Transitions to synchronization in adaptive multilayer networks with higher-order interactions

Real-world networks are often characterized by simultaneous interactions between multiple agents that adapt themselves due to feedback from the environment. In this article, we investigate the dynamics of an adaptive multilayer network of Kuramoto oscillators with higher-order interactions. The dynamics of the nodes within the layers are adaptively controlled through the global synchronization order parameter with the adaptations present alongside both pairwise and higher-order interactions. We first explore the dynamics with a linear form of the adaptation function and discover a tiered transition to synchronization, along with continuous and abrupt routes to synchronization. Multiple routes to synchronization are also observed due to the presence of multiple stable states. We investigate the bifurcations behind these routes and illustrate the basin of attraction to attain a deeper understanding of the multistability that is born as a consequence of the adaptive interactions. When nonlinear adaptation is infused in the system, we observe three different kinds of tiered transition to synchronization, viz., continuous tiered, discontinuous tiered, and tiered transition with a hysteretic region. Our study provides an overview of how inducting order parameter adaptations in higher-order multilayer networks can influence dynamics and alter the route to synchronization in dynamical systems.