Hydrodynamic Lyapunov modes in coupled map lattices

In this paper, numerical and analytical results are presented which indicate that hydrodynamic Lyapunov modes (HLMs) also exist for coupled map lattices (CMLs). The dispersion relations for the HLMs of CMLs are found to fall into two different universality classes. It is characterized by $\lambda \sim k$ for coupled standard maps and $\lambda \sim k^2$ for coupled circle maps. The conditions under which HLMs can be observed are discussed. The role of the Hamiltonian structure, conservation laws, translational invariance, and damping is elaborated. Our results are as follows: (1) The Hamiltonian structure is not a necessary condition for the existence of HLMs. (2) Conservation laws or the translational invariance alone cannot guarantee the existence of HLMs. (3) Including a damping term in the system of coupled Hamiltonian maps does not destroy the HLMs. The $\lambda \text{−}k$ dispersion relation of HLMs, however, changes to the universality class with $\lambda \sim k^2$ under damping. In contrast, no HLMs survives in the system of coupled circle maps under damping. (4) An on-site potential destroys the HLMs. (5) The study of zero-value Lyapunov exponents (LEs) and associated Lyapunov vectors (LVs) shows that translational invariance and conservation laws play different roles in the tangent space dynamics. (6) The dynamics of the coordinate and momentum parts of LVs in Hamiltonian systems are related but different. Furthermore, numerical results for a two-dimensional system show that the appearance of HLMs in CMLs is not restricted to the one-dimensional case.

Figure 1

Lyapunov spectrum for coupled standard maps Eq. (8) with $ϵ=0.6$. The enlargement in the inset shows that there is only one pair of zero-value Lyapunov exponents for this system. This implies that there is only one sort of HLMs in this simple system. The system size used is $L=128$.

Figure 2

Snapshots of the $\delta u$ components of typical LVs for coupled standard maps Eq. (8) with parameters as in Fig. 1. The LV corresponding to the largest LE $(\alpha =1)$ is highly localized while the one corresponding to a near zero LE $(\alpha =126)$ is extended.

Figure 3

Contour plot of the static LV structure factors (upper panel) and the dominant wave number $k_{\mathit{max}}$ versus the index $\alpha$ of Lyapunov vectors (lower panel) for coupled standard maps Eq. (8) with the same parameter setting as in Fig. 1. The sharp dominant peaks in the static LV structure factors with $k_{\mathit{max}}\to 0$ as $\alpha \to 128$ indicate the existence of HLMs in this system. The unit $2\pi$ is used for the wave number $k$ in all concerned figures throughout this paper.

Figure 4

(Color online) Lyapunov spectra (upper panel) and the peak wave number $k_{\mathit{max}}$ (lower panel) versus the index $\alpha$ for coupled standard maps Eq. (8) with coupling strength $ϵ=0.1$, 0.3, 0.6, and 0.9. Changing the value of $ϵ$ does not change the fact that $k_{\mathit{max}}\to 0$ as $\alpha \to 128$, i.e., the existence of HLMs does not change. The system size used is $L=128$.

Figure 5

Lyapunov spectrum (top panel), the contour plot of the static LV structure factors $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ (middle panel), and the peak wave-number $k_{\mathit{max}}$ (bottom panel) for coupled circle maps with force-like coupling given in Eq. (10) with $ϵ=1.3$. The sharp dominant peaks in the spectra $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ with $k_{\mathit{max}}\to 0$ as $\alpha \to 164$ imply the existence of HLMs in this system. The system size used is $L=256$.

Figure 6

(a) The Lyapunov spectrum, (b) contour plot of $S_u^{\left(\alpha \alpha \right)}\left(k\right)$, and (c) $k_{\mathit{max}}$ versus the index $\alpha$ of the LVs for the Bohr model Eq. (11) with $ϵ=1.0$. The system size used is $L=256$.

Figure 7

Similar to Fig. 6 but with $ϵ=0.58$. In the bottom panel the wave number $k_{\mathit{max}}$ for the dominant peak of the spectrum of each LV is plotted versus $\alpha$. It can be easily seen that $k_{\mathit{max}}$ attains a finite value as ${\lambda }^{\left(\alpha \right)}\to 0$. This shows that there exist no HLMs in this system.

Figure 8

(Color online) (a)–(c) Contour plot of $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ for the coupled circle map with boundary condition {$u_t^0=u_t^L$, $u_t^{L+1}=u_t^1$} (force-pb), {$u_t^0=0$, $u_t^{L+1}=u_t^L$} (force-fib-pb), respectively, and for the Bohr model Eq. (11) with {$x_t^0=0$, $x_t^{L+1}=0$} (diffusive). The coupling strength used is $ϵ=1.3$. Panel (d) shows the peak wave number $k_{\mathit{max}}$ for the three cases. An obvious difference between the case of force-fib-pb and diffusive can be seen. Panel (e) shows the Lyapunov spectra for the three cases. They are nearly identical. The system size used is $L=256$.

