Effects of uncommon non-isochronicities on remote synchronization

Remote synchronization (RS) is referred to as the weak synchronization between indirectly coupled nodes (among leaf nodes) while asynchronous between the directed coupled nodes (i.e., hub and leaf nodes). The underlying mechanisms are related to (i) coupling terms indirectly connecting two nearly identical oscillators while having a much different mediator in between, (ii) non-isochronicity of the oscillators. Here we show the effects of uncommon non-isochronicity values of oscillators on the occurrence of RS. In addition, we show clear difference of using either the first order phase approximations of the Stuart–Landau oscillators or Kuramoto–Sakaguchi (KS) models to study RS. By both theoretical and numerical analysis, we show that the critical coupling for synchronization between the directly connected nodes is inversely proportional to the difference value of non-isochronicity. On the contrary, the critical coupling for RS between indirectly connected leaf nodes is determined by the sum of non-isochronicity. In addition, RS may experience fading out after the first appearance when the coupling is progressively increased, which is indeed time delayed phase synchronization.