Universal Features of Hydrodynamic Lyapunov Modes in Extended Systems with Continuous Symmetries

Numerical and analytical evidence is presented to show that hydrodynamic Lyapunov modes (HLMs) do exist in lattices of coupled Hamiltonian and dissipative maps. More importantly, we find that HLMs in these two classes of systems are different with respect to their spatial structure and their dynamical behavior. To be concrete, the corresponding dispersion relations of Lyapunov exponent versus wave number are characterized by $\lambda \sim k$ and $\lambda \sim k^2$, respectively. The HLMs in Hamiltonian systems are propagating, whereas those of dissipative systems show only diffusive motion. Extensive numerical simulations of various systems confirm that the existence of HLMs is a very general feature of extended dynamical systems with continuous symmetries and that the above-mentioned differences between the two classes of systems are universal in large extent.

Figure 1

(a) Lyapunov spectrum and (b) contour plot of the static LV structure factors for coupled standard maps Eq. (1) with $ϵ=0.6$. In (b), the graph of $k_{\max }^{(\alpha )}$ is overlaid on the contour plot, where $k_{\max }^{(\alpha )}$ is the wave number of the dominant peak in each spectrum $S_u^{(\alpha \alpha )}(k)$. (c) and (d) are similar to (a) and (b) but for coupled circle maps, Eq. (2), with $ϵ=2.3$. For both contour plots there is a ridge structure in the regime $\lambda \gtrsim 0$ with $k_{\max }^{(\alpha )}\to 2\pi /L$ as ${\lambda }^{(\alpha )}\to 0$. This implies the existence of HLMs.

Figure 2

The $\lambda \mathrm{\text{−}}k$ dispersion relations for various extended systems with continuous symmetries. The normalized data for different systems collapse on two master curves. These results strongly support our conjecture that there are two classes of systems with $\lambda \sim k$ and $\lambda \sim k^2$ respectively. Here, systems in the former group include Eq. (1) with $f(z)=\frac{1}{2\pi }\sin (2\pi z)$ (SM), Eq. (1) with $f(z)=2z$ (mod 1) (HBM), and the $1d$ $XY$ model ($XY$) [11]. Systems belonging to the class $\lambda \sim k^2$ are Eq. (2) with $f(z)=\frac{1}{2\pi }\sin (2\pi z)$ (CM), Eq. (2) with $f(z)=2z$ (mod 1) (BM), Eq. (1) with $f(z)=\frac{1}{2\pi }\sin (2\pi z)$, and $\gamma =0.7$ (SM-D), Eq. (1) with $f(z)=2z$ (mod 1) and $\gamma =0.7$ (HBM-D) and the $1d$ Kuramoto-Sivashinsky equation (KS).

Figure 3

Dynamic LV structure factors $S_u^{(\alpha \alpha )}(k,\omega )$ for (a) coupled standard maps, Eq. (1) with $ϵ=1.3$; (b) coupled circle maps, Eq. (2) with $ϵ=1.3$; (c) Eq. (1) with $ϵ=1.3$, $\gamma =0.2$; and (d) Eq. (1) with $ϵ=1.3$, $\gamma =0.7$.