Figure 9

(Color online) Spectral entropy [38] $H_S\equiv -{\sum }_{k_i}S\left(k_i\right)\ln S\left(k_i\right)$ vs. $\alpha$ of force-like coupled and diffusively coupled circle maps with $ϵ=1.3$. The smaller value of $H_S$ for the force-like coupled model means the HLMs are more significant there than in the case with diffusive coupling.

Figure 10

(Color online) Comparison of the two quantities $S_x^{\left(\alpha \alpha \right)}\left(k\right)$ and $k^2{S}_u^{\left(\alpha \alpha \right)}\left(k\right)$ for coupled circle maps (upper panel) and coupled skewed tent maps (lower panel). The agreement of the two quantities in the regime $k\simeq 2\pi ∕L$ implies that the assumption in Eq. (12) used for deducing Eq. (17) is plausible.

Figure 11

(Color online) Static LV structure factor $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ for LV with $\alpha =163$ of force-like coupled circle maps [Eq. (10)]. Here $k_{\mathit{max}}=2\pi ∕L$ and the power law fitting gives $S_u^{\left(\alpha \alpha \right)}\left(k\right)\sim k^{-1.4}$ in the regime $k⩾k_{\mathit{max}}$. According to Eq. (17), the values of $k_{\mathit{max}}$ extracted from the two models with different couplings should be different (see Fig. 8) and there exist no HLMs in the model with diffusive couplings. The zero-value Lyapunov exponent is at $\alpha =164$.

Figure 12

(Color online) (a) Peak wave number $k_{\mathit{max}}$ vs. $\alpha$ for the force-like and diffusively coupled skewed tent maps [40] with $r=5$. (b) Static structure factor $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ for the LV with $\alpha =179$. Here $k_{\mathit{max}}=2\pi ∕L$ and the power law fitting gives that $S_u^{\left(\alpha \alpha \right)}\left(k\right)\sim k^{-2.2}$ in the regime $k⩾k_{\mathit{max}}$. According to the criterion [Eq. (17)], the values of $k_{\mathit{max}}$ extracted from the two models with different couplings should be identical (see the numerical results shown in panel a). The zero-value Lyapunov exponent is at $\alpha =178$. For this case, the HLMs only exist for the negative branch of Lyapunov spectrum $(\alpha >178)$. The reason for this and the jump in $k_{\mathit{max}}$ is not important for the current discussion.

Figure 13

(a) The Lyapunov spectrum, (b) contour plot of $S_u^{\left(\alpha \alpha \right)}\left(k\right)$, and (c) $k_{\mathit{max}}$ versus $\alpha$ for the model Eq. (18) with $ϵ=1.0$. There is no sharp peak in the spectra $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ and the $k_{\mathit{max}}$ extracted attains a finite value $k_0⪢2\pi ∕L$ as ${\lambda }^{\left(\alpha \right)}\to 0$, which implies that there are no HLMs in this model. The system size used here is $L=128$.

Figure 14

(Color online) (a) The Lyapunov spectrum, (b) contour plot of $S_u^{\left(\alpha \alpha \right)}\left(k\right)$, and (c) $k_{\mathit{max}}$ versus $\alpha$ for coupled standard maps under damping with $ϵ=2.3$, ${\gamma }_0=0.7$, and $L=128$. Although the Lyapunov spectrum is no longer symmetric with respect to ${\lambda }^{\left(\alpha \right)}=0$, there is still $k_{\mathit{max}}\to 2\pi ∕L$ as ${\lambda }^{\left(\alpha \right)}\to 0$, i.e., HLMs survive damping in this system. Note that, in panel (c), the two curves for the coordinate and momentum parts of the Lyapunov vectors are mutual mirror images due to the $\mu$-symplectic structure of the system treated here [39].

Figure 15

(a) The Lyapunov spectrum, (b) contour plot of $S_u^{\left(\alpha \alpha \right)}\left(k\right)$, and (c) $k_{\mathit{max}}$ versus $\alpha$ for coupled circle maps under damping with $ϵ=2.3$, ${\gamma }_0=0.4$, and $L=256$. In contrast to Fig. 14 HLMs are not observed in this system since $k_{\mathit{max}}$ attains a finite $k_0⪢2\pi ∕L$ as ${\lambda }^{\left(\alpha \right)}\to 0$.

