Spatiotemporal stability of synchronized coupled map lattice states

TL;DR

This work develops a spatiotemporal stability analysis for synchronized states in coupled map lattices by constructing a spatiotemporal orbit Jacobian in the first Brillouin zone. By treating space and time on equal footing, it derives eigenvalue spectra and Bravais stability exponents that quantify perturbations across all time-space frequencies, including incoherent perturbations. For steady states, the stability landscape shows that stronger coupling generally reduces instability to incoherent perturbations and can lead to stable bands as $a\to1$, while period-2 synchronized states exhibit nonmonotonic, regime-dependent stability and can disappear at $a=1/3$. The framework unifies coherent and incoherent stability analyses and connects to partition-function weights in deterministic chaotic field theories, offering a path to systematic spatiotemporal analyses of dissipative lattice dynamics.

Abstract

In the realm of spatiotemporal chaos, unstable periodic orbits play a major role in understanding the dynamics. Their stability changes and bifurcations in general are thus of central interest. Here, coupled map lattice discretizations of nonlinear partial differential equations, exhibiting a variety of behaviors depending on the coupling strength, are considered. In particular, the linear stability analysis of synchronized states is performed by evaluating the Bravais lattice orbit Jacobian in its reciprocal space first Brillouin zone, with space and time treated on equal grounds. The eigenvalues of the orbit Jacobian operator, computed as functions of the coupling strength, tell us about the stability of the periodic orbit under a perturbation of a certain time- and space frequency. Moreover, the stability under aperiodic, that is, incoherent perturbations, is revealed by integrating the sum of the stability exponents over all space-time frequencies.

Figures (4)

• Figure 1: The range of $a$ and $k$ that makes $\Lambda_k$ unstable is the region where the plane $z=T_N'(\phi_p)$ is: (a) higher than the surface $h(a,k)= \frac{1-a\cos k}{1-a}$ ($T_2$ model is considered with steady state $\phi_1=1$); (b) lower than the surface $h(a,k)= -\frac{1+a\cos k}{1-a}$ ($T_2$ model with steady state $\phi_0=-1/2$).
• Figure 2: The range of $a$ and $k_2$ that make $|f(a,k_2)|<1$ is the interval where the function $F(a,k_2)=|f(a,k_2)|-1$ is negative. (a) $T_2$ model and $\phi_1=1$. (b) $T_2$ model and $\phi_0=-1/2$.
• Figure 3: The stability exponent versus strength of coupling $a$ (solid blue line). Beyond the red vertical line the condition $|f(a,k_2)|>1$ no longer holds and thus the stability exponent is evaluated numerically (gray dots). (a) $T_2$ model for the steady state $\phi_1=1$ (b) $T_2$ model for the steady state $\phi_0=-1/2$.
• Figure 4: (a) The expressions $V'_{01}$ (blue) and $V'_{10}$ (yellow) vs. coupling $a$. (b) The range of $a$ and $k_2$ that make $|z_{\pm}(a,k_2)|<1$ is determined by the intersection of the function $G(a,k_2)=\sqrt{g^2(a,k_2)-\Delta V'}/2$ with the plane $z=1$ ($T_2$ model).