Hyperbolic decoupling of tangent space and effective dimension of dissipative systems

We show, using covariant Lyapunov vectors, that the tangent space of spatially extended dissipative systems is split into two hyperbolically decoupled subspaces: one comprising a finite number of frequently entangled “physical” modes, which carry the physically relevant information of the trajectory, and a residual set of strongly decaying “spurious” modes. The decoupling of the physical and spurious subspaces is defined by the absence of tangencies between them and found to take place generally; we find evidence in partial differential equations in one and two spatial dimensions and even in lattices of coupled maps or oscillators. We conjecture that the physical modes may constitute a local linear description of the inertial manifold at any point in the global attractor.

Figure 1

Spectrum of the Lyapunov exponents ${\Lambda }^{\left(j\right)}$ for the 1D KS equation with $L=96$. (a) Full spectra. Different symbols correspond to different cutoff wave numbers $k_{\mathrm{cut}}$ and different boundary conditions, either PBC or RBC, as indicated in the legend. (Inset) Close-up around the threshold. (b) Close-up of (a) showing the positive end of the spectra. (c) The same Lyapunov spectra shown against $j^4$. The data for $k_{\mathrm{cut}}=42\times 2\pi /L$ with PBC (black upward triangles) are omitted for the sake of clarity. The value of the Kaplan-Yorke dimension is about $21.6$ for $k_{\mathrm{cut}}=42\times 2\pi /L$ and PBC.

Figure 2

Typical spatiotemporal structure of the trajectory $u(x,t)$ (a) and CLVs $\delta u^{\left(j\right)}(x,t)$ (b), (c) in the physical and the spurious regions, respectively, for the 1D KS equation ($L=96,k_{\mathrm{cut}}=42\times 2\pi /L$, PBC). The index of the CLV shown here is $j=8$ (b) and $j=46$ (c).

Figure 3

Spatial power spectrum of the CLVs for the 1D KS equation ($L=96,k_{\mathrm{cut}}=42\times 2\pi /L$, PBC). (a) Power spectrum of CLVs of index $j=1,16,32,38,44,52,60,68,76$, and 84 (from left to right at the peak positions). (b) Peak wave number $k_{\mathrm{peak}}$ of the power spectra (red circles) and $k=[j/2]\times 2\pi /L$ (blue line).

Figure 4

Distributions ${\rho }_{\mathrm{v}}\left(\theta \right)$ of the angles $\theta$ between pairs of CLVs for the 1D KS equation ($k_{\mathrm{cut}}=42\times 2\pi /L$, PBC). (a) ${\rho }_{\mathrm{v}}\left(\theta \right)$ versus $\theta$ for $L=96$. The indices of the pairs are indicated on the figure. (b) ${\rho }_{\mathrm{v}}\left(\theta \right)$ versus $1/\theta$ for $L=96$ and for pairs (41,42), (43,44), (61,62), (81,82), (20,60) from right to left. The ordinate is averaged over both sides of the distribution. (Inset) Close-up of the distributions. (c) Angle distributions ${\rho }_{\mathrm{v}}\left(\theta \right)$ for the pair (41,42) with varying system size $L=96,97,\cdots ,105$ (from inside to outside in the figure). The distribution is obtained from shorter simulations for $L>99$, recorded over time ${10}^5$. (d) Probability density ${\rho }_{\mathrm{v}0}$ of the distribution ${\rho }_{\mathrm{v}}\left(\theta \right)$ at $\theta =0$ and $\pi$, as a function of $L$ (inset) and of $1/(L-L_{\mathrm{c}})$ with a given $L_{\mathrm{c}}$ (main panel). Pairs of the vectors are fixed at $(41,42)$ (black open symbols) and $(43,44)$ (red full symbols). The value of $L_{\mathrm{c}}$ is $L_{\mathrm{c}}=93.0$ for the pair $(41,42)$ and $L_{\mathrm{c}}=97.5$ for $(43,44)$. The circles and squares correspond to the recording times ${10}^5$ and ${10}^6$, respectively.

