Dynamical behavior of hydrodynamic Lyapunov modes in coupled map lattices

In our previous study of hydrodynamic Lyapunov modes (HLMs) in coupled map lattices, we found that there are two classes of systems with different $\lambda \text{−}k$ dispersion relations. For coupled circle maps we found the quadratic dispersion relations $\lambda \sim k^2$ and $\lambda \sim k$ for coupled standard maps. Here, we carry out further numerical experiments to investigate the dynamic Lyapunov vector (LV) structure factor which can provide additional information on the Lyapunov vector dynamics. The dynamic LV structure factor of coupled circle maps is found to have a single peak at $\omega =0$ and can be well approximated by a single Lorentzian curve. This implies that the hydrodynamic Lyapunov modes in coupled circle maps are nonpropagating and show only diffusive motion. In contrast, the dynamic LV structure factor of coupled standard maps possesses two visible sharp peaks located symmetrically at $\pm {\omega }_u$. The spectrum can be well approximated by the superposition of three Lorentzian curves centered at $\omega =0$ and $\pm {\omega }_u$, respectively. In addition, the $\omega \text{−}k$ dispersion relation takes the form ${\omega }_u=c_{u}k$ for $k\to 2\pi ∕L$. These facts suggest that the hydrodynamic Lyapunov modes in coupled standard maps are propagating. The HLMs in the two classes of systems are shown to have different dynamical behavior besides their difference in spatial structure. Moreover, our simulations demonstrate that adding damping to coupled standard maps turns the propagating modes into diffusive ones alongside a change of the $\lambda \text{−}k$ dispersion relation from $\lambda \sim k$ to $\lambda \sim k^2$. In cases of weak damping, there is a crossover in the dynamic LV structure factors; i.e., the spectra with smaller $k$ are akin to those of coupled circle maps while the spectra with larger $k$ are similar to those of coupled standard maps.

Figure 1

(Color online) Two snapshots of the instantaneous profile of LV No. 116 (upper row) and the corresponding spatial power spectra $s_u^{\left(\alpha \alpha \right)}(k,t)$ (lower row) for coupled standard maps with $ϵ=1.3$. The system size $L=128$ is set for all figures concerning the coupled standard maps unless otherwise stated.

Figure 2

(Color online) Intermittent time evolution of the peak wave number $k^{*}$ (upper panel) and the spectral entropy $H_s\left(t\right)$ (lower panel) of coupled standard maps with $ϵ=1.3$.

Figure 3

Two snapshots of the instantaneous profile of LV No. 156 (upper row) and the corresponding spatial power spectra $s_u^{\left(\alpha \alpha \right)}(k,t)$ (lower row) for coupled circle maps, Eq. (2), with $ϵ=1.3$. The system size $L=256$ is set for all figures concerning the coupled circle maps unless otherwise stated. The zero-value Lyapunov exponent is at $\alpha =164$.

Figure 4

(Color online) Intermittent time evolution of the peak wave number $k^{*}$ (upper panel) and the spectral entropy $H_s\left(t\right)$ (lower panel) of coupled circle maps, Eq. (2), with $ϵ=1.3$.

Figure 5

(Color online) Dynamic LV structure factors $S_u^{\left(\alpha \alpha \right)}(k,\omega )$ of LV No. 163 of coupled circle maps, Eq. (2), with $ϵ=1.3$. The peak at $\omega =0$ becomes weaker as the wave number $k$ increases.

Figure 6

(Color online) Approximations of the dynamic LV structure factors $S_u^{\left(\alpha \alpha \right)}(k,\omega )$ with the Lorentzian curve [see Eq. (10)]. Here the results of LV No. 163 of coupled circle maps are shown with a representation of only the positive $\omega$ part of $S_u^{\left(\alpha \alpha \right)}(k,\omega )$.

Figure 7

(Color online) The fitting parameter $a_1$ vs the wave number $k$ for several cases with various $ϵ$ and $\alpha$. A straight line with the slope 2.0 is plotted to show the parabolic dependence of $a_1$ on $k$.

