Hyperbolicity and the Effective Dimension of Spatially Extended Dissipative Systems

Using covariant Lyapunov vectors, we reveal a split of the tangent space of standard models of one-dimensional dissipative spatiotemporal chaos: A finite extensive set of $N$ dynamically entangled vectors with frequent common tangencies describes all of the physically relevant dynamics and is hyperbolically separated from possibly infinitely many isolated modes representing trivial, exponentially decaying perturbations. We argue that $N$ can be interpreted as the number of effective degrees of freedom, which has to be taken into account in numerical integration and control issues.

Figure 1

Lyapunov spectrum ${\Lambda }^{(j)}$ made of exponents arranged by decreasing value for the one-dimensional KS equation with $L=96$. Black upward triangles: $k_{\mathrm{cut}}=422\pi /L$ and PBC. Red downward triangles: $k_{\mathrm{cut}}=85\times 2\pi /L$ and PBC. Blue dots: $k_{\mathrm{cut}}=170\times \pi /L$ and RBC [$u(L,t)=u(0,t)=0$]. In all three cases, the spectrum yields a positive region with maximum Lyapunov exponent ${\Lambda }^{(1)}\approx 0.09$. Top inset: ${\Lambda }^{(j)}$ vs $j^4$ for the large-$j$ (isolated) modes. Bottom inset: Close-up around the threshold.

Figure 2

Properties of CLVs for the KS system ($L=96$, $k_{\mathrm{cut}}=42\times 2\pi /L$, and PBC). (a) Spatiotemporal plot of a typical vector in the isolated region ($j=46$, total time 100). (b) Spatial power spectra of vectors of indices $j=1$, 16, 32, 38, 44, 52, 60, 68, 76, and 84 (from left to right in peak position). (c) Top panel: Peak wave number in the power spectra (red circles) and $k=[j/2]2\pi /L$ (black line). Bottom panel: DOS violation fraction ${\nu }_{\tau }^{(j,j+1)}$ for pairs of neighboring vectors (pairs within the same step are omitted). (d) Angle distributions between pairs of vectors. (e) The same as (d) but with a different abscissa.

Figure 3

CGL equation in the amplitude turbulence regime ($L=64$, $k_{\mathrm{cut}}=31\times 2\pi /L$, and PBC). (a),(b) Spatiotemporal plots of the phase component of a typical vector $j=91$ in the isolated region (same trajectory, but at two distant periods of time, during a total time of 20 for each plot). (c) Lyapunov spectrum; inset: close-up around threshold. (d) Time fraction ${\nu }_{\tau }^{(j,j+1)}$ of DOS violation, as a function of $\tau$ ($j=78$, 82, 86, 90, and 94, from top to bottom).

Figure 4

Extensivity of the Lyapunov spectrum for the KS equation with PBC. Inset: Quantities indicating effective dimensions of the system: number of non-negative exponents (circles), Kaplan-Yorke dimension (squares), metric entropy (diamonds, multiplied by 50), and number of physical modes (triangles).