Dynamics of nonlocally coupled oscillators with time-delayed interactions

Abstract

Populations of coupled oscillators have been investigated as models of many physical, chemical, and biological systems. Interactions between the elements of the systems are generally time delayed due to the finite speed of signal transmission or signal processing. When the time delays are comparable to the characteristic time scale of the systems, they can affect the dynamics significantly. In this study, motivated by neuronal systems, we investigate the effects of time-delayed interactions on the dynamics of nonlocally coupled oscillators. We show that time delays can induce various patterns, and multistabilities of patterns and in-phase synchronous oscillations. In our models, we introduce a finite interaction radius $γ_0$ to incorporate the nonlocality of interactions. Each oscillator is allowed to interact with its neighbors within $γ_0$. With this coupling topology, we study two kinds of time delays : uniform time delays and distance-dependent time delays. We also extend our study to the case with random coupling topologies. Through numerical simulations, we find that uniform time delays with the finite interaction radius can induce various kinds of patterns including plane waves, spirals, antispirals, and transient squarelike pinwheels in addition to in-phase synchronous oscillations. We show that a quantity Ωτ, where Ω is the synchronization frequency of a state and τ is the time delay, characterizes the state. The system exhibits multistabilities of patterns and in-phase synchronous oscillations. With distance-dependent time delays due to a finite and constant signal transmission speed υ, the system shows plane waves, squarelike or rhombuslike pinwheels, spirals, and targets in addition to in-phase synchronous oscillations. It is shown that the oscillators cannot be perfectly synchronized and can exhibit only patterns if $θ=Ωγ_0/υ$, the maximum virtual phase difference between oscillators due to delays, is greater than a constant value $θ_c$. As in the...