Figure 16

(Color online) (a)–(c) Results of simulations for coupled circle maps with on-site potential with $ϵ=1.3$, $\kappa =2.0$, and $L=256$. (d) Spectral entropy $H_S$ for the coupled standard maps with $ϵ=0.3$, $L=128$, and $\kappa =2$, 20, and 80, respectively. These results show that the on-site potential strongly influences the behavior of LVs and destroys the hydrodynamic Lyapunov modes.

Figure 17

(Color online) Dispersion relation ${\lambda }^{\left(\alpha \right)}$ vs. $k$ for (a) coupled standard maps Eq. (8) and (b) coupled circle maps Eq. (10). Fitting of the data to a power law function ${\lambda }^{\left(\alpha \right)}\sim k^{\beta }$ yields $\beta \approx 1.0$ for the coupled standard maps and $\beta \approx 2.0$ for the coupled circle maps.

Figure 18

(Color online) The dispersion relation ${\lambda }^{\left(\alpha \right)}$ vs. $k$ for the model Eq. (10) with $f\left(z\right)=2z$ (mod 1) and $ϵ=-1.3$. The analytical expression $\lambda =\ln [1-4ϵ(1-\cos k)]$ has a perfect agreement with the numerical simulations.

Figure 19

(Color online) The dispersion relation ${\lambda }^{\left(\alpha \right)}$ vs. $k$ for the model Eq. (8) with $f\left(z\right)=2z$ (mod 1) and $ϵ=-1.3$. The agreement of the numerical simulations and the analytical estimation, which is $\lambda \equiv \ln \left[{\mu }_1\left(k\right)\right]=\ln [(\eta +2+\sqrt{{\eta }^2+4\eta })∕2]$, is perfect.

Figure 20

(Color online) Dispersion relation ${\lambda }^{\left(\alpha \right)}$ vs. $k$ for the model Eq. (8) with $f\left(z\right)=2z$ (mod 1) and ${\gamma }_t^l={\gamma }_0$. The open symbols represent the results of numerical simulations using Benettin’s method while the filled symbols are for the analytical result stated in Eq. (25). The agreement between them is perfect. The parameter setting used is $L=256$ and $ϵ=-1.3$.

Figure 21

(Color online) Dispersion relation ${\lambda }^{\left(\alpha \right)}$ vs. $k$ for the coupled standard maps Eq. (8) with ${\gamma }_t^l={\gamma }_0\ne 0$. The results are similar to those shown in Fig. 20. This simulation confirms the validity of Eqs. (26, 27) for more general systems beyond the one with $f\left(z\right)=2z$ (mod 1).

Figure 22

$\cos {\theta }_{uv}^{\left(\alpha \right)}$ [see Eq. (28)] for the coupled standard maps Eq. (8) with $ϵ=0.6$. From the plot one can infer that the $u$ and $v$ components of LVs with $\alpha \approx 128$, i.e., ${\lambda }^{\left(\alpha \right)}\approx 0$, are nearly orthogonal.

Figure 23

(Color online) The peak wave number $k_{\mathit{max}}$ vs. $\alpha$ for the $u$ and $v$ components of the LVs for the coupled standard maps Eq. (8) with $ϵ=0.6$. The two curves are mutual mirror image due to the symmetry of Hamiltonian system ${\Gamma }_v^{\left(\alpha \right)}=-{\Gamma }_u^{(2L+1-\alpha )}$.

Figure 24

(Color online) $S_{\mathit{tot}}^{\left(\alpha \right)}$ vs. $\alpha$ for the coupled standard maps Eq. (8) with $ϵ=0.6$. Here $S_{\mathit{tot}}^{\left(\alpha \right)}$ [see Eq. (29)] is the total power in the $u$ or $v$ component of a LV with index $\alpha$. For $\alpha <128$, i.e., ${\lambda }^{\left(\alpha \right)}>0$, the total power in the $v$ component of a LV is always smaller than in the $u$ component.

Figure 25

(Color online) The normalized height $S^{\left(\alpha \alpha \right)}\left(k_{\mathit{max}}\right)∕S_{\mathit{tot}}^{\left(\alpha \right)}$ of the highest peak in the static LV structure factors of the coupled standard maps Eq. (8) with $ϵ=0.6$. This quantity is a rough measure of the significance of the HLMs. Numerical results show that coherent wave structures are more significant in $u$ components than in $v$ components.

Figure 26

(Color online) Similar to Fig. 24 but with $f\left(z\right)=2z$ (mod 1) and $ϵ=-1.3$. Again for ${\lambda }^{\left(\alpha \right)}>0$ the total power in the $v$ component of a LV is smaller than in the $u$ component especially for the regime ${\lambda }^{\left(\alpha \right)}\approx 0$ where HLMs were found in the $u$ component.

