UC Davis

The interplay between the structure of a networked system and the dynamics of its constituent elements, including their interactions, leads to non-trivial emergent behaviors. These behaviors can be essential to the function of the system as a whole, and thus mathematical frameworks that allow careful analysis of such behaviors are valuable. The contribution of this dissertation is to use the ideas of symmetries and balanced equivalence relations to show how network structure can be used to find the admissible patterns of synchronization (e.g., cluster synchronization states), track their dynamics, perform their linear stability analysis, and finally to retrieve the dynamics from data. An additional contribution is extending these principles to the analysis of cluster synchronization on hypergraphs. Hypergraphs allow capturing higher order interactions beyond the pairwise interactions captured in strictly dyadic network systems.

Symmetries are ubiquitous in nature. A networked dynamical system can have symmetries in its coupling structure, the dynamics of its constituent elements, or both. The first contribution of this dissertation is demonstrating how such symmetries present themselves in the structure and spectral properties of the Koopman operator, which is a linear infinite dimensional operator that exactly reproduces the dynamics of the system in the space of observables. This can be put into practice as the Koopman operator can be approximated via data driven methods. We demonstrate how the knowledge of the symmetries can be incorporated into such approximations to speed up the analysis and make it more accurate.

Cluster synchronization is a type of synchronization that is characterized by a subset of nodes in the system having fully synchronized trajectories (i.e., forming a cluster), while following a distinct trajectory from all the other clusters. Such behavior arises when all the nodes in the same cluster receive the same dynamical input from all the other nodes in the system. Therefore, symmetries as well as equitable partitions are useful tools to find the admissible cluster synchronization states for a given system. The second contribution of this dissertation is generalizing the results related to cluster synchronization in systems of coupled oscillators to study intricate patterns of synchronization, such as the family of states where cluster synchronization and splay states coexist. In such states, due to the interaction between the nodal dynamics and network structure, groups of oscillators become effectively decoupled despite the existence of physical coupling between them.

Networks capture pairwise (dyadic) interactions between elements, yet some systems are inherently higher order. For instance, a 3-species chemical reaction or a publication with three coauthors involve triadic interactions. The final contribution of this dissertation is to advancing the methodology of studying dynamics on systems with higher order interactions (e.g., triadic). Since such interactions can not be represented as a sum of dyadic interactions, their analysis requires new tools. Up to now, full synchronization on hypergraphs (which encode higher order interactions) has been the main focus of in the literature, since the dyadic projection of the adjacency tensor can be sufficient in stability calculations. We show that this approach is not sufficient for more intricate dynamics such as cluster synchronization with respect to determining admissible states and performing stability calculations. To address that, we introduce a formalism based on node and edge clusters, and demonstrate how to apply it to admissibility and stability analysis. This formalism provides a principled way to organize the analysis of dynamics on hypergraphs and serves as a tool to investigate the role of higher order interactions in stabilizing and destabilizing different states.