Heteroclinic and homoclinic connections in a Kolmogorov-like flow

Recent studies suggest that unstable recurrent solutions of the Navier-Stokes equation provide new insights into dynamics of turbulent flows. In this study, we compute an extensive network of dynamical connections between such solutions in a weakly turbulent quasi-two-dimensional Kolmogorov flow that lies in the inversion-symmetric subspace. In particular, we find numerous isolated heteroclinic connections between different types of solutions—equilibria, periodic, and quasiperiodic orbits—as well as continua of connections forming higher-dimensional connecting manifolds. We also compute a homoclinic connection of a periodic orbit and provide strong evidence that the associated homoclinic tangle forms the chaotic repeller that underpins transient turbulence in the symmetric subspace.

Figure 1

Low-dimensional projection of the state space generated using data from numerical simulation of a weakly turbulent Kolmogorov-like flow. Instantaneous snapshots of the flow in the physical space are associated with points in the state space. Turbulent evolution is represented by a trajectory (black curve) passing through neighborhoods of unstable equilibria (spheres) and periodic orbits (loops). Stable and unstable manifolds (green curves and light blue surface) of these solutions constrain the dynamics in their neighborhoods. The projection method is described in Appendix pp1.

Figure 2

Rotationally symmetric equilibrium solutions (a) ${\mathrm{EQ}}_2$, (b) ${\mathrm{EQ}}_0$, (c) ${\mathrm{EQ}}_1$, (d) ${\mathrm{EQ}}_3$. Color (black arrows) indicates vorticity (velocity).

Figure 3

Trajectories $\mathbf{u}(\theta ,t)$ (gray curves) approximating the unstable manifold of ${\mathrm{EQ}}_2$ (red sphere). $\mathbf{u}(\theta ,t)$ originate on a small circle around ${\mathrm{EQ}}_2$ that lies in a plane spanned by the complex conjugate pair of unstable eigenvectors ${\widehat{\mathbf{e}}}_1$ and ${\widehat{\mathbf{e}}}_2$. Coordinates $c_1,c_2$ are projections of $\mathbf{u}(\theta ,t)$ onto orthonormal vectors constructed from ${\widehat{\mathbf{e}}}_1, {\widehat{\mathbf{e}}}_2$.

Figure 4

State space speeds $s(\theta ,t)$ along two trajectories in the unstable manifold of ${\mathrm{EQ}}_2$ that exhibit contrasting dynamics. Solid (dashed) curve is speed along a trajectory that approaches (does not approach) an equilibrium closely after leaving the neighborhood of ${\mathrm{EQ}}_2$. The lowest speed $s_m\left(\theta \right)={min}_{t}s(\theta ,t)$ for $t/{\tau }_c\in [10,25]$ for each curve is marked using an open circle.

Figure 5

Minimum state space speed $s_m\left(\theta \right)={min}_{t}s(\theta ,t)$ along each trajectory $\mathbf{u}(\theta ,t)$ in the 2D unstable manifold of ${\mathrm{EQ}}_2$. Trajectories corresponding to $s_m\ll 1$ are possible dynamical connections from ${\mathrm{EQ}}_2$ to an equilibrium.

Figure 6

Heteroclinic connections from ${\mathrm{EQ}}_2$ to various ECSs. (a) A connection (gray curve) to ${\mathrm{EQ}}_0$ that belongs to the family ${\mathrm{DC}}_1$ and an isolated connection ${\mathrm{DC}}_3$ (black curve) to ${\mathrm{EQ}}_1$ (b) Family of connections ${\mathrm{DC}}_2$ (gray curves) that terminate at ${\mathrm{EQ}}_0$ and isolated connections ${\mathrm{DC}}_6, {\mathrm{DC}}_7$ (black curves) to ${\mathrm{PO}}_1$.

Figure 7

State space speed along trajectories $\mathbf{u}({\theta }_4^{-},t)$ (black) and $\mathbf{u}({\theta }_4^{+},t)$ (gray) that separate after shadowing a periodic orbit.

Figure 8

Geometry of state space around ${\mathrm{PO}}_1$. Trajectories $\mathbf{u}({\theta }_4^{\pm },t)$ approach ${\mathrm{PO}}_1$ indistinguishably but depart from its neighborhood in opposite directions (inset) guided by the manifold trajectories ${\mathbf{u}}^{\pm }(\eta ,t)$ (dashed red curves). $\mathbf{u}({\theta }_4^{-},t)$ (black curve) converges to ${\mathrm{EQ}}_0$ and $\mathbf{u}({\theta }_4^{+},t)$ (gray curve) becomes turbulent. Similar behavior is manifested by $\mathbf{u}({\theta }_3^{\mp },t)$.

