4.1 Connections identified and their properties

Details, specifically the dimension of the unstable manifold of the initial equilibrium, and the passage time $T$ , about the heteroclinic connections are given in table 1, where the equilibria are indicated following the same nomenclature employed by Gibson et al. (Reference Gibson, Halcrow and Cvitanović2009). The considered initial and final equilibria possess discrete symmetries which are part of the equivariance under a symmetry group. To construct a connecting trajectory within the symmetry subspace shared between initial and final equilibria, we explicitly constrain the time evolution to the shared symmetry subspace. The symmetry group considered here is $S=\{1,s_{1},s_{2},s_{3}\}$ and coincides with that of Gibson et al. (Reference Gibson, Halcrow and Cvitanović2008); $s_{1}$ and $s_{2}$ indicate the ‘shift-reflect’ and ‘shift-rotate’ symmetries, respectively, and $s_{3}=s_{1}s_{2}$ . Four of the considered equilibria have all the symmetries in $S$ , and the remaining ones have only $s_{3}$ . The first three connections coincide with those described by Halcrow et al. (Reference Halcrow, Gibson, Cvitanović and Viswanath2009).

Table 1.

Computed heteroclinic connections for $Re=400$ and corresponding symmetry subspaces. The dimension of the unstable manifold of the initial equilibrium ( $EQ_{out}$ ) is $d(W^{u})$ , while $d(W_{S}^{u})$ is the dimension of the intersection of the unstable manifold with the symmetry-invariant subspace reported here for the initial and final equilibrium ( $EQ_{in}$ ). The last column provides the passage time for the different orbits with shell around the initial state of radius $r_{out}=10^{-4}$ ( $r_{out}=10^{-3}$ for underlined ones).

Two manifolds generically intersect if the sum of their dimensions is higher than or equal to the dimension of the state space they are embedded in (Halcrow et al. Reference Halcrow, Gibson, Cvitanović and Viswanath2009). An intersection between the unstable manifold of the initial state and the stable one of the final state can thus generically exist if the codimension of the stable manifold in the $S$ -invariant space of the final state $d(W_{S}^{u}(EQ_{in}))$ is smaller than or equal to the dimension of the unstable manifold of the initial state $d(W_{S}^{u}(EQ_{out}))$ . This geometric condition is satisfied for all the computed heteroclinic connections (compare the fourth and fifth columns in table 1) and indicates the structural stability of all presented connections.

The convergence histories of the computed heteroclinic connections are shown in figure 2. For validation, panel (a) provides the distance of the velocity field $\boldsymbol{u}(t)$ to the target equilibrium $\boldsymbol{u}_{EQ_{in}}$ (continuous line) and to the initial equilibrium $\boldsymbol{u}_{EQ_{out}}$ (dashed line) for the three connections already obtained by Halcrow et al. (Reference Halcrow, Gibson, Cvitanović and Viswanath2009). For a fixed passage time $T$ and time increments, $\unicode[STIX]{x0394}T$ , chosen between 1 and 5 time units, the optimization algorithm typically requires 5 to 20 forward/backward time integrations to converge. The convergence curves are in good agreement with those obtained in the literature, with the convergence threshold achieved by Halcrow et al. (Reference Halcrow, Gibson, Cvitanović and Viswanath2009) indicated by a dashed line in figure 2(b).

Figure 2.

(a) Distances of the velocity field $\boldsymbol{u}(t)$ to the final equilibrium $\boldsymbol{u}_{EQ_{in}}$ (continuous line) and to the initial one $\boldsymbol{u}_{EQ_{out}}$ (dashed line) along three heteroclinic connections found by Halcrow et al. (Reference Halcrow, Gibson, Cvitanović and Viswanath2009) versus time. (b) Distances to the final equilibrium $\boldsymbol{u}_{EQ_{in}}$ for the new computed heteroclinic connections (continuous lines). The dashed line represents the highest residual value for the heteroclinic connections computed in Halcrow et al. (Reference Halcrow, Gibson, Cvitanović and Viswanath2009). The L2-norm of the considered equilibria ranges from 0.1259 to 0.4049. (Gibson et al. Reference Gibson, Halcrow and Cvitanović2009).

