Self-organised dynamics beyond scaling of avalanches: Cyclic stress fluctuations in critical sandpiles

Abstract

Recognising changes in collective dynamics in complex systems is essential for predicting potential events and their development. Possessing intrinsic attractors with laws associated with scale invariance, self-organised critical dynamics represent a suitable example for quantitatively studying changes in collective behaviour. We consider two prototypal models of self-organised criticality, the sandpile automata with deterministic (Bak-Tang-Wiesenfeld) and probabilistic (Manna model) dynamical rules, focusing on the nature of stress fluctuations induced by driving - adding grains during the avalanche propagation, and dissipation through avalanches that hit the system boundary. Our analysis of stress evolution time series reveals robust cycles modulated by collective fluctuations with dissipative avalanches. These modulated cycles are multifractal within a broad range of time scales. Features of the associated singularity spectra capture the differences in the dynamic rules behind the self-organised critical states and their response to the increased driving rate, altering the process stochasticity and causing a loss of avalanche scaling. In the related sequences of outflow current, the first return distributions are found to follow modified laws that describe different pathways to the gradual loss of cooperative behaviour in these two models. The spontaneous appearance of cycles is another characteristic of self-organised criticality. It can also help identify the prominence of self-organisational phenomenology in an empirical time series when underlying interactions and driving modes remain hidden.