Figure 9

Heteroclinic connections ${\mathrm{DC}}_8, {\mathrm{DC}}_9$ (black curves) from ${\mathrm{EQ}}_2$ to ${\mathrm{PO}}_1$ that sandwich the family of connections ${\mathrm{DC}}_1$ (gray curves) from ${\mathrm{EQ}}_2$ to ${\mathrm{EQ}}_0$. The solid (dashed) red curve is the connection ${\mathrm{DC}}_{10}$ (${\mathrm{DC}}_{11}$) from ${\mathrm{EQ}}_2$ to ${\mathrm{EQ}}_3$ (${\mathrm{EQ}}_3$ to ${\mathrm{EQ}}_0$).

Figure 10

Minimum state space speed (black curve) $s_m$ along trajectories ${\mathbf{u}}^{-}(\eta ,t)$ in the 1D unstable manifold of ${\mathrm{PO}}_1$. $\eta \in [0,T)$ parametrizes points along ${\mathrm{PO}}_1$.

Figure 11

Family of heteroclinic connections (${\mathrm{DC}}_{14}$, gray curves) from ${\mathrm{PO}}_1$ to ${\mathrm{EQ}}_0$. The black curve is a homoclinic orbit of ${\mathrm{PO}}_1$ (${\mathrm{DC}}_{15}$) and the blue curve is a heteroclinic connection from ${\mathrm{PO}}_1$ to ${\mathrm{EQ}}_3$ (${\mathrm{DC}}_{16}$). The red curve is ${\mathrm{DC}}_{11}$ from ${\mathrm{EQ}}_3$ to ${\mathrm{EQ}}_0$.

Figure 12

Heteroclinic connections from ${\mathrm{PO}}_2$ (green loop) to ${\mathrm{EQ}}_0$ (${\mathrm{DC}}_{20}$, black curve) and ${\mathrm{PO}}_1$ (${\mathrm{DC}}_{21}$, gray curve). ${\mathrm{DC}}_{21}$ eventually converges to ${\mathrm{PO}}_1$ tracing out a helical path similar to that shown in Fig. 8.

Figure 13

Heteroclinic connection ${\mathrm{DC}}_{23}$ from ${\mathrm{PO}}_2$ (green loop) to ${\mathrm{PO}}_0$ (blue loop). The inset shows the unstable quasiperiodic orbit ${\mathrm{QP}}_1$ (red curve) that lies near stable ${\mathrm{PO}}_0$.

Figure 14

Topology of connections. Double lines represent multiple distinct connections. Solid (dashed) lines represent short (long) connections.

Figure 15

Instantaneous distances from a long transient turbulent trajectory to ${\mathrm{PO}}_1$ ($D_1$, red) and ${\mathrm{EQ}}_0$ ($D_0$, blue). Repeated close passes to ${\mathrm{PO}}_1$ suggest that the homoclinic tangle between the stable and unstable manifolds of ${\mathrm{PO}}_1$ underpins transient turbulence in symmetric subspace. Vanishing of $D_0$ implies convergence to ${\mathrm{EQ}}_0$.

Figure 16

Recurrence analysis to detect signatures of both EQs and POs (a) Recurrence (black curve) and speed (gray curve) plots for a trajectory that originates at ${\mathrm{PO}}_1$ and visits the neighborhoods of ${\mathrm{EQ}}_0$ ($t/{\tau }_c=30$) and ${\mathrm{PO}}_2$ ($t/{\tau }_c\in [33,40]$). (b) Minimum recurrence $r_m$ (black curve) and minimum speed $s_m$ (gray curve) for each trajectory ${\mathbf{u}}^{-}\left(\eta ,t\right)$ from ${\mathrm{PO}}_1$. Trajectories with $r_m\ll 1$ ($s_m\ll 1$) correspond to dynamical connections from ${\mathrm{PO}}_1$ to EQs/POs (EQs). $s_m$ was scaled such that $r_m, s_m$ have the same mean. Black circle indicates $r_m$ for the trajectory shown in panel (a).