Figure 8

$k$ dependence of the fitting parameter $a_0$ of the LV Nos. 159, 162, and 163 (from top to bottom) of coupled circle maps with $ϵ=1.3$. The corresponding static structure factors are shown in the same plot, respectively. The agreement between curves and symbols implies a linear relation $a_0\simeq (1∕\pi )S_u^{\left(\alpha \alpha \right)}\left(k\right)$.

Figure 9

(Color online) Dynamic LV structure factors of LV No. 126 of coupled standard maps with $ϵ=0.6$. The upper panel shows those for the $u$ component of LVs and the lower panel depicts those for the $v$ component. A distinct feature of the spectra is the existence of two sharp peaks centering symmetrically at $\omega =\pm {\omega }_u$.

Figure 10

(Color online) three- and two-pole approximations of the dynamic LV structure factor $S_u^{\left(\alpha \alpha \right)}(k,\omega )$ of LV No. 126 of coupled standard maps with $ϵ=0.6$ and $k=2\pi ∕L$. Obviously the three-pole fit is better than the two-pole fit although the central peak of the three-pole fit is quite weak and flat.

Figure 11

(Color online) The fitting parameters of the three-pole approximations of dynamic structure factors of five LVs of coupled standard maps with $ϵ=0.6$.

Figure 12

(Color online) The velocity $c_u$ vs the index $\alpha$ of LVs. The slow increase of $c_u$ with $\alpha$ from 118 to 126 reflects the weak dependence of $c_u$ on LVs.

Figure 13

(Color online) Dynamic LV structure factors $S_u^{\left(\alpha \alpha \right)}(k,\omega )$ of LV No. 126 with several larger wave numbers $k$ (upper panel) and the $k$ dependence of ${\omega }_u$ (lower panel). The different meanings of ${\omega }_u$ for larger and smaller $k$ are described in the main text.

Figure 14

(Color online) $k$ dependence of the fitting parameters $a_2$ and $a_0$ and the static LV structure factors $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ of the LV No. 122 (upper panel) and No. 118 (lower panel) of coupled standard maps with $ϵ=0.6$. In case $\alpha =118$ we assign $a_0=a_2$ for the three-pole approximation.

Figure 15

(Color online) Ratio $a_2∕a_0$ vs $k$ of the LV Nos. 122, 124, and 126 of coupled standard maps with $ϵ=0.6$. It is nearly constant for $k⩾6\pi ∕L$ but hold different values for various LVs.

Figure 16

(Color online) Log-log plot of the fitting parameter $a_1$ of coupled standard maps with $ϵ=0.6$. Numerical fitting of the data to a power-law function $a_1\sim k^{-\beta }$ yields $\beta \simeq 1.0$.

Figure 17

(Color online) Dynamic LV structure factors $S_u^{\left(\alpha \alpha \right)}(k,\omega )$ of coupled damped standard maps under damping. The damping coefficient is ${\gamma }_0=0.0$, 0.2, 0.5, and 0.7 (from top to bottom), respectively. The coupling strength is $ϵ=1.3$. The side peaks in the spectra fade out gradually with increasing ${\gamma }_0$.

Figure 18

(Color online) The fitting parameter ${\omega }_u$ vs $k$ for the three cases ${\gamma }_0=0.0$, 0.2, and 0.5. Fitting data to a linear function ${\omega }_u=c_{u}k$ gives $c_u=0.136$, 0.161, and 0.175, respectively.

Figure 19

(Color online) The fitting parameters $a_i$ for the dynamic LV structure factors of coupled damped standard maps with $\gamma =1.0$ and 0.5.

Figure 20

(Color online) Approximation of the spectrum $S_u^{\left(\alpha \alpha \right)}(k,\omega )$ of LV No. 76 of the overdamped case ${\gamma }_0=0.7$. The upper panel is the log-linear plot for the case $k=2\pi ∕L$, and the lower panel is the log-log plot for four cases with $k=2\pi ∕L$ to $8\pi ∕L$. The agreement between the numerical data and the fitting curves is perfect for all cases shown. The zero-value Lyapunov exponent is at $\alpha =78$.

