Dynamics of the chain of forced oscillators with long-range interaction: From synchronization to chaos

We consider a chain of nonlinear oscillators with long-range interaction of the type $1∕l1+α$⁠, where $l$ is a distance between oscillators and $0<α<2$⁠. In the continuous limit, the system’s dynamics is described by a fractional generalization of the Ginzburg-Landau equation with complex coefficients. Such a system has a new parameter $α$ that is responsible for the complexity of the medium and that strongly influences possible regimes of the dynamics, especially near $α=2$ and $α=1$⁠. We study different spatiotemporal patterns of the dynamics depending on $α$ and show transitions from synchronization of the motion to broad-spectrum oscillations and to chaos.