Geometry of the edge of chaos in a low-dimensional turbulent shear flow model

We investigate the geometry of the edge of chaos for a nine-dimensional sinusoidal shear flow model and show how the shape of the edge of chaos changes with increasing Reynolds number. Furthermore, we numerically compute the scaling of the minimum perturbation required to drive the laminar attracting state into the turbulent region. We find this minimum perturbation to scale with the Reynolds number as ${\mathrm{Re}}^{-2}$.

Figure 1

The histogram shows the distance of the edge from the laminar attractor for 100 000 randomly chosen directions in the nine-dimensional state space, at $\mathrm{Re}=400$. The bin size is 0.0003. The largest bin count of 2620 is at the bin centered at 0.005 76.

Figure 2

The distance of the periodic orbit on the edge from the laminar attractor on the $y$ axis vs Reynolds number on the $x$ axis is plotted, for $\mathrm{Re}=200,300,...,1000,1500,2000$.

Figure 3

Panels (a), (c), and (e) show the contour plots for the lifetimes, in the plane 1 formed using orthogonal vectors $v1$ and $v2$, for Reynolds numbers 200, 400, and 1000, respectively. Panels (b), (d), and (f) show the contour plots for the lifetimes, in the plane 2, formed using orthogonal vectors $v1$ and $v3$, for Reynolds numbers 200, 400, and 1000, respectively. In each of the plots, the points with lifetimes exceeding the maximum value indicated by the color bar are color coded with the color corresponding to the maximum value. The figure is shifted so that the origin corresponds to the laminar attractor.

Figure 4

$1/\tau$ vs $\mathrm{Re}$ is plotted for $\mathrm{Re}=200,300,600,700,800,900,1000,1500,2000$. Here $\tau$ is the average lifetime. Note that we exclude $\mathrm{Re}=400,500$. Because of the presence of a nontrivial attractor at $\mathrm{Re}=400,500$ (sustained turbulence instead of transient turbulence corresponding to a chaotic saddle), the average lifetime for convergence to the laminar attractor is not defined.

Figure 5

For Re 600, we compute the lifetimes for points in plane 1 starting from the laminar attractor until radius values of 0.05. The figure considers those points lying in the turbulent region and plots the corresponding radial distance on the $x$ axis vs lifetime on the $y$ axis. Observe that in the transient turbulence region, the distribution of lifetimes is largely independent of radial distance.

Figure 6

The basin of attraction corresponding to $P_{\mathrm{att}}$ is plotted in red (dark gray), where $P_{\mathrm{att}}\approx (0.877,0.115).$ The black rectangle denotes the region which is magnified in Fig. 7.

Figure 7

Panels (a), (b), and (c) are magnifications of the black rectangle shown in Fig. 6, close to $P_{\mathrm{att}}$ and plot the basins of attraction corresponding to $P_{\mathrm{att}}$ in red (dark gray), to $P_{\mathrm{attsym}}$ in black, and to the laminar attractor in blue (dark gray), respectively.

Figure 8

For 10 000 randomly chosen vectors, the minimum distance from the edge of chaos vs Reynolds number is plotted for $\mathrm{Re}=200,300,...,1000,1500,2000$.

Figure 9

For 10 000 randomly chosen vectors, the average distance from the edge of chaos vs Reynolds number is plotted, for $\mathrm{Re}=200,300,...,1000,1500,2000$.

Figure 10

For 10 000 randomly chosen vectors, the maximum distance from the edge of chaos vs Reynolds number is plotted, for $\mathrm{Re}=200,300,...,1000,1500,2000$. We believe that the deviation from the scaling of the maximum boundary distance is caused by the bending of the long tendril-like structures seen in Figs. 3 to 3. This deviation from the scaling does not feature in Fig. 9 (which plots the average distance from the edge of chaos vs Reynolds number), as there are few points on the edge which have a large distance from the laminar attractor. (See the histogram in Fig. 1.)