISNI:       0000 0004 8498 4368
Awarding Body: University of Aberdeen
Current Institution: University of Aberdeen
Date of Award: 2019
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In this thesis we present a comprehensive mathematical review about Lyapunov exponents (LEs) in both, discrete and continuous dynamical systems, clarifying all the intricate mathematical approaches required to make a correct calculation of LEs. Moreover, we bring two scenarios where the LEs can be used to understand the complex behaviour in a class of dynamical complex networks with constant Jacobian. In the first scenario, we explore the relationship between the LEs and the atypical set of LEs, called conditional Lyapunov exponents (CLEs), that are the LEs of the synchronization manifold and its tranversal directions. The CLEs are tremendously important for technological systems, since they provide the stability of the synchronous state, important for the correct functioning of several man made systems. Since the CLEs are easier to be calculated, it is interesting to find relationships between them and the usual asymptotic LEs of dynamical networks. Therefore, we present a class of dynamical networks for which these exponents are the same, allowing a trivial calculation. As another application for the LEs, we investigate, analytically and numerically, the relationship between the coupling strengths and the extensive behaviour of the sum of the positive Lyapunov exponents in multiplex networks formed by coupled dynamical units. We show which are the relevant parameters leading to extensivity, providing exact formulas about how they are related. We also demonstrate that it is always possible to construct infinitely large extensive networks by attaching, with rescaled inter-connections, infinitely many smaller networks with arbitrary topology.
Supervisor: Baptista, Murilo S. Sponsor: University of Aberdeen
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Lyapunov exponents ; System analysis