Figure 5

Distribution ${\rho }_{\mathrm{s}}\left(\phi \right)$ of the angle $\phi$ between the physical and spurious subspaces for the 1D KS equation ($k_{\mathrm{cut}}=42\times 2\pi /L$, PBC). (a) ${\rho }_{\mathrm{s}}\left(\phi \right)$ versus $\phi$ for $L=96$. The physical subspace is spanned by CLVs of index 1 to $N_{\mathrm{ph}}=43$, while the rest spans the spurious subspace. Note that $\phi$ is by definition equal to or smaller than $\pi /2$, in contrast to $\theta$ in Fig. 4. (Inset) The same data shown against $1/\phi$. (b) Probability density ${\rho }_{\mathrm{s}0}$ of the distribution ${\rho }_{\mathrm{s}}\left(\theta \right)$ at $\phi =0$ for varying system size $L$. It is measured between two subspaces spanned by CLVs of indices given in the legend, and shown against $L$ (inset) and $1/(L-L_{\mathrm{c}})$ (main panel) with the same $L_{\mathrm{c}}$ as in Fig. 4d. The circles and squares correspond to the recording times ${10}^5$ and ${10}^6$, respectively.

Figure 6

Violation of DOS in the KS equation ($L=96,k_{\mathrm{cut}}=42\times 2\pi /L$, PBC). (a) Time fraction of the DOS violation, ${\nu }_{\tau }^{(j,j+1)}$, for pairs of the neighboring modes with $\tau =0.2$. The pairs within the same step are omitted. (Inset) ${\nu }_{\tau =0.2}^{(j,j+1)}$ with $j=41$ (black open symbols) and $j=43$ (red solid symbols), shown against $1/(L-L_{\mathrm{c}})$ with the same $L_c$ as in Fig. 4d. The circles and squares correspond to the recording times ${10}^5$ and ${10}^6$, respectively. (b) ${\nu }_{\tau =0.2}^{(j_1,j_2)}$ for arbitrary pairs. The black color indicates ${\nu }_{\tau }^{(j_1,j_2)}=0$, that is, hyperbolically isolated pairs. The red dashed lines indicate the threshold $N_{\mathrm{ph}}=43$ between the physical and spurious modes.

Figure 7

Extensivity of the Lyapunov spectrum (main panel) and effective dimensions (inset) of the KS equation (with PBC). Shown in the inset are the number of the non-negative Lyapunov exponents (black circles), the Kaplan-Yorke dimension (red squares), the metric entropy (green diamonds, multiplied by 50), and the number of the physical modes $N_{\mathrm{ph}}$ (blue triangles). $N_{\mathrm{ph}}$ is estimated here from finite-time simulations of given sizes, so that it could be slightly underestimated. The dashed line indicates linear regression for each quantity.

Figure 8

Spatiotemporal dynamics of the phase turbulence at $\alpha =2.0$ and $\beta =-0.91$ (a) and the amplitude turbulence at $\alpha =2.5$ and $\beta =-2.0$ (b) in the CGL equation with PBC. The system size is $L=64$ for both cases. The gray scale covers the whole range of the values taken by $\left|W\right|$ and $\mathrm{d}\Phi /\mathrm{d}t$ except for the phase velocity $\mathrm{d}\Phi /\mathrm{d}t$ in the amplitude turbulence (b) showing frequent phase slips (bright and dark spots). Notice the different time scales between (a) and (b).

Figure 9

Lyapunov spectrum for the CGL amplitude turbulence. The stepwise structure starts at $j=71$. The red dashed line indicates ${\Lambda }^{\left(j\right)}=1-k^2-{2\langle \left|W\right|}^2\rangle$ derived by Garnier and Wójcik for pure Fourier perturbations [36]. (Inset) The same data against $j^2$. The Kaplan-Yorke dimension in this case is about $28.6$.