Figure 27

(Color online) $c_1^2\left(k\right)$ vs. $k$ for the force-like coupled Hamiltonian system Eq. (8) with $f\left(z\right)=2z$ (mod 1) and $ϵ=-1.3$. The analytical result from Eq. (22) and the two numerical estimations, where NM1 is the ratio $S_v^{\left(\alpha \alpha \right)}\left(k_{\mathit{max}}\right)∕S_u^{\left(\alpha \alpha \right)}\left(k_{\mathit{max}}\right)$ and NM2 is ${\sum }_l{\mid \delta u^{\left(\alpha \right)l}\mid }^2∕{\sum }_l{\mid \delta v^{l\left(\alpha \right)}\mid }^2$, agree quite well.

Figure 28

(Color online) Ratios $S_v^{\left(\alpha \alpha \right)}\left(k\right)∕S_u^{\left(\alpha \alpha \right)}\left(k\right)$ versus $k$ for several LVs of coupled standard maps Eq. (8) with $ϵ=0.6$. For comparison, the value of $c^2\left(k\right)$ for the case of coupled Bernoulli shift maps with $ϵ=-0.6$ [see Eq. (22)] is plotted as a dashed line in the same figure. As $k\to 0$, the ratios follow the trend of $c_1^2\left(k\right)\sim k^2$ quite well.

Figure 29

(Color online) (a) Peak wave number $k_{\mathit{max}}$ and (b) $S\left(k_{\mathit{max}}\right)$ vs. $\alpha$ for the $u$ and $v$ components of LVs for the coupled skewed tent maps with $ϵ=1.3$ and $r=0.2$ [40]. The coincidence of $k_{\mathit{max}}$ for both components of LVs in the regime $\lambda \approx 0$ supports our statement that it is possible to observe the indication of HLMs in the momentum part of Lyapunov vectors if the HLMs in the corresponding coordinate part is significant enough. The comparision of $S\left(k_{\mathit{max}}\right)$ for the parts of LVs shows that the long wavelength structures in the momentum part are much weaker than in the corresponding coordinate part.

Figure 30

(Color online) (a) Profile of $u$ and $v$ components of LV nos. 128 and 129 of coupled standard maps Eq. (8) and (b) that for LV no. 164 of coupled circle maps Eq. (10). The LVs shown are associated with translational invariance.

Figure 31

(Color online) Similar to Fig. 30 but for the case with asymmetric interaction with $f\left(z\right)=(1∕2\pi )\sin \left(2\pi z\right)$ and $g\left(z\right)=(1∕2\pi )\cos \left(2\pi z\right)$ [cf. Eq. (18)]. Note in contrast to Fig. 30 the LVs associated with translational invariance are now featureless.

Figure 32

Profile of LV no. 164 for the diffusively coupled circle maps with $ϵ=1.3$ [see Eq. (11)]. This LV corresponds to the conservation law of the system considered.

Figure 33

(a) The fluctuations $\sigma \left({\lambda }_{\tau }\right)$ vs. $\alpha$ show strong variation in the neighborhood of the zero LEs ($\alpha =127$,128). This is demonstrated by the temporal evolution of the finite-time Lyapunov exponent ${\lambda }_{\tau }$ for $\alpha =126$ and $\alpha =127$. The results shown are for the force-like coupled standard maps Eq. (8) with $ϵ=1.3$.

Figure 34

Similar to Fig. 33 but for the diffusively coupled circle maps Eq. (11) with $ϵ=1.3$. Here the zero-value LE behaves quite similar to its nonzero neighbours.

Figure 35

Lyapunov spectrum for the force-like coupled circle maps on a two-dimensional $16\times 16$ lattice with $ϵ=0.8$ [see Eq. (33)]. The inset shows the part about $\lambda =0$.

Figure 36

(Color online) Contour plot of the static LV structure factor for LV nos. 130 and 146 of the force-like coupled circle maps on a two-dimensional $16\times 16$ lattice with $ϵ=0.8$. The existence of sharp peaks in the spectra is obvious.

Figure 37

(Color online) Dispersion relation ${\lambda }^{\left(\alpha \right)}$ vs. $k_{\mathit{max}}$ of the force-like coupled circle maps on a two-dimensional $32\times 32$ lattice with $ϵ=2.0$. Fitting the numerical data to a power-law ${\lambda }^{\left(\alpha \right)}\sim k_{\mathit{max}}^{\beta }$ yields $\beta \simeq 2.04$, i.e., the two-dimensional lattice of dissipative maps is also characterized by a quadratic dispersion relation as the one-dimensional case shown in Fig. 17.