3. Overview of the bifurcation diagram

Even if only solutions preserving the $\mathit{R}_{\pi }$ symmetry are considered, the 4LDSSC problem features as many as two equilibria (E) and three limit cycles (C) inter-related by nine bifurcations. Figure 2 shows the bifurcation diagram containing all solution branches and bifurcation points that are accessible through $\mathit{R}_{\pi }$ -restricted time integration in terms of $u_{\backslash }$ as a function of $ \textit{Re}$ . Instabilities breaking the $\mathit{R}_{\pi }$ invariance are not considered in the description that follows. The base, fully symmetric solution $\mathit{E}_{\textit{S}}$ (red) issues a pair of stable, asymmetric but $\mathit{M}_{\backslash }$ -conjugate-symmetric $\mathit{E}_{\textit{A}}$ solution branches (blue) at a supercritical pitchfork bifurcation $\mathit{P}_{\textit{1S}}$ (leftmost red square). The stability lost in this first pitchfork is later recovered in a subcritical pitchfork bifurcation $\mathit{P}_{\textit{2S}}$ (rightmost red square) whence a second pair of $\mathit{E}_{\textit{A}}$ solution branches, this time unstable saddles, emerge. The nodal and saddle pairs of $\mathit{E}_{\textit{A}}$ branches meet in two symmetry-conjugate saddle-node bifurcations ( $\mathit{SN}_{\textit{A}}$ , not shown) at higher $ \textit{Re}$ that are not accessible with mere time-integration. While these solution branches and bifurcations had been previously reported in the literature (Chen et al. Reference Chen, Tsai and Luo2013), the remainder of the bifurcation diagram is new; with one notable exception, the long-period space–time symmetric periodic orbit, which was already known. Back to the $\mathit{E}_{\textit{S}}$ branch, the solution loses again stability, this time irreversibly, in an $\mathit{R}_{\pi }$ -breaking supercritical Hopf bifurcation ( $\mathit{H}_{\textit{S}}$ , red diamond). From the Hopf bifurcation, a stable branch of space–time-symmetric periodic orbits $\mathit{C}_{\textit{S}}$ (pink) emerges, which disappear eventually in a heteroclinic bifurcation ( $\mathit{Het}_{\textit{S}}$ , pink stars) involving a symmetric collision with the two saddle $\mathit{E}_{\textit{A}}$ solutions. Meanwhile, the nodal $\mathit{E}_{\textit{A}}$ solutions remain stable and become focuses as $ \textit{Re}$ is increased until the advent of two $\mathit{M}_{\backslash }$ -symmetry-conjugate supercritical Hopf bifurcations ( $\mathit{H}_{\textit{A}}$ , blue diamonds). From this point on, the unstable focuses are no longer traceable with time evolution. The two stable, $\mathit{M}_{\backslash }$ -asymmetric but mutually symmetric cycles $\mathit{C}_{\textit{A}}$ (cyan) grow in amplitude over a short $ \textit{Re}$ range (see the magnification in figure 2 b) and become unstable in $\mathit{M}_{\backslash }$ -conjugate-symmetric cyclic folds ( $\mathit{FC}_{\textit{A}}$ , cyan triangles), reversing their progress towards decreasing $ \textit{Re}$ . Once unstable, the $\mathit{C}_{\textit{A}}$ cycles grow large and collide with the saddle $\mathit{E}_{\textit{A}}$ solutions in mutually symmetric homoclinc bifurcations ( $\mathit{Hom}_{\textit{A}}$ , cyan stars). At sufficiently high $ \textit{Re}$ , a seemingly unconnected space–time-symmetric periodic orbit $\mathit{C}_{\textit{S}^\prime}$ (orange) of large amplitude and long period exists. As $ \textit{Re}$ is decreased, however, the cycle disappears in a boundary crisis, again involving the saddle branches of $\mathit{E}_{\textit{A}}$ , in the guise of yet another heteroclinic bifurcation ( $\mathit{Het}_{\textit{S'}}$ , orange stars). Here, $\mathit{C}_{\textit{S}^\prime}$ , which is the aforementioned exception, was known (Wahba Reference Wahba2011), but its origin at $\mathit{Het}_{\textit{S'}}$ had not been established.

Figure 2.