Figure 21

$k$ dependence of the fitting parameter $a_0$ and the corresponding static LV structure factors $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ of the LV Nos. 76, 72, and 68, respectively (from top to bottom). The agreement between the two sets yields $a_0=(1∕2\pi )S_u^{\left(\alpha \alpha \right)}\left(k\right)$.

Figure 22

(Color online) The width $a_1$ of the peak in the spectra $S_u^{\left(\alpha \alpha \right)}(k,\omega )$ vs $k$ for various LVs. Numerical fitting of the data for the case $\alpha =76$ gives $a_1\sim k^2$. Curves for other LVs follow the same trend for large $k$, although they depart from the parabolic function $a_1\sim k^2$ in the small-$k$ regime.

Figure 23

(Color online) The upper panel shows the contour plot of the static LV structure factors of the dissipative system, Eq. (2), with the nonlinear function $f\left(z\right)$ in Eq. (15). Here, $ϵ=1.3$ and $r=0.2$. The lower panel displays the static LV structure factor of LV No. 230. The dominant peak is very sharp here, which implies that the LV profile is rather close to a pure plane wave.

Figure 24

(Color online) Time evolution of several quantities characterizing the LV dynamics. The erratic dip-relaxation events in $s_u^{\left(\alpha \alpha \right)}(k_{\mathit{max}},t)$ are a manifestation of the diffusive motion of ${\mathcal{U}}^{\left(\alpha \right)}(r,t)$. The phase $\varphi (k_{\mathit{max}},t)$ is nearly constant during the intervals between the dip-relaxation events. This implies that the LVs are of nonpropagating nature in this system.

Figure 25

(Color online) The upper panel shows the dip-relaxation event in $s_u^{\left(\alpha \alpha \right)}(k_{\mathit{max}},t)$ at $t\approx 7720$. Four snapshots of the LV profile ${\mathcal{U}}^{\left(\alpha \right)}(r,t)$ during this dip-relaxation event are represented in the lower panels. It becomes clear from the plot that the dip-relaxation event is characterized by the sudden deformation and slow recovering of the plane-wave LV profile.

Figure 26

(Color online) The coherence between the dip-relaxation events in the time evolution of $s_u^{\left(\alpha \alpha \right)}(k_{\mathit{max}},t)$ and the spikelike fluctuations in $\lambda (\tau ,t)$ for several LVs shows that the dip-relaxation events are caused by the interaction among LVs.

Figure 27

(Color online) Same as Fig. 24, but for the Hamiltonian system, Eq. (1). Here, $ϵ=1.3$ and $r=0.15$. The dip-relaxation events are similar to those in Fig. 24. Note that, in contrast to the dissipative case shown in Fig. 24, the phase $\varphi (k_{\mathit{max}},t)$ changes continuously during the intervals between dip-relaxation events. This implies that the LVs in this system are propagating. The apparent stochastic motion of the phase variable $\varphi (k_{\mathit{max}},t)$ means that the propagating Lyapunov modes are of short lifetime.

Figure 28

(Color online) The upper panel shows the enlargement of the dip-relaxation event at $t\approx 1010$. Four snapshots of the LV profile ${\mathcal{U}}^{\left(\alpha \right)}(r,t)$ during this dip-relaxation event are presented in the lower panels. As can be clearly seen from the plot, the dip-relaxation event resembles those in the dissipative case; i.e., they are characterized by the sudden deformation and slow recovering of the plane-wave LV profile.

Figure 29

(Color online) Same as Fig. 27, but for the case $ϵ=1.3$ and $r=0.1$. Here, the wild dip-relaxation events disappear completely, which causes the continuous variation of the phase variable $\varphi (k_{\mathit{max}},t)$ to become more evident. Note that the fluctuations of the finite-time Lyapunov exponents are very small compared to the differences between neighboring Lyapunov exponents. This is the reason why the dip-relaxation events disappear here.

Figure 30

Same as Fig. 27, but for the case $ϵ=1.3$ and $r=0.2$. Here, the dip-relaxation events become more frequent in comparison with the case $ϵ=1.3$ and $r=0.15$. Note that the sudden jump in the phase variable $\varphi (k_{\mathit{max}},t)$ and the spikelike fluctuations in $\lambda (\tau ,t)$ also take place more frequently.