No abstract

We report the discovery of a discrete hierarchy of micro-transitions occurring in models of continuous and discontinuous percolation. The precursory micro-transitions allow us to target almost deterministically the location of the transition point to global connectivity. This extends to the class of intrinsically stochastic processes the possibility to use warning signals anticipating phase transitions in complex systems.

Finding conditions that support synchronization is a fertile and active area of research with applications across multiple disciplines. Here we present and analyze a scheme for synchronizing chaotic dynamical systems by transiently uncoupling them. Specifically, systems coupled only in a fraction of their state space may synchronize even if fully coupled they do not. Although, for many standard systems, coupling strengths need to be bounded to ensure synchrony, transient uncoupling removes this bound and thus enables synchronization in an infinite range of effective coupling strengths. The presented coupling scheme thus opens up the possibility to induce synchrony in (biological or technical) systems whose parameters are fixed and cannot be modified continuously.Synchronization is one of the most prevalent collective phenomena in coupled dynamical systems [1]. Synchronization and related consensus phenomena have been frequently found in biological, ecological, physical, engineering and social systems such as in predator-prey dynamics, the spread of epidemics, the migration of large populations, systems of self-driven particles and systems of social or technical dynamics [2][3][4][5][6][7][8][9][10][11][12]. For chaotic systems, synchronization typically emerges only within a specific range of coupling strengths and is impossible otherwise [1,[13][14][15].In this letter, we propose and analyze a way of inducing synchronization between coupled chaotic oscillators by transient uncoupling: If the system is in a certain predefined subset of its state space, coupling is active; otherwise it is inactive. We systematically study the dependence of successful synchronization on the fraction of state space where coupling is active. Synchronization may emerge even for systems that coupled continuously in time (i.e., standard coupling) do not synchronize. Furthermore, the system may synchronize for an infinite range of coupling strengths, although this is often not possible for ordinarily coupled chaotic systems.A systematic numerical analysis reveals how transverse stability properties vary across the attractor with the location of active coupling, not only between more or less stable synchrony but all the way from stability to instability for the same system. This demonstrates that transient uncoupling modifies the collective dynamics in a non-trivial way. These results may find applications in inducing synchrony in systems whose local coupling parameters cannot be continuously varied with ease but only switched on or off. * malte@nld.ds.mpg.de †

Crackling noise is a common feature in many systems that are pushed slowly, the most familiar instance of which is the sound made by a sheet of paper when crumpled. In percolation and regular aggregation, clusters of any size merge until a giant component dominates the entire system. Here we establish 'fractional percolation', in which the coalescence of clusters that substantially differ in size is systematically suppressed. We identify and study percolation models that exhibit multiple jumps in the order parameter where the position and magnitude of the jumps are randomly distributed-characteristic of crackling noise. This enables us to express crackling noise as a result of the simple concept of fractional percolation. In particular, the framework allows us to link percolation with phenomena exhibiting non-self-averaging and power law fluctuations such as Barkhausen noise in ferromagnets.