Space Representation of Stochastic Processes with Delay

Abstract

We show that a time series $x_t$ evolving by a non-local update rule $x_t=f(x_{t-n},x_{t-k})$ with two different delays $k<n$ can be mapped onto a local process in two dimensions with special time-delayed boundary conditions provided that $n$ and $k$ are coprime. For certain stochastic update rules exhibiting a non-equilibrium phase transition this mapping implies that the critical behavior does not depend on the short delay $k$. In these cases, the autocorrelation function of the time series is related to the critical properties of directed percolation.

6 pages, 8 figures

References in corpus (6)

• Universality classes in nonequilibrium lattice systems
• Universal scaling behavior of non-equilibrium phase transitions
• Noise-induced dynamics in bistable systems with delay
• Synchronization of mutually coupled chaotic lasers in the presence of a shutter
• Universal Critical Behavior of the Synchronization Transition in Delayed Chaotic Systems
• Stochastic differential equations with time-delayed feedback and multiplicative noise

Cited by in corpus (2)

• Spatio-temporal phenomena in complex systems with time delays
• Evidence of a critical phase transition in a purely temporal dynamics with long-delayed feedback