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Zigzag patterns in one dimension or checkerboard patterns in two dimensions occur in a variety of pattern-forming systems. We introduce an order parameter 'phase defect' to identify this transition and help to recognize the associated universality class on a discrete lattice. In one dimension, if x i (t) is a variable value at site i at time t. We assign spin s i (t) = 1 for x i (t) > x i − 1 (t) , s i (t) = − 1 if x i (t) < x i − 1 (t) , and s i (t) = 0 if x i (t) = x i − 1 (t). The phase defect D (t) is defined as D (t) = ∑ i = 1 N | s i (t) + s i − 1 (t) | 2 N for a lattice of N sites with periodic boundary conditions. It is zero for a zigzag pattern. In two dimensions, D (t) is the sum of row-wise as well as column-wise phase defects and is zero for the checkerboard pattern. The persistence P (t) is the fraction of sites whose spin value did not change even once at all even times till time t. We find that D (t) ∼ t − δ and P (t) ∼ t − θ for the parameter range over which the zigzag or checkerboard pattern is realized with δ = 0. 5 and θ = 3 / 8 for 1-d coupled logistic or Gauss maps, and δ = 0. 45 and θ = 0. 22 for 2-d logistic or Gauss maps. The exponent θ matches with the persistence exponent at zero temperature for the Ising model and δ matches with the exponent for the Ising model at the critical temperature. This power-law decay is observed over a range of parameter values and not just the critical point. • We define an order parameter D (t) for zigzag and checkerboard patterns. • D (t) shows a power-law decay over a range of parameter values. • Persistence also shows a power-law decay over the same range. • The exponents are similar to those for the Ising model. [ABSTRACT FROM AUTHOR]

• We observe directed percolation (DP) transition to the absorbing state in a coupled Gaussian maps. • It reaches a coarse-grained period-2 state in time. It has no long-range spatial order even in a coarse-grained sense. • We propose flip rate F(t) (fraction of sites which don't return to the same band after two time-steps) as an order parameter. • Estimates of decay exponent of F(t), persistence exponent as well as of z and v‖ put the transition firmly in DP class. We study coupled Gauss maps in one dimension and observe a transition to band periodic state with 2 bands. This is a periodic state with period-2 in a coarse-grained sense. This state does not show any long-range order in space. We compute two different order parameters to quantify the transition a) Flipping rate F (t) which measures departures from period-2 and b) Persistence P (t) which quantifies the loss of memory of initial conditions. At the critical point, F (t) shows a power-law decay with exponent 0.158 which is close to 1-D directed percolation (DP) transition. The persistence exponent at the critical point is found to be 1.51 which matches with several models in 1-D DP class. We also study the finite-size scaling and off-critical scaling to estimate other exponents z and ν ∥. We observe excellent scaling for both F (t) as well as P (t) and the exponents obtained are clearly in DP class. We believe that DP transition could be observed in systems where activity goes to zero even if the spatial profile could be inhomogeneous and lacking any long-range order. [ABSTRACT FROM AUTHOR]

The concept of q -deformation has been expanded in a variety of situations including q -deformed nonlinear maps. Coupled q -deformed logistic maps with positive and negative q -diffusive coupling are investigated. The transition to state of synchronized fixed point in the thermodynamic limit for random initial conditions is studied as dynamic phase transitions. Though rare for one-dimensional maps, q -deformed maps show multistability. For weak coupling, the synchronized state may not be observed in the thermodynamic limit with random initial conditions even if it is linearly stable. Thus multistability persists. However, for large lattices with random initial conditions, five well-defined critical points are observed where the transition to a synchronized state occurs. Two of these show continuous phase transition. One of them is in the directed percolation universality class. In another case, the order parameter shows power-law decay superposed with oscillations on a logarithmic scale at the critical point and does not match with known universality classes. • We investigate q -diffusively coupled q -deformed logistic maps. • We study transitions to synchronized fixed points as dynamic phase transitions. • Transition to a nonzero fixed point is in the directed percolation universality class. • Order parameter decays as power-law with logarithmic oscillations for transition to fixed point coexisting with two bands. [ABSTRACT FROM AUTHOR]

• We study transition to two band attractor state in coupled logistic maps with linear/nonlinear coupling and delay. • For nonlinear coupling with even delay/ linear coupling with odd delay, we find antiferromagnetic order at critical point. • For linear coupling with even delay/ nonlinear coupling with odd delay, we observe ferromagnetic order at critical point. • The persistence exponent for antiferromagnetic ordering is 0.375. For ferromagnetic ordering it is 0.285. We study one-dimensional coupled logistic maps with delayed linear or nonlinear nearest-neighbor coupling. Taking the nonzero fixed point of the map x * as reference, we coarse-grain the system by identifying values above x * with the spin-up state and values below x * with the spin-down state. We define persistent sites at time T as the sites which did not change their spin state even once for all even times till time T. A clear transition from asymptotic zero persistence to non-zero persistence is seen in the parameter space. The transition is accompanied by the emergence of antiferromagnetic, or ferromagnetic order in space. We observe antiferromagnetic order for nonlinear coupling and even delay, or linear coupling and odd delay. We observe ferromagnetic order for linear coupling and even delay, or nonlinear coupling and odd delay. For symmetric coupling, we observe a power-law decay of persistence. The persistence exponent is close to 0.375 for the transition to antiferromagnetic order and close to 0.285 for ferromagnetic order. The number of domain walls decays with an exponent close to 0.5 in all cases as expected. The persistence decays as a stretched exponential and not a power-law at the critical point, in the presence of asymmetry. [ABSTRACT FROM AUTHOR]