This talk focuses on the phenomenon of synchronization in the context of high-dimensional dynamical systems, particularly those described by coupled oscillator models, with the couplings encompassing a wide range of different interactions. Primarily, the discussion covers two interaction
types: (a) multilayer and (b) higher-order. Multilayer interactions are inspired by the inherent layered
topology observed in many physical systems, where oscillators exhibit distinct interlayer and intralayer
interactions. Higher-order interactions (i.e., involving ≥ 3 state variables) are introduced to address
situations where purely pairwise interactions are unable to capture complex dynamical phenomena.
Specific instances of these interactions are examined in the context of the Kuramoto model of phase
oscillators, which is a paradigmatic model of synchronization and is renowned for its analytical convenience and wide applicability. By adopting mathematical tools like the Ott-Antonsen ansatz, we derive
analytical insights into the synchronization transitions in each system and validate the findings using
large-scale numerical simulations. Additionally, we also (i) develop a GPU-optimized solver for fast
and efficient simulations of large-scale dynamical systems, and (ii) investigate the impacts of altering
network connectivity in a discrete dynamical system.