Lyapunov Analysis Captures the Collective Dynamics of Large Chaotic Systems

We show, using generic globally coupled systems, that the collective dynamics of large chaotic systems is encoded in their Lyapunov spectra: most modes are typically localized on a few degrees of freedom, but some are delocalized, acting collectively on the trajectory. For globally coupled maps, we show, moreover, a quantitative correspondence between the collective modes and some of the so-called Perron-Frobenius dynamics. Our results imply that the conventional definition of extensivity must be changed as soon as collective dynamics sets in.

Figure 1

Collective chaos in globally coupled limit-cycle oscillators [Eq. (1)] with $c_1=-2.0$, $c_2=3.0$, $K=0.47$, $N={10}^7$. (a) Typical snapshots of the population in the complex plane; note the stretching and folding of the supporting line. (b) Typical distribution $\rho (\theta \equiv \arg W)$ for a configuration where no fold is present. (c) Time series $|⟨W⟩|$.

Figure 2

Spectra of $\lambda$ vs $h$ (a), $Y_2$ vs $h$ (b), and $Y_2$ vs $\lambda$ (c) for the system depicted in Fig. 1 for different sizes $N$. Insets: close-ups of the positive branch. (Arrows indicate increasing $N$.)

Figure 3

Size dependence for the system depicted in Figs. 1, 2. (a) $\lambda$ and $Y_2$ vs $N$ around $\lambda =0$. The two modes with $\lambda =0$ are shown in filled symbols. (b) same as (a) for the first 3 modes. (c) $Y_2$ vs $N$ for the last four modes. (d) Distribution of instantaneous $Y_2$ values for the first mode vs $N$. Inset: rescaling showing that eventually $Y_2\sim 1/N$. [The dashed lines in (a),(c) correspond to the power law $Y_2\sim 1/N$.]

Figure 4

Lyapunov analysis for globally coupled noisy logistic maps [Eq. (2)] with $a=1.57$, $K=0.28$, $\sigma =0.1$, and the corresponding PF equation [Eq. (3)]. (a) ($Y_2$, $\lambda$) spectra for different $N$. The vertical lines indicate the location of the first two LE of the PF dynamics. (b) First three LE vs system size $N$ (symbols) and for first PF LE (dashed line). (c) $Y_2$ vs $N$ for the first three modes [same symbols as in (b), the dashed line indicates the power law $Y_2\sim 1/N$]. (d) Typical snapshot for the distribution $\rho (x)$ (top) and for the first CLV $\delta \rho (x)$ (bottom). Solid lines and circles are the results from PF and maps of size $N={10}^7$, respectively.