Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics

We undertake an exploration of recurrent patterns in the antisymmetric subspace of the one-dimensional Kuramoto-Sivashinsky system. For a small but already rather “turbulent” system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set of equilibria offers a coarse geometrical partition of this manifold. The Newton descent method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin—its backbone consists of several Smale horseshoe repellers, well approximated by intrinsic local one-dimensional return maps, each with an approximate symbolic dynamics. The dynamics appears decomposable into chaotic dynamics within such local repellers, interspersed by rapid jumps between them.

Figure 1

(Color online) Long-time evolution of a typical “sustained turbulence” trajectory for $L=38.5$. (a) Space-time representation of $u(x,t)$ in the $(x,t)$ plane, $x∊[0,L∕2]$ horizontally, $t∊[0,7600]$, in vertical segments. The color (gray scale) represents the magnitude of $u(x,t)$. (b) $(a_1,a_2)$ Fourier mode projection, (c) $(a_3,a_4)$ projection. The typical time scale is set by the shortest periods of the unstable periodic orbits embedded in the central, “wobbly” and side, “traveling wave” patterns of order $T=20–25$ (see Fig. 5), so this is a very long simulation, over 300 “turnover” times. The goal of this paper is to describe the characteristic unstable wobble and traveling wave patterns in terms of a hierarchy of invariant periodic orbit solutions.

Figure 2

(Color online) (a) The orientation of a tangent $\tilde{v}\left(\tilde{x}\right)$ of the guess loop $L\left(\tau \right)$ does not coincide with the orientation of the velocity field $v\left(\tilde{x}\right)$; for a periodic orbit $p$ it does so at every $x∊p$. The Newton descent method aligns the closed loop tangent to the given vector field by driving the loop to a periodic orbit. Newton descent at work for a Kuramoto-Sivashinsky system: (b) a near return extracted from a long-time orbit, (c) initial guess loop crafted from it, and (d) the periodic orbit $p$ reached by the Newton descent. $N=512$ points representation of the loop, $(a_1,a_2)$ Fourier mode projection.

Figure 3

Equilibria in Michelson $(u,u_x)$ representation [23], and as $u\left(x\right)$ spatially periodic profiles: (a),(b) $C_1$; (c),(d) $C_2$; (e),(f) $R_1$; (g),(h) $R_2$; (i),(j) $T$. $L=38.5$, antisymmetric subspace.

Figure 4

Spatial profile of $u(x,t)$ at an instant $t$ when a typical trajectory passes (a),(b) $S_C$, (c),(d) or $S_R$, (e),(f) or equilibrium $T$, the transition region between the central $S_C$ and side regions $S_R$. Compare with Fig. 3.

Figure 5

$(a_1,a_2)$ Fourier mode projection of several shortest periodic orbits in $S_C$. The periods are $T=\left(\mathrm{a}\right)$ 25.6095; (b) 25.6356; (c) 36.7235; (d) 37.4083.

Figure 6

(a) Shortest periodic orbit $p_{R2}$ in $S_R$, $T=12.0798$, $(a_3,a_4)$ Fourier mode projection. (b) Second shortest periodic orbit $p_{R1}$ in $S_R$, $T=20.0228$, $(a_3,a_4)$ projection. (c) $(a_1,a_2)$ projection and (d) $(a_3,a_4)$ projection of a long periodic orbit connecting $S_C$ and $S_R$, $T=355.34$.

Figure 7

(Color online) Projection of a typical state space trajectory segment onto the linearized stability eigenplanes of equilibria $C_1$ and $R_1$. The projection of part of (a) the center orbit to the eigenplane $(e_1,e_2)$ of $C_1$, and (b) the right-sided orbit to the eigenplane $(e_1,e_2)$ of $R_1$.

Figure 8

(a)–(d) Fourier mode projection of the unstable manifold segment $L_C$ of $C_1$ on the Poincaré section ${\mathcal{P}}_C$. (a) ($a_1$, $a_2$) projection of the unstable manifold on ${\mathcal{P}}_C$. (b) ($a_3$, $a_4$) projection ($L_C$ is represented by 564 points). (c) Projection of the first iteration of $L_C$. (d) Projection of all computed unstable periodic orbits of the full domain topological length up to $n=8$. (e),(f) Poincaré return map on ${\mathcal{P}}_C$, the intrinsic coordinate. (e) Bimodal return map in the fundamental domain, with the symbolic dynamics given by three symbols {0, 1, 2}. × marks the position of $C_1$ and ○ mark the turn-back points. (f) Periodic points in the fundamental domain, overlaid over the return map (e).

Figure 9

($a_1$,$a_2$) Fourier mode projection of two cycles in $S_C$ listed in Table : (a) cycle 0011, (b) cycle 0012. Two short typical cycles not listed in the table: (c) a symmetric unstable periodic orbit in $S_C$ with the period $T=77.4483$, (d) an asymmetric unstable periodic orbit in $S_C$ with the period $T=81.3345$.

Figure 10

Fourier mode projections of the unstable manifold $L_R$ of $0\text{cycle}$ on the Poincaré section ${\mathcal{P}}_R$: (a) ($a_1$,$a_2$) and (b) ($a_3$,$a_4$) projection of the unstable manifold on ${\mathcal{P}}_R$. Projection of (c) the first iteration of $L_R$ and (d) the second iteration of $L_R$. (e),(f) The Poincaré return map on ${\mathcal{P}}_R$, intrinsic coordinate: (e) Unimodal return map, with alphabet {0,1}. (f) The return map reconstructed from the periodic points (diamonds). For comparison, the unstable manifold return map is indicated by the dots.

Figure 11

(a) ($a_3$,$a_4$) Fourier mode projection of the attracting periodic orbit 01 within $S_R$. (b) A nearby unstable periodic orbit within $S_R$.