Coupled dynamical systems are omnipresent in everyday life. In general, interactions between
individual elements composing the system are captured by complex networks. The latter
greatly impact the way coupled systems are functioning and evolving in time. An important
task in such a context, is to identify the most fragile components of a system in a fast and
efficient manner. It is also highly desirable to have bounds on the amplitude and duration
of perturbations that could potentially drive the system through a transition from one equi-
librium to another. A paradigmatic model of coupled dynamical system is that of oscillatory
networks. In these systems, a phenomenon known as synchronization where the individual
elements start to behave coherently may occur if couplings are strong enough. We propose
frameworks to assess vulnerabilities of such synchronous states to external perturbations. We
consider transient excursions for both small-signal response and larger perturbations that can
potentially drive the system out of its initial basin of attraction.
In the first part of this thesis, we investigate the robustness of complex network-coupled
oscillators. We consider transient excursions following external perturbations. For ensemble
averaged perturbations, quite remarkably we find that robustness of a network is given by
a family of network descriptors that we called generalized Kirchhoff indices and which are
defined from extensions of the resistance distance to arbitrary powers of the Laplacian matrix
of the system. These indices allow an efficient and accurate assessment of the overall vulnera-
bility of an oscillatory network and can be used to compare robustness of different networks.
Moreover, a network can be made more robust by minimizing its Kirchhoff indices. Then for
specific local perturbations, we show that local vulnerabilities are captured by generalized
resistance centralities also defined from extensions of the resistance distance. Most fragile
nodes are therefore identified as the least central according to resistance centralities. Based on
the latter, rankings of the nodes from most to least vulnerable can be established. In summary,
we find that both local vulnerabilities and global robustness are accurately evaluated with
resistance centralities and Kirchhoff indices. Moreover, the framework that we define is rather
general and may be useful to analyze other coupled dynamical systems.
In the second part, we focus on the effect of larger perturbations that eventually lead the sys-
tem to an escape from its initial basin of attraction. We consider coupled oscillators subjected
to noise with various amplitudes and correlation in time. To predict desynchronization and
transitions between synchronous states, we propose a simple heuristic criterion based on the
distance between the initial stable fixed point and the closest saddle point. Surprisingly, we
find numerically that our criterion leads to rather accurate estimates for the survival probability and first escape time. Our criterion is general and may be applied to other dynamical
systems.
EPFL2020