Figure 10

Spatiotemporal evolution of a typical CLV in the spurious region ($j=78$) for the CGL amplitude turbulence. The phase component $\arg \delta W^{\left(j\right)}$ is plotted. The two plots are from the same trajectory but at two distant periods of time, showing pure upward traveling waves and patches of upward and downward waves, respectively.

Figure 11

Spatial power spectrum $S_j\left(k\right)$ of the CLVs for the CGL amplitude turbulence, shown as a function of the wave number $k$ and the Lyapunov index $j$. Given that $S_j\left(k\right)$ and $S_j(-k)$ are statistically equivalent, their average is shown in the figure. The black dashed line separates the physical and spurious regions at $N_{\mathrm{ph}}=74$. The white solid line indicates $k=\left[\right(j+1)/4]2\pi /L$.

Figure 12

Distributions of the angle between neighboring CLVs (a), (b) and between the physical and spurious subspaces (c), (d) in the CGL amplitude turbulence. (a) Distribution function ${\rho }_{\mathrm{v}}\left(\theta \right)$ for CLV pairs $(66,67),(70,71),(74,75),(86,87),(94,95)$ from outside to inside. (b) ${\rho }_{\mathrm{v}}\left(\theta \right)$ versus $1/\theta$ for the data shown in (a). The ordinate is averaged over both sides of the distribution. (c) Angle distribution ${\rho }_{\mathrm{s}}\left(\phi \right)$ between the physical subspace spanned by CLVs of indices $1\le j\le N_{\mathrm{ph}}=74$ and the spurious one formed by the remaining CLVs. (d) The same data shown against $1/\phi$.

Figure 13

Violation of DOS in the CGL amplitude turbulence. (a) Time fraction ${\nu }_{\tau }^{(j,j+1)}$ of the DOS violation of the neighbor exponents as a function of $\tau$. The dashed line is a guide for the eyes showing an exponential decay. (b) ${\nu }_{\tau }^{(j_1,j_2)}$ for arbitrary pairs with $\tau =16$. The black color indicates ${\nu }_{\tau }^{(j_1,j_2)}=0$, that is, hyperbolically isolated pairs. The red dashed lines indicate the threshold $N_{\mathrm{ph}}=74$ between the physical and spurious modes.

Figure 14

Lyapunov spectrum ${\Lambda }^{\left(j\right)}$ for the CGL phase turbulence. Different symbols correspond to different regions of the Lyapunov modes (see text). The Kaplan-Yorke dimension in this case is about $14.6$. (Inset) Lyapunov spectra with different spatial resolutions.

Figure 15

Spatial power spectrum $S_j\left(k\right)$ of the CLVs for the CGL phase turbulence, shown as a function of the wave number $k$ and the Lyapunov index $j$. Given that $S_j\left(k\right)$ and $S_j(-k)$ are statistically equivalent, their average is shown in the figure. The peak wave number for each $j$ is indicated by the white rectangles. The black dashed lines separate the four regions of the Lyapunov modes. The white solid lines indicate $k=[j/2]2\pi /L$ (upper line) and $k=\left[\right(j+1)/4]2\pi /L$ (lower line), which are the trivial peak wave numbers found for the spurious modes in the KS equation and in the CGL amplitude turbulence, respectively.

Figure 16

Spatiotemporal evolution of a typical KS-like and AT-like mode in the CGL phase turbulence. (a) Amplitude $\left|W\right|$ of the dynamics. (b), (c) Phase component $\delta {\Phi }^{\left(j\right)}$ of the CLV for the KS-like spurious mode at $j=28$ (b) and for the AT-like one at $j=91$ (c).

