Scaling behavior of nonhyperbolic coupled map lattices

Coupled map lattices of nonhyperbolic local maps arise naturally in many physical situations described by discretized reaction diffusion equations or discretized scalar field theories. As a prototype for these types of lattice dynamical systems we study diffusively coupled Tchebyscheff maps of $N\text{th}$ order which exhibit strongest possible chaotic behavior for small coupling constants $a$. We prove that the expectations of arbitrary observables scale with $\sqrt{a}$ in the low-coupling limit, contrasting the hyperbolic case which is known to scale with $a$. Moreover we prove that there are log-periodic oscillations of period $\ln N^2$ modulating the $\sqrt{a}$ dependence of a given expectation value. We develop a general 1st order perturbation theory to analytically calculate the invariant one-point density, show that the density exhibits log-periodic oscillations in phase space, and obtain excellent agreement with numerical results.

Figure 1

The function $\sqrt{a}{F}^{\left(N\right)}\left(\ln a\right)$ of Eq. (7) with $h\left(\Phi \right)=V_N\left(\Phi \right)$ as function of $a$ for $N=2,3$ in a double-logarithmic plot with basis $N^2$.

Figure 2

Log-periodic oscillations of the rescaled invariant density $f\left(x\right)$ as observed for $N=2$ (top) and $N=3$ (bottom) and different values for $a$.

Figure 3

Scaling behavior of the invariant one-point density near the left and right edges of the interval $[-1,1]$ for $N=2$ and $a=0.00007625$. Shown is the density at the left (upper diagram) and right edges (lower diagram) in comparison with the exact results (dashed-dotted curve, covered by the numerical result) and the density ${\rho }_0\left(\varphi \right)$ (dotted curve) together with several thresholds, i.e., points with nonanalytic behavior (dashed lines).