Bifurcation diagram for the $\mathit{R}_{\pi }$ symmetry-restricted 4LDSSC flow problem. (a) Full diagram. Solution branches are labelled (in colour) on the top half of the diagram and bifurcations (in black) on the lower half to avoid cluttering. (b) Magnification of the boxed region (grey rectangle) of panel (a). Shown are solution branches for the symmetric $\mathit{E}_{\textit{S}}$ (red) and asymmetric $\mathit{E}_{\textit{A}}$ (blue) steady states, as well as for the space–time symmetric $\mathit{C}_{\textit{S}}$ (pink) and $\mathit{C}_{\textit{S}^\prime}$ (orange), and asymmetric $\mathit{C}_{\textit{A}}$ (cyan) periodic orbits. Superscripts, when present, refer to stability properties of the solution along a specific sub-branch. Line styles are assigned according to the number of unstable eigenmodes (within the $\mathit{R}_{\pi }$ -restricted subspace): solid for none (node or focus, superscripts n or f), dashed for one (saddle, superscript s), dotted for two (unstable focus, superscript uf). Note that the sub-branch $\mathit{E}_{\textit{A}}^{\textit{uf}}$ cannot be computed with time-evolution and has been extrapolated from $\mathit{E}_{\textit{A}}^{\textit{f}}$ to guide the eye. Pitchfork (P, squares), Hopf (H, diamonds), fold of cycles (FC, triangles) and homo/heteroclinic bifurcations (Hom/Het, stars) are indicated and labelled. The subscript indicates the bifurcating solution and a numeral is used when more than one bifurcation of the same type has been identified along a branch. The solutions shown in figures 3, 4 and 5 correspond to $ \textit{Re}=500$ (grey dashed line) and those in figures 6, 7, 8 and 9 to $ \textit{Re}=744$ (grey dash-dotted line).

All above-mentioned bifurcations are listed in table 1, indicating the type, bifurcating solution, critical value of the Reynolds number and imaginary part of the bifurcating eigenvalue, along with additional observations as required. We will argue later that, provided the $\mathit{R}_{\pi }$ rotational symmetry is preserved (either naturally or enforced), the full dynamics of the 4LDSSC problem is encoded in the 2-D normal form of a degeneration of the double zero bifurcation subject to Z $_2$ symmetry. Accordingly, only two modes are at stake, and the type and stability of the solutions are dictated by the real or complex nature of the eigenmodes as well as the sign of the real part of the respective eigenvalues $\lambda$ in the case of equilibria or the modulus of the only relevant multiplier $\mu$ in the case of cycles. We will use superscripts to classify steady states as stable node (n: two negative real eigenvalues), stable focus (f: a complex pair of eigenvalues with negative real part), saddle (s: a negative and a positive real eigenvalues), unstable node (un: two positive real eigenvalues) or unstable focus (uf: a complex pair with positive real part). Similarly, limit cycles will be denoted as nodal (n: dominant multiplier within the unit circle) or saddle (s: dominant multiplier outside of the unit circle). The superscripts will be omitted when referring to the full solution branch, regardless of stability. It must be borne in mind that this stability classification concerns only the $\mathit{R}_{\pi }$ invariant subspace. For instance, we will refer to $\mathit{C}_{\textit{S}}$ as a stable/nodal cycle, although it is actually linearly unstable to a $\mathit{R}_{\pi }$ -breaking mode.

Table 1.

List of R $_\pi$ -invariant bifurcations of the 4LDSSC problem. Given are the name and type of the bifurcation, the solution concerned, the critical $ \textit{Re}$ and the imaginary part of the bifurcating eigenmode. Additionally, the symmetry broken for symmetry-breaking bifurcations or the saddle solution responsible for the boundary crisis in the case of global bifurcations are provided in the Observations column. The errors have been estimated from the variance of fitting parameters as twice the standard deviation (95 % confidence interval). Note that they do not express the uncertainty associated with the numerical discretisation of the problem.

The symmetries of the problem imply that all $\mathit{R}_{\pi }$ -preserving solutions breaking the $\mathit{M}_{\backslash }$ invariance must, of necessity, break also $\mathit{M}_{\textit{/}}$ . Further, solutions breaking the mirror symmetry appear in pairs, mutually related by reflection about either of the diagonals, and so do the bifurcations they undergo. All flow fields and bifurcation analyses of non-mirror-symmetric solutions will be presented throughout the paper for the ${u_{\backslash }}\lt 0$ version of the pair, the other being accessible through mirror reflection.