Figure 17

Distributions of the angle between neighboring CLVs (a)–(d) and between pairs of subspaces (e), (f) in the CGL phase turbulence. (a) Distribution function ${\rho }_{\mathrm{v}}\left(\theta \right)$ for CLV pairs $(34,35),(25,26),(33,34),(27,28),(29,30)$ from outermost to innermost. They determine the thresholds for the physical, KS-like, and mixed regions. (b) ${\rho }_{\mathrm{v}}\left(\theta \right)$ versus $1/\theta$ for the data shown in (a). The ordinate is averaged over both sides of the distribution. (c) ${\rho }_{\mathrm{v}}\left(\theta \right)$ for CLV pairs $(82,83),(85,86),(86,87),(90,91)$ as indicated by the labels. They are relevant to the threshold between the mixed and AT-like regions. (d) ${\rho }_{\mathrm{v}}\left(\theta \right)$ versus $1/\theta$ for the data shown in (c). The ordinate is averaged over both sides of the distribution. (e) Subspace angle distributions ${\rho }_{\mathrm{s}}\left(\phi \right)$. The abbreviations P, K, M, and A refer to the physical modes ($1\le j\le 27$), the KS-like spurious modes ($28\le j\le 33$), the mixed spurious modes ($34\le j\le 86$), and the AT-like spurious modes ($87\le j\le 126$), respectively. The rightmost two curves overlap almost perfectly. (f) The same data as (e) shown against $1/\phi$.

Figure 18

Violation of DOS in the CGL phase turbulence. (a) Time fraction ${\nu }_{\tau }^{(j,j+1)}$ of the DOS violation for pairs of neighboring Lyapunov exponents as a function of $\tau$. Pairs with and without tangencies in the angle distributions [Figs. 17a and 17b] are denoted by open and solid symbols, respectively. All pairs in the KS-like spurious region ($28\le j\le 33$ excluding those in the same doublet) perfectly satisfy DOS, that is, ${\nu }_{\tau }^{(j_1,j_2)}=0$, even with such a small value of $\tau$ as $0.2$. (b) ${\nu }_{\tau }^{(j_1,j_2)}$ for arbitrary pairs with $\tau =4$. The black color indicates ${\nu }_{\tau }^{(j_1,j_2)}=0$, that is, hyperbolically isolated pairs. The red dashed lines indicate the thresholds of the four regions.

Figure 19

Density plots showing the structure of the trajectory (a) and of the CLVs (b)–(f) in the CGL phase turbulence. (a) Density plot for the amplitude $A$ and the phase curvature $\Psi$ of the trajectory. The red dashed curve indicates the averaged value of $A$ for each bin of $\Psi$. (b)–(f) Density plots for the CLVs, comparing the actual amplitude component $\delta A$ with that expected from the assumption of the slaved amplitude, $\delta A_{\mathrm{p}}$ [Eq. (12)]. The red dashed lines indicate $\delta A_{\mathrm{p}}=\delta A$. The abbreviations P, K, M, and A denote a physical, KS-like, mixed, and AT-like mode, respectively. The upward and downward arrows next to M denote the upper and lower branches, respectively, in the plot of the peak wave number [Fig. 14b]. All the data shown here are recorded during two periods of length ${10}^3$ separated by an interval of roughly ${10}^6$.

Figure 20

Lyapunov spectrum for the coupled tent maps (13) with varying coupling constant $K$. (Inset) Close-up of the end of the spectrum for $K=0.65$. The green lines show the spectrum estimated under the Fourier mode approximation (see text). The Kaplan-Yorke dimension for $K=0.65$ is about $39.9$.

Figure 21

Spatiotemporal evolution of the dynamics $x_i^t$ (a), of a physical Lyapunov mode at $j=1$ (b), and of a spurious Lyapunov mode at $j=251$ (c) in the coupled tent maps with $K=0.65$. The spatial profiles are plotted every eight time steps in order to capture the internal structure of the period-4 collective behavior.

