  Fixed-energy sandpile (FES) models, introduced to understand the
self-organized criticality, show a continuous phase transition between
absorbing and active phases. In this work, we study the dynamics of the
deterministic FES models on random networks. We observe that close to absorbing
transition the density of active nodes oscillates and nodes topple in
synchrony. The deterministic toppling rule and the small-world property of
random networks lead to the emergence of sustained oscillations. The amplitude
of oscillations becomes larger with increasing the value of network randomness.
The bifurcation diagram for the density of active nodes is obtained. We use the
activity-dependent rewiring rule and show that the interplay between the
network structure and the FES dynamics leads to the emergence of a bistable
region with a first-order transition between the absorbing and active states.
Furthermore during the rewiring, the ordered activation pattern of the nodes is
broken, which causes the oscillations to disappear.
