Lyapunov Spectral Gap and Branch Splitting of Lyapunov Modes in a Diatomic System

Lyapunov instability of a “diatomic” system of coupled map lattices is studied and the dynamics of Lyapunov modes (LMs) is compared with phonon dynamics. Similar to the phonon case mass differences between neighboring sites induce gaps in the Lyapunov spectrum and LMs split into two types correspondingly. An unexpected finding is that contrary to the phonon case a nontrivial threshold value for the mass difference is required for the occurrence of the spectral gap and the splitting of LMs. A possible origin of such a nontrivial threshold value of mass differences is suggested.

Figure 1

Lyapunov exponents as function of the Lyapunov index $\alpha$ and the mass ratio $\kappa \equiv {ϵ}_2/{ϵ}_1$. Here $L=1024$ and $r=0.2$. Notice that the gaps in the Lyapunov spectra disappear as $\kappa$ increases beyond a certain threshold value ${\kappa }_c$.

Figure 2

The Lyapunov spectral gap size $\delta \lambda \equiv {\lambda }^{(L/2-1)}-{\lambda }^{(L/2)}$ vs the mass ratio $\kappa$ with $r=0.2$. Obviously, $\delta \lambda$ has different system size dependencies as $\kappa$ is below or beyond the threshold value ${\kappa }_c\approx 0.45$.

Figure 3

Coordinate component of the offset vector for the first lattice site $\delta u_1^{(\alpha )}$ vs that for the second site $\delta u_2^{(\alpha )}$ for Lyapunov modes with (a) $\alpha =L/2-1$ and (b) $\alpha =L/2$, which belong to the acoustic and optical branch, respectively. Here $L=128$, $\kappa =0.125$, and $r=0.3$. Note that the distributions of phase points are highly anisotropic for both LMs. (c) The probability distributions of the quantity $\theta (t)\equiv \frac{1}{\pi }\arctan (\delta u_2^{(\alpha )}/\delta u_1^{(\alpha )})$. Two additional cases with $\kappa =0.25$ and 0.5 are also shown. Note that the peak in $P[\theta (t)]$ fades away gradually with increasing $\kappa$.

Figure 4

(a) Phase diagram for the diatomic system Eq. (1). Appearance of spectral gaps is judged in a system with $L=128$ by using the criterion $\delta \lambda >{\lambda }^{(L/2-2)}-{\lambda }^{(L/2-1)}+{\lambda }^{(L/2+1)}-{\lambda }^{(L/2+2)}$. Notice that the appearance of a gap in the Lyapunov spectrum becomes more difficult to observe with increasing $r$. (b) Lyapunov spectra for several cases with $\kappa =1$, $r=0.05$, 0.1, 0.15, and 0.2, respectively. See that the regime of the Lyapunov spectrum with step structures shrinks with increasing $r$.