Approach to zigzag and checkerboard patterns in spatially extended systems

TL;DR

This study introduces phase defect $D(t)$ and phase persistence $P(t)$ as order parameters to quantify zigzag and checkerboard patterns in 1D and 2D coupled map lattices using Gauss and logistic maps. It shows that both $D(t)$ and $P(t)$ decay as power laws over broad ranges of the coupling parameter, with exponents matching Ising-like universality classes across dimensions and map types. Finite-size scaling confirms consistent dynamic scaling, revealing $z\approx 2$ in 1D and $z\approx 2.16$ in 2D (particularly for the logistic map), and highlights metastable behaviors in some Gauss-map cases. The results imply that these non-equilibrium patterns exhibit universal, parameter-range power-law dynamics independent of the microscopic details, offering a practical framework for identifying zigzag/checkerboard phases without detailed equation knowledge.

Abstract

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 $D(t)\sim t^{-δ}$ and $P(t)\sim t^{-θ}$ for the parameter range over which the zigzag or checkerboard pattern is realized. We observe that $δ=0.5$ and $θ=3/8$ for 1-d coupled logistic maps or Gauss maps, and $θ=0.22$ and $δ=0.45$ in 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 critical point.

Figures (19)

• Figure 1: (a) Gauss map is plotted for $\nu=5, \beta=-0.8$, $\nu=7,\beta=-0.45$ and $\nu=16,\beta=0$. For smaller $\beta$, we observe 3 fixed points while for larger $\beta$, we observe only one. b) Bifurcation diagram for single Gauss map as function of $\beta$ is plotted for $\nu=5$. We observe reverse period-doubling. c) Same plot for $\nu=7$ d) Bifurcation diagram for $\nu=16$. We observe period adding, interspersed by the chaotic attractor.
• Figure 2: Bifurcation diagram of coupled Gauss map for $\nu=7.5$ and $\beta=0.8$ in which all sites $x_{i}(t)$ are plotted as a function repulsive coupling $\epsilon$ at $t=1000$ and $N=100$.
• Figure 3: Coupled Gauss maps are simulated for $N=3\times 10^5$ and averaged over a $20$ configurations for values of coupling $\epsilon$ such that $\epsilon_1<\epsilon<\epsilon_2$ where $\epsilon_1=-2.61$ and $\epsilon_2=-1.26$. (a) We plot phase defect $D(t)$ as a function of time $t$ in the above $\epsilon$ range. We observe, $D(t) \sim t^{-\delta}$ over this entire range and the decay exponent is $\delta=0.5$. (b) We plot phase persistence $P(t)$ as a function of time $t$ in this range. We note that $P(t) \sim t^{-\theta}$ with decay exponent of $\theta=0.375$.
• Figure 4: We plot spatial profile $x_{i}(T)$ versus $i$ fort $T=10^5$, and $T=10^5+1$ for a) $\epsilon=-2.2$ b) $\epsilon=-1.3$. The lattice size is $N=100$. However, only 30 sites are plotted for clarity.
• Figure 5: We plot spatial return map $x_{i+2}(T)$ versus $x_{i}(T)$ for both even as well odd sites. We consider $N=300$ and $T=2\times10^5$ and simulate for a) $\epsilon=-1.3$, b) $\epsilon=-2.2$.

Theorems & Definitions (4)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4