Figure 22

Spatial power spectrum of the CLVs for the tent coupled-map lattice with $K=0.65$, shown as a function of the (integer) wave number $k$ and the Lyapunov index $j$. The dashed line indicates $k=87$, which gives the smallest eigenvalue for the coupling operator $1+(K/2)\mathcal{D}$ (see text). The panel (b) is a close-up of (a) near the peak wave numbers of the CLVs at the end of the Lyapunov spectrum.

Figure 23

Hyperbolicity properties for the tent coupled-map lattice with $K=0.65$. (a) Time fraction ${\nu }_{\tau }^{(j_1,j_2)}$ of the DOS violation for arbitrary pairs with $\tau =2$. The black color indicates ${\nu }_{\tau }^{(j_1,j_2)}=0$, that is, hyperbolically isolated pairs. The red dashed lines indicate the threshold $N_{\mathrm{ph}}=250$ between the physical and spurious modes. (b) Close-up of (a) near the end of the spectrum. (c) Angle distributions ${\rho }_{\mathrm{v}}\left(\theta \right)$ for neighboring CLVs of indices $(246,247),(248,249),...,(254,255)$ from upper left to lower right. (d) ${\rho }_{\mathrm{v}}\left(\theta \right)$ against $1/\theta$. The ordinate is averaged over both sides of the distribution. (e) Angle distribution ${\rho }_{\mathrm{s}}\left(\phi \right)$ between the physical subspace ($1\le j\le 250$) and the spurious one ($251\le j\le 256$). (f) ${\rho }_{\mathrm{s}}\left(\phi \right)$ against $1/\phi$.

Figure 24

Lyapunov spectrum for the CGL lattice (15) with varying $h$. The numerically obtained spectra are shown by the symbols, while the black line indicates the spectrum of the contribution from the coupling term under the Fourier mode approximation for $h=0.5$ (see text). The Kaplan-Yorke dimension is about $8.8$, $11.6$, and $14.7$ for $h=0.5$, $h=0.65$, and $h=0.8$, respectively.

Figure 25

Spatial power spectrum $S_j\left(k\right)$ of the CLVs for the CGL lattice with $h=0.5$ (a) and $h=0.8$ (b), shown as a function of the (integer) wave number $k$ and the Lyapunov index $j$. Given that $S_j\left(k\right)$ and $S_j(-k)$ are statistically equivalent, their average is shown in the figure. Note that the color plots are in different linear scales for the sake of clearity.

Figure 26

Hyperbolicity properties for the CGL lattice with $h=0.5$. (a) Angle distributions ${\rho }_{\mathrm{v}}\left(\theta \right)$ for neighboring CLVs of indices $(22,23),(46,47),(50,51)$, and $(48,49)$ from innermost to outermost curve. (b) ${\rho }_{\mathrm{v}}\left(\theta \right)$ against $1/\theta$. The ordinate is averaged over both sides of the distribution. The order of the curves is the same as in (a) from lower left to upper right. (c) Angle distribution ${\rho }_{\mathrm{s}}\left(\phi \right)$ between the physical subspace ($1\le j\le N_{\mathrm{ph}}$) and the spurious one ($N_{\mathrm{ph}}+1\le j$) with $N_{\mathrm{ph}}=22$ (black solid line). The red dashed line shows the angle distribution between the subspaces of indices $1\le j\le 46$ and $47\le j\le 64$, showing the hyperbolic isolation of the tangled spurious subspace. (d) The same plots as (c) but shown against $1/\phi$ (black circles for the physical and spurious subspaces, and red squares for the other). (e) Time fraction ${\nu }_{\tau }^{(j,j+1)}$ of the DOS violation for pairs of neighboring Lyapunov exponents as a function of $\tau$. (f) Time fraction ${\nu }_{\tau }^{(j_1,j_2)}$ of the DOS violation for arbitrary pairs with $\tau =3$. The black color indicates ${\nu }_{\tau }^{(j_1,j_2)}=0$, that is, hyperbolically isolated pairs. The red dashed lines indicate the threshold $N_{\mathrm{ph}}=22$ between the physical and spurious modes and that between the stepwise spurious modes $1\le j\le 46$ and the tangled spurious modes $47\le j\le 64$.