4. Invariant solutions

In this section, we discuss all invariant solutions of the problem mainly in terms of symmetries and flow physics as regards the topology and dynamics of large-scale vortical structures within the cavity. For an extended discussion on jet dynamics and wall boundary layers, we refer the reader to Appendix A, where all exact solutions are revisited with the focus set on kinetic energy fields and wall shear stress distributions.

Figure 3.

Steady states at $ \textit{Re}=500$ : (a) symmetric $\mathit{E}_{\textit{S}}^{\textit{uf}}$ , (b) asymmetric focus $\mathit{E}_{\textit{A}}^{\textit{f}}$ and (c) asymmetric saddle ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ . Shown are vorticity $\omega _z=\partial _x \ {v}-\partial _y{u}\in [-2.5,2.5]$ colourmaps (blue for negative, yellow for positive) and streamfunction $\psi$ contours (black for positive, white for negative; the zero contour in red), equispaced in steps $\Delta \psi =0.01$ . The frame denotes solution (colour) and stability within the $\mathit{R}_{\pi }$ symmetry subspace (linestyle: solid for zero-dimensional unstable manifold, dashed for 1-D, dotted for 2-D).

The base solution is a steady state that has all the symmetries of the problem and exists for all values of the Reynolds number. Figure 3(a) depicts vorticity ( $\omega _z$ ) colourmaps and streamfunction ( $\psi$ ) contours for the symmetric steady state $\mathit{E}_{\textit{S}}$ at $ \textit{Re}=500$ . The $\mathit{R}_{\pi }$ symmetry is clear from the $\pi$ -rotational invariance of both $\psi$ and $\omega _z$ , while the $\mathit{M}_{\backslash }$ and $\mathit{M}_{\textit{/}}$ symmetries are evinced by the sign inversion of $\psi$ and $\omega _z$ upon reflection with respect to either of the cavity diagonals. As a result of the aforementioned symmetries, the flow is exactly quiescent in the centre of the cavity, whence four straight streamlines are issued symmetrically along the diagonals towards the four corners. These streamlines divide the cavity into four triangular recirculation cells that do not exchange fluid. The cells adjacent to the top and bottom (left and right) walls rotate clockwise (counter clockwise) dragged by their sliding motion. Symmetric ejections of vorticity can be observed from the top-right and bottom-left corners along the diagonal, which are due to the confluence of contiguous lids running into each other. The two jets (see Appendix A) meet at the centre of the cavity along the diagonal and collide head-on preserving all symmetries. The $\mathit{E}_{\textit{S}}$ solution depicted at $ \textit{Re}=500$ is unstable, but the instability being associated with a mode that breaks the $\mathit{M}_{\backslash }$ and $\mathit{M}_{\textit{/}}$ invariance, enforcing either of the symmetries, along with $\mathit{R}_{\pi }$ , allows computation of the state with just a time stepping code. Actually, enforcing the $\mathit{R}_{\pi }$ symmetry is also necessary to stabilise $\mathit{E}_{\textit{S}}^{\textit{uf}}$ beyond $ \textit{Re}\gtrsim Re_{{\mathit{P}_{\textit{2S}}}}$ due to an $\mathit{R}_{\pi }$ -breaking pitchfork bifurcation occurring very shortly after $\mathit{P}_{\textit{2S}}$ (see Meseguer et al. Reference Meseguer, Alonso, Batiste, An and Mellibovsky2024).

Figure 4.

Phase map projection, on the $({u_{\backslash }},v_{\backslash })$ plane, at $ \textit{Re}=500$ . Steady states are represented with circles, the pattern (filled, half-filled or empty) denoting the dimensionality of the unstable manifold (null, one or two) within the $\mathit{R}_{\pi }$ subspace. Colour lines closing in loops represent limit cycles (solid for stable, dashed for unstable). Grey lines are connecting manifolds approximated by time evolution. Arrow heads indicate time direction. Labelled bullets indicate snapshots in figure 5. The region within the box has been enlarged in the inset.

