When Can One Observe Good Hydrodynamic Lyapunov Modes?

Inspired by recent results on differences in fluctuations of finite-time Lyapunov exponents between hard-core and soft-potential systems we surmise that partial domination of the Oseledec splitting (DOS) with respect to subspaces associated with near-zero Lyapunov exponents is essential for observing good hydrodynamic Lyapunov modes (HLMs). Numerical results for coupled map lattices are presented to show the importance of DOS for observing good HLMs. This is achieved by relating splitting parameters to the maximum value of the Lyapunov mode structure factor.

Figure 1

(a) Lyapunov spectrum ${\Lambda }^{(\alpha )}$ with bars indicating the standard deviations of LLEs ${\lambda }^{(\alpha )}$, (b) the splitting parameters $p_{\mathrm{split}}$ and $\nu$ [18], (c) the minimal angle between Oseledec subspaces $\theta$, (d) $S^{(\alpha )}(k_{\max })/S_{\mathrm{total}}^{(\alpha )}$, and $s_{\min }^{(\alpha )}$ for a case with $f_T(z)$, $ϵ=1.3$, $r=0.02$, and $L=256$.

Figure 2

The significance measures of HLMs $S^{(\alpha )}(k_{\max })/S_{\mathrm{total}}^{(\alpha )}$ and $s_{\min }^{(\alpha )}$ as a function of the splitting parameter $p_{\mathrm{split}}$. The cases presented include $f_B(z)$ with $ϵ=-1.3$, $r=0.1$ (B1) and 0.2 (B2), $f_T(z)$ with $ϵ=1.3$, $r=0.02$ (T1), 0.05 (T2), 0.1 (T3), and 0.2 (T4), respectively. Filled and open symbols are for $S^{(\alpha )}(k_{\max })/S_{\mathrm{total}}^{(\alpha )}$ and $s_{\min }^{(\alpha )}$, respectively.

Figure 3

The significance measure of HLMs $S^{(\alpha )}(k_{\max })/S_{\mathrm{total}}^{(\alpha )}$ as a function of the fraction of time with DOS violation for the cases as shown in Fig. 2.