Decoupled synchronized states in networks of linearly coupled limit cycle oscillators

Networks of limit cycle oscillators can show intricate patterns of synchronization such as splay states and cluster synchronization. Here we analyze dynamical states that display a continuum of seemingly independent splay clusters. Each splay cluster is a block splay state consisting of sub-clusters of fully synchronized nodes with uniform amplitudes. Phases of nodes within a splay cluster are equally spaced, but nodes in different splay clusters have an arbitrary phase difference that can be fixed or evolve linearly in time. Such coexisting splay clusters form a decoupled state in that the dynamical equations become effectively decoupled between oscillators that can be physically coupled. We provide the conditions that allow the existence of particular decoupled states by using the eigendecomposition of the coupling matrix. Additionally, we provide an algorithm to search for admissible decoupled states using the external equitable partition and orbital partition considerations combined with symmetry groupoid formalism. Unlike previous studies, our approach is applicable when existence does not follow from symmetries alone and also illustrates the differences between adjacency and Laplacian coupling. We show that the decoupled state can be linearly stable for a substantial range of parameters using a simple eight-node cube network and its modifications as an example. We also demonstrate how the linear stability analysis of decoupled states can be simplified by taking into account the symmetries of the Jacobian matrix. Some network structures can support multiple decoupled patterns. To illustrate that, we show the variety of qualitatively different decoupled states that can arise on two-dimensional square and hexagonal lattices.Comment: 19 pages, 11 figure

Koopman Operator and its Approximations for Systems with Symmetries

Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, including complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and to simplify their analysis. The Koopman operator is an infinite dimensional linear operator that fully captures a system's nonlinear dynamics through the linear evolution of functions of the state space. Importantly, in contrast with local linearization, it preserves a system's global nonlinear features. We demonstrate how the presence of symmetries affects the Koopman operator structure and its spectral properties. In fact, we show that symmetry considerations can also simplify finding the Koopman operator approximations using the extended and kernel dynamic mode decomposition methods (EDMD and kernel DMD). Specifically, representation theory allows us to demonstrate that an isotypic component basis induces block diagonal structure in operator approximations, revealing hidden organization. Practically, if the data is symmetric, the EDMD and kernel DMD methods can be modified to give more efficient computation of the Koopman operator approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out the development, we discuss the effect of measurement noise

Exotic states in a simple network of nanoelectromechanical oscillators.

Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions

Structure and dynamics of networks with dyadic and higher order interactions

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