Figure 2(b) shows the new six heteroclinic connections computed here. Three of the new connections lead to the edge state (Nagata Reference Nagata1990; Schneider et al. Reference Schneider, Gibson, Lagha, De Lillo and Eckhardt2008), two of them to $EQ3$ , and one to $EQ9$ . All connections reported by Halcrow et al. (Reference Halcrow, Gibson, Cvitanović and Viswanath2009) land on the codimension-1 stable manifold of the edge state, which simplifies identifying those connections using a one-dimensional search algorithm. The variational method proposed here yields additional connections not involving the edge state.

4.2 Confirming that computed orbits shadow heteroclinic connections

Heteroclinic connections lie in the intersection of the unstable manifold of the initial equilibrium and the stable manifold of the final equilibrium. The use of finite energy shells renders the passage time along the orbit finite but may potentially lead to spurious results. A specific problem is due to transient growth effects that may allow a trajectory to leave the neighbourhood of an equilibrium along an algebraically amplified direction bypassing the unstable manifold (Farano et al. Reference Farano, Cherubini, Robinet, De Palma and Schneider2018). We thus analyse the dynamics close to the equilibria where curved stable and unstable manifolds can be approximated by their respective tangent spaces, the stable and unstable subspaces spanned by the stable or unstable eigenvectors of the Navier–Stokes equations linearized around the equilibrium. The analysis of the orbits connecting finite energy shells technically involves two steps: first, along the orbit we determine the range in the vicinity of both equilibria, where the dynamics is well approximated by the Navier–Stokes equations linearized around $EQ_{out}$ or $EQ_{in}$ , respectively. This is achieved by explicitly comparing the nonlinear and the linear time evolution. Within the validity range of the linearization, we then project onto an orthonormal basis of the unstable tangent spaces to the invariant manifolds of both equilibria and thereby separate the energetic contributions in the stable and unstable subspaces approximating the respective manifolds. (Note that we technically only use projections on the unstable subspace spanned by the orthogonalized unstable eigenvectors. These leading eigenvectors can be accurately constructed by our iterative Arnoldi method, while most of the stable eigenvectors are challenging to compute accurately using matrix-free techniques.)

Figure 3.

Time evolution of contributions in the stable and unstable subspaces of $EQ_{out}$ during the initial part of the trajectory $EQ5\rightarrow EQ1$ . (a) L2-norms $\Vert \boldsymbol{u}^{\prime \,s/u}\Vert _{2}$ of the projections onto the unstable ( $\boldsymbol{u}^{\prime \,u}$ , solid line) and stable ( $\boldsymbol{u}^{\prime \,s}$ , dashed line) subspace. The difference between the linearized and the full nonlinear evolution $\unicode[STIX]{x1D702}=(\Vert \boldsymbol{u}_{lin}^{\prime }(t)-\boldsymbol{u}^{\prime }(t)\Vert _{2}^{2}/\Vert \boldsymbol{u}_{lin}^{\prime }(t)\Vert _{2}^{2})$ (dotted line, right axis) characterizes the deviation of the invariant manifolds from the approximating linear subspaces due to manifold curvature in the direction of the trajectory. (b) Relative contribution measured in terms of relative energy in the unstable (solid line) and stable (dashed line) subspace. Vertical red dotted lines indicates when the relative contribution of the stable subspace $E^{s}(t)/E(t)$ drops below 1 %. Vertical blue dotted line indicates when the difference between the linearized and the full nonlinear evolution $\unicode[STIX]{x1D702}$ reaches 1 %.