Figure 27

Hyperbolicity properties for the CGL lattice with $h=0.8$. (a) Angle distributions ${\rho }_{\mathrm{v}}\left(\theta \right)$ for all pairs of neighboring CLVs. Indicated by the arrows are the distributions for the pairs including either or both of the two Lyapunov modes with ${\Lambda }^{\left(j\right)}=0$. All the distributions except the one for the two zero Lyapunov modes, shown by the bold arrow, have finite densities at $\theta =0$ and $\pi$. (b) Time fraction ${\nu }_{\tau }^{(j_1,j_2)}$ of the DOS violation for arbitrary pairs with $\tau =5$. The black color indicates ${\nu }_{\tau }^{(j_1,j_2)}=0$, that is, hyperbolically isolated pairs.

Figure 28

Lyapunov spectrum for the 2D KS equation. The red dashed line shows the values of ${\Lambda }^{\left(j\right)}$ estimated from the dominating wave number $(k_x,k_y)$ assigned trivially by the index (see text). (Left inset) Close-up of the beginning of the spectrum. (Right inset) Lyapunov exponent ${\Lambda }^{\left(j\right)}$ vs $k^2-k^4$, where $k^2\equiv k_x^2+k_y^2$ is the square of the peak wave number for the power spectrum of the $j$th CLV (see Fig. 29). Blue crosses and black circles correspond to the physical and spurious modes, respectively. The red dashed line indicates ${\Lambda }^{\left(j\right)}=k^2-k^4$. The Kaplan-Yorke dimension in this case is about $25.1$.

Figure 29

Power spectrum $S(k_x,k_y)$ of the trajectory (a) and of the CLVs (b)–(f) for the 2D KS equation. The spectra are shown for $k_x,k_y\in [-k_{\mathrm{cut}},k_{\mathrm{cut}}]$ with $k_{\mathrm{cut}}=10\times 2\pi /L$.

Figure 30

Hyperbolicity properties for the 2D KS equation. (a) Time fraction ${\nu }_{\tau }^{(j_1,j_2)}$ of the DOS violation for arbitrary pairs with $\tau =0.2$. The black color indicates ${\nu }_{\tau }^{(j_1,j_2)}=0$, that is, hyperbolically isolated pairs. The red dashed lines indicate the threshold $N_{\mathrm{ph}}=121$ between the physical and spurious modes. (b) Time fraction ${\nu }_{\tau }^{(j,j+1)}$ of the DOS violation for pairs of neighboring Lyapunov exponents as a function of $\tau$. (c) Angle distributions ${\rho }_{\mathrm{v}}\left(\theta \right)$ for neighboring CLVs of indices $(89,90),(129,130),(109,110),(121,122)$, and $(137,138)$ from outermost to innermost curve. Notice that they are not in order with respect to the index. (d) ${\rho }_{\mathrm{v}}\left(\theta \right)$ against $1/\theta$. The ordinate is averaged over both sides of the distribution. The order of the curves is the same as in (c) from upper right to lower left. (e) Angle distribution ${\rho }_{\mathrm{s}}\left(\phi \right)$ between the physical subspace ($1\le j\le N_{\mathrm{ph}}$) and the spurious one ($N_{\mathrm{ph}}+1\le j$) with $N_{\mathrm{ph}}=121$ (green solid line). The red dashed line shows the angle distribution ${\rho }_{\mathrm{s}}\left(\phi \right)$ between the subspaces of indices $1\le j\le 109$ and $110\le j$. (f) The same plots as (e) but shown against $1/\phi$ (green diamonds for the physical and spurious subspaces, and red squares for the other). Each color corresponds to the same index pair throughout in the panels (b)–(f).