Emergence of Order in Dynamical Phases in Coupled Fractional Gauss Map

TL;DR

This paper explores how fractional-order coupled map lattices exhibit more ordered spatiotemporal patterns, including synchronization and periodic states, across various topologies, highlighting the impact of long-term memory effects.

Contribution

It extends the concept of fractional maps to coupled spatiotemporal systems and demonstrates enhanced order and synchronization phenomena compared to integer-order systems.

Findings

Fractional systems show more ordered patterns and synchronization.

Periodic states with periods greater than one are observed in fractional systems.

Power-law decay of standard deviation indicates long-term memory effects.

Abstract

Dynamical behaviour of discrete dynamical systems has been investigated extensively in the past few decades. However, in several applications, long term memory plays an important role in the evolution of dynamical variables. The definition of discrete maps has recently been extended to fractional maps to model such situations. We extend this definition to a spatiotemporal system. We define a coupled map lattice on different topologies, namely, one-dimensional coupled map lattice, globally coupled system and small-world network. The spatiotemporal patterns in the fractional system are more ordered. In particular, synchronization is observed over a large parameter region. For integer order coupled map lattice in one dimension, synchronized periodic states with a period greater than one are not obtained. However, we observe synchronized periodic states with period-3 or period-6 in one dimensional coupled fractional maps even for a large lattice. With nonlocal coupling, the synchronization is reached over a larger parameter regime. In all these cases, the standard deviation decays as power-law in time with the power same as fractional-order. The physical significance of such studies is also discussed.