Periodic solutions and chaos in the Barkley pipe model on a finite domain

Barkley's bipartite pipe model is a continuous two-state reaction-diffusion system that models the transition to turbulence in pipes, and reproduces many qualitative features of puffs and slugs, localized turbulent structures seen during the transition. Extensions to the continuous model, including the incorporation of time delays and constraining the system to finite open domains—a trigger for convective instability—reveal additional solutions to the system, including periodic solutions and chaos unseen in the original $1+1$-dimensional system. It is found that the nature of solutions depends strongly on the size of the domain under study as well as choice of boundary conditions: on a finite domain for a particular window of parameter space, period doubling and chaos are observed.

Figure 1

Solution space at $N=140$ as a function of time delay $\tau$ and $r$, where $\tau$ is number of time steps $\mathrm{\Delta }t$. In this length of domain a variety of solutions is observed. A Hopf bifurcation first occurs in the undelayed system near $(\tau ,r)=(0,0.71)$; with increasing $\tau$, the Hopf bifurcation occurs at increasing values of $r$, establishing a curve that defines the critical value ${\tau }_c$ above which periodic solutions are observed. Near $r=0.839$, independently of $\tau$, a final saddle-node bifurcation takes place whereby periodic solutions are destroyed and a homogeneous steady state (HSS) emerges (Fig. 2).

Figure 2

Near the saddle-node bifurcation at $r_c\approx 0.839$, the frequencies of oscillation for $q$ increase in accordance with the universal inverse square-root scaling law for saddle-node bifurcations. Shown is the best fit curve $\mathrm{frequency}=a+b{(r_c-r)}^{-0.501}$, where $a=1520$ and $b=50.6$. This figure was drawn using data from $N=500$.

Figure 3

Spacetime evolution of $q(z,t)$ in the $\overline{U}=0$ comoving frame with the spatial variable $z$ on the horizontal axis and time $t$ on the vertical. The inlet of the pipe ($z\sim 0$) can show relatively large initial turbulent transients. (a) Periodic solutions at $r=0.8274$, $N=500$, $\tau =0$. In a shorter domain, these same parameters result in canards (see Fig. 8). (b) A chaotic trajectory at $r=0.82$, $N=140$, $\tau =21$.

Figure 4

Period-doubling route to chaos demonstrated in phase space, with $q$ on the horizontal axis and $\dot{q}$ on the vertical. $N=142$, $\tau =7$, and (a) $r=0.780$, (b) $r=0.787$, (c) $r=0.789$, and (d) $r=0.79035$.

Figure 5

Period-doubling route to chaos demonstrated in phase space with $u$ on the horizontal axis and $\dot{u}$ on the vertical. $N=142$, $\tau =7$, and (a) $r=0.780$, (b) $r=0.787$, (c) $r=0.789$, and (d) $r=0.79035$.

Figure 6

Bifurcation diagram for the strongly chaotic region at $N=140$; initial conditions are $q(t=0)=0.70$.

Figure 7

Top: A chaotic time series at $r=0.8176$ and $\tau =22$ for $N=140$. Bottom: The corresponding phase portraits: $(q,\dot{q})$ (red, bottom left) and $(u,\dot{u})$ (blue, bottom right).

Figure 8

Coexistence of small- and large-amplitude solutions at $r=0.8274$, $N=150$, $\tau =0$. Canard behavior is observed when a trajectory, initially following the small-amplitude limit cycle, undergoes a fast excursion to the large-amplitude limit cycle before returning to the original orbit. (a) The spacetime evolution of $q(z,t)$; and the (b) phase space diagrams of $(q,\dot{q})$ (top) and $(u,\dot{u})$ (bottom) during the canard cycle. (c) A canard cycle in $(u,q)$ space demonstrating orbits around the small-amplitude limit cycle with a quick excursion to a large-amplitude relaxation cycle. The small-amplitude limit cycle is destroyed at the saddle-node bifurcation at $r\approx 0.839$.

Figure 9

Nearest-neighbor reaction-diffusion dynamics from time step $t$ (top row) to $t+dt$ (bottom).