Approach to zigzag and checkerboard patterns in spatially extended systems

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 $D(t)={\frac{\sum_{i=1}^N \vert s_i(t)+s_{i-1}(t)\vert} {2N}}$ 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 till time $t$. We find that $D(t)\sim t^{-\delta}$ and $P(t)\sim t^{-\theta}$ for the parameter range over which the zigzag or checkerboard pattern is realized. We observe that $\delta=0.5$ and $\theta=3/8$ for 1-d coupled logistic maps or Gauss maps, and $\theta=0.22$ and $\delta=0.45$ in 2-d logistic or Gauss maps. The exponent $\theta$ matches with the persistence exponent at zero temperature for the Ising model, and $\delta$ 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 critical point.