For all identified orbits connecting finite energy shells of radius $r_{out}=10^{-4}$ , $r_{in}=10^{-3}$ , the identified initial condition lies predominantly in the stable subspace of $EQ_{out}$ , with relative energy in the stable versus unstable space ranging from 82 % to 98 %. To follow the evolution of the contributions we decompose $\boldsymbol{u}^{\prime }(t)=\boldsymbol{u}^{\prime \,u}(t)+\boldsymbol{u}^{\prime \,s}(t)$ into components in the stable/unstable subspace. Figure 3(a) shows the time evolution of the L2-norm of the contributions within the stable/unstable subspace of the orbit while leaving $EQ_{out}$ for the connection $EQ5\rightarrow EQ1$ . Data for other connections is provided in the supplementary material available at https://doi.org/10.1017/jfm.2018.860. Both $\Vert \boldsymbol{u}^{\prime \,u}\Vert _{2}(t)$ and $\Vert \boldsymbol{u}^{\prime \,s}\Vert _{2}(t)$ initially increase. The growth of the stable contribution slows down and starts to decay, while in the unstable subspace sustained approximately exponential growth continues. If the dynamics were linear, we would expect the stable contribution to decay to zero. However, due to the curvature of the invariant manifolds, the projection on the stable subspace may grow again, when the stable subspace no longer well approximates the attracting curved stable manifold. This is evidenced by the subsequent growth of $\Vert \boldsymbol{u}^{\prime \,s}\Vert _{2}(t)$ starting at $t\approx 30$ when the deviation between linear and nonlinear evolution $\unicode[STIX]{x1D702}=(\Vert \boldsymbol{u}_{lin}^{\prime }(t)-\boldsymbol{u}^{\prime }(t)\Vert _{2}^{2}/\Vert \boldsymbol{u}_{lin}^{\prime }(t)\Vert _{2}^{2})$ starts to increase. We thus observe transient growth in the stable subspace which is superseded by exponential growth along the unstable subspace. Consequently, the trajectory shows a transient excursion along the stable subspace, but eventually the orbit approaches the unstable subspace, as evidenced by the decaying relative contribution of the stable subspace (see figure 3 b). This decay is observed during the initial part of the orbit and while the unstable subspace remains a good approximation of the unstable manifold describing the nonlinear dynamics (see vertical blue dotted line in figure 3 b, and dotted line in figure 3 a). For all connections found for a shell of radius $r_{out}=10^{-4}$ the contribution of $\boldsymbol{u}^{\prime }$ in terms of energy in the unstable subspace reaches more than 99 % while the relative difference between the linear and nonlinear evolution $\unicode[STIX]{x1D702}$ remains bounded by 0.858 %. (Alternatively, when the ratio of L2-norms reaches $\Vert \boldsymbol{u}^{\prime \,u}\Vert _{2}>0.99\Vert \boldsymbol{u}^{\prime }\Vert _{2}$ , $\sqrt{\unicode[STIX]{x1D702}}$ remains bounded by 7.23 %.) The analysis consisting of projecting onto the stable subspace and monitoring the deviation between linear and nonlinear time evolution thus strongly supports the following interpretation: during the initial part, the orbit identified by the optimization algorithm does not follow the unstable manifold of $EQ_{out}$ , but rather exploits algebraic growth along transiently amplified directions predominantly in the stable manifold. However, after a short excursion along the stable manifold, the orbit approaches the unstable manifold and shadows a heteroclinic connection.

For two orbits $EQ9\rightarrow EQ1$ and $EQ11\rightarrow EQ1$ the passage time is too large for the optimization to converge if a small energy shell of radius $r_{out}=10^{-4}$ is chosen. For the connection $EQ9\rightarrow EQ1$ and shell of radius $r_{out}=10^{-3}$ , the contributions of $\boldsymbol{u}^{\prime }$ in terms of relative energy in the unstable subspace reach 99 % while $\unicode[STIX]{x1D702}$ remains bounded by 0.763 %. For the orbit $EQ11\rightarrow EQ1$ a strongly curved unstable manifold renders the linearization invalid after a short time and before the relative contribution in terms of energy (L2-norm) of the unstable manifold has reached 61 % (78 %). However, the contraction rate along the stable subspace is similar to that observed for all other orbits. This suggests that despite the importance of nonlinear effects the orbit follows the strongly curved unstable manifold when leaving $EQ_{out}$ .

An analogous analysis of the dynamics linearized around $EQ_{in}$ confirms that all orbits approach the final equilibria along their stable manifold. Consequently, the orbits connecting finite shells around two equilibria follow the unstable manifold of $EQ_{out}$ and the stable one of $EQ_{in}$ . The initial part of the orbit is affected by transient amplification in the stable manifold, but the excursion decays before an L2-distance from $EQ_{out}$ of order $10^{-2}$ – $10^{-3}$ is reached. More importantly, this distance is small compared to the typical L2-norm of the distance between both connected equilibria. The ratio of the distance affected by transient amplification to distance between both equilibria ranges from $0.016$ to $0.100$ . Consequently, the orbit shadows the heteroclinic connections along at least 90 % of the distance between equilibria. Data for the individual connections are given in the supplementary material.

