This talk will highlight some new phenomena and analysis techniques
for networks of oscillators. First are decoupled states, where the
interactions of the nodal dynamics and the network topology give rise
to intricate patterns of synchronization resulting in physically
coupled oscillators appearing to act independently of one another.
Next, the situation where oscillators may be coupled via higher-order
interactions (e.g., triadic, quadratic) is considered. The focus is on
cluster synchronization on hypergraphs. We show how the projection of
the hypergraph onto dyadic matrices, which is sufficient for analyzing
full synchronization, fails for cluster synchronization. We develop
instead a framework based on node and edge partitions that enables us
to represent the dynamics and simplify stability calculations taking
advantage of recent advances in simultaneous block
diagonalization. Finally, we consider the fact that additional
dynamics, beyond just synchronization, can be happening on networks of
oscillators. For instance, both power grids and brain networks consist
of networks of oscillators upon which load arrives and where cascades
of load shedding are observed. Often the sizes of the cascades follow
a power-law degree distribution and the dynamics are modeled using the
Bak-Tang-Weisenfeld (BTW) model of self-organized criticality,
although this neglects the oscillatory nature of the nodes. We couple
the seminal BTW model to the seminal model of oscillators, the
Kuramoto model, and show how this can lead to a long-time oscillatory
behavior at an emergent timescale. The bulk of the cycle is spent in a
build-up phase where oscillators are fully synchronized and cascades
largely avoided, while overall load increases. Then the system reaches
a tipping point where a large cascade triggers a "cascade of larger
cascades" (a dragon king), after which there is a short transient
phase restoring full synchrony. We show that the interplay of the
oscillatory and cascading dynamics gives rise to a self-amplifying
mechanism, which is not present in either model alone, leading to the
dragon king events.

This is joint work with Anastasiya Salova and Guram Mikaberidze.