Dynamical phases and phase transition in simplicially coupled logistic maps

Coupled map lattices are a popular and computationally simpler model of pattern formation in nonlinear systems. In this work, we investigate three-site interactions with linear multiplicative coupling in one-dimensional coupled logistic maps that cannot be decomposed into pairwise interactions. We observe the transition to synchronization and the transition to long-range order in space. We coarse-grain the phase space in regions and denote them by spin values. We use two quantifiers the flip rate F(t) that quantify departure from expected band-periodicity as an order parameter. We also study a non-Markovian quantity, known as persistence P(t) to study dynamic phase transitions. Following transitions are observed. (a) Transition to two band attractor state: At this transition F(t) as well as P(t) shows a power-law decay in the range of coupling parameters. Here all sites reach one of the bands. The F(t) as well as P(t) decays as power-law with the decay exponent δ1=0.46 and η1=0.28, respectively. (b) The transition from a fluctuating chaotic state to a homogeneous synchronized fixed point: Here both the quantifiers F(t) and P(t) show power-law decay with decay exponent δ2=1 and η2=0.11, respectively. We compare the transitions with the case, where pairwise interactions are also present. The spatiotemporal evolution is analyzed as the coupling parameter is varied.