We verify that the computed orbits are robust to changes of the size of the initial energy shell. All orbits available for $r_{out}=10^{-4}$ were successfully recomputed for a larger shell of radius $r_{out}=10^{-3}$ . For connections where the dimension of the intersection of the unstable manifold of the initial equilibrium with the stable manifold of the final state is greater than one, e.g. $EQ4\rightarrow EQ1$ , there is a continuous family of connections between these two states. Consequently, varying the energy shell may yield slightly different connections within the continuum. Likewise, the constrained search in Halcrow et al. (Reference Halcrow, Gibson, Cvitanović and Viswanath2009) yields a specific connection, while the variational algorithm may find a different connection within the continuum. For all connections with one-dimensional intersection, independently of the energy shell, the algorithm always converges to the only existing connection. This is demonstrated in figure 4 for the orbit $EQ4\rightarrow EQ3$ .

Figure 4.

State-space projection (Gibson et al. Reference Gibson, Halcrow and Cvitanović2008) of the orbit connecting $EQ5\rightarrow EQ1$ (blue) and $EQ4\rightarrow EQ1$ (green), for which the dimension of the intersection is three and two, respectively. For a specific connection, changing the energy shell around the initial equilibrium from $E_{0_{out}}=10^{-6}$ (solid line) to $E_{0_{out}}=10^{-8}$ (dashed line) results in finding different orbits. For those connections for which the dimension of the intersection is one, g.e. $EQ4\rightarrow EQ3$ (cyan), the algorithm finds the only existing connection.

4.3 Connectivity of equilibrium states

When visualizing the computed heteroclinic connections in the energy input ( $I$ ) versus dissipation rate ( $D$ ) plane (Kawahara & Kida Reference Kawahara and Kida2001) (figure 5 a) one observes that most orbits are located in the vicinity of the edge state equilibrium $EQ1$ . With the exception of $EQ10$ and $EQ11$ all other connected equilibria have a lower dissipation than the average value $D=I\approx 2.5$ for long turbulent transients (Cherubini & De Palma Reference Cherubini and De Palma2014). Along the majority of all connections – with exceptions $EQ3\rightarrow EQ1$ , $EQ9\rightarrow EQ1$ – the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ decreases, indicating that turbulent fluctuations are reduced along the orbit. This suggests that the presented heteroclinic connections are more relevant for relaminarization processes than the turbulent dynamics itself. Of special interest is the trajectory $EQ11\rightarrow EQ1$ , involving $EQ11$ with dissipation well above the turbulent mean. This trajectory captures features of turbulent bursting.

To better visualize how the EQs are mutually connected, the heteroclinic connections are also plotted using the 3D orthonormal projection of the state space proposed by Gibson et al. (Reference Gibson, Halcrow and Cvitanović2008). The time series of the projection coefficients $(a_{1}(t),a_{2}(t),a_{3}(t))$ are obtained using the inner product $a_{i}(t)=\langle \boldsymbol{u}(t),\boldsymbol{e}_{i}\rangle$ defined above, where $i=1,2,3$ and $\boldsymbol{e}_{i}$ are the orthonormal basis functions based on $EQ2$ and its half-cell translated siblings (Gibson et al. Reference Gibson, Halcrow and Cvitanović2008). The resulting trajectories projected onto this reduced basis are reported in figure 5(b). Within the network of heteroclinic connections, the edge state $EQ1$ is centrally located. For all other equilibrium states there is a direct connection landing on $EQ1$ . This indicates that all the considered equilibria are states on the edge of chaos, characterizing the dynamics on the stable manifold of the edge state. This is consistent with the edge of chaos and the chaotic saddle being dynamically linked for the considered domain size and Reynolds number (Chantry & Schneider Reference Chantry and Schneider2014). The flow evolution along the newly constructed heteroclinic connections are discussed in the supplementary material, where we also provide flow visualizations for all connections (see the supplementary movie).

Figure 5.

(a) Projection of the heteroclinic connections onto the energy input, $I$ , and the dissipation rate, $D$ , normalized by their value in the laminar flow. (b) State-space projection onto the 3D orthonormal basis $\boldsymbol{a}_{1,2,3}$ of all the computed heteroclinic connections. Colours are the same as in figure 2.