Figure 4 shows a phase map representation, on the $({u_{\backslash }},v_{\backslash })$ plane, of the 4LDSSC problem at $ \textit{Re}=500$ . The symmetric solution $\mathit{E}_{\textit{S}}^{\textit{uf}}$ of figure 3(a) appears in this depiction as a point at the origin (red empty circle). The $\mathit{R}_{\pi }$ -restricted time evolution starting from its immediate vicinity spirals away (grey lines), revealing its nature as an unstable focus, i.e. having a complex-conjugate pair of unstable modes, hence the superscript.

At sufficiently low values of the Reynolds number ( $ \textit{Re}\lt Re_{{\mathit{P}_{\textit{1S}}}}$ ), $\mathit{E}_{\textit{S}}^{\textit{n}}$ is stable and, being the only stable state, acts as a global attractor. In this regime, the solution may be converged in the full square domain without enforcing any of its symmetries. The stability is lost in a supercritical pitchfork bifurcation $\mathit{P}_{\textit{1S}}$ at $ \textit{Re}_{{\mathit{P}_{\textit{1S}}}}\simeq 133.0$ , whence a pair of symmetry-conjugate branches of stable nodal asymmetric states $\mathit{E}_{\textit{A}}^{\textit{n}}$ are issued that eventually evolve into stable focuses $\mathit{E}_{\textit{A}}^{\textit{f}}$ in a node-focus transition. At the transition, the leading eigenvalue of $\mathit{E}_{\textit{A}}^{\textit{n}}$ , which is real and stable, collides with a second real eigenvalue and together form a complex pair. As a result, trajectories cease to approach the solution monotonically and start spiralling instead. The stable node $\mathit{E}_{\textit{A}}^{\textit{n}}$ becomes a stable focus $\mathit{E}_{\textit{A}}^{\textit{f}}$ , with no change in its stability, and no new solutions are created either. The transition is not a bifurcation and is therefore not shown in figure 2(a), where $\mathit{E}_{\textit{A}}^{\textit{n}}$ and $\mathit{E}_{\textit{A}}^{\textit{f}}$ are two consecutive portions of the stable $\mathit{E}_{\textit{A}}$ sub-branch (solid blue). Here, $\mathit{E}_{\textit{A}}^{\textit{f}}$ , shown in figure 3(b) at $ \textit{Re}=500$ , preserves the $\mathit{R}_{\pi }$ invariance, but breaks both the $\mathit{M}_{\backslash }$ and $\mathit{M}_{\textit{/}}$ symmetries. As already pointed out, there exist two E $_{\textit{A}}^{\textit{n/f}}$ states which are symmetry conjugates of one another. The state conjugate to that shown in figure 3(b) may be obtained from it by applying either of the broken reflection symmetries to $\mathit{E}_{\textit{A}}$ : ${\mathit{M}_{\backslash }}{\mathit{E}_{\textit{A}}} = {\mathit{M}_{\textit{/}}}{\mathit{E}_{\textit{A}}}$ . It may also be obtained from time evolution in the full square domain, without applying any symmetry restriction, but starting from an appropriate choice of initial condition. For $\mathit{E}_{\textit{A}}^{\textit{f}}$ , the jets, shy of colliding at the cavity centre, deviate from the diagonal to one side and the other (see Appendix A). As a result, the jets avoid mutual collision, but impinge on opposite walls instead – opposite from the vertex whence the jets are issued and opposite from one another to avoid crossing. Two of the isolated recirculation regions bounded by opposing walls that were characteristic of $\mathit{E}_{\textit{S}}$ have merged across the cavity centre into one large vortex, while the other two remain separate and have shrunk on account of their having been pushed towards their respective walls. In the phase map projection of figure 4, the two symmetry-conjugate $\mathit{E}_{\textit{A}}^{\textit{f}}$ states (filled blue circles) are related by a $180^\circ$ rotation about the origin. They are stable focuses, as indicated by the inward spiralling behaviour of their stable manifold (grey oriented lines).

Figure 5.

Space–time symmetric periodic orbit $\mathit{C}_{\textit{S}}$ at $ \textit{Re}=500$ . Snapshots of the solution at (a) $t_0$ , (b) $t_1=t_0+3.32$ (closest approach to ${\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}$ ), (c) $t_2=t_0+T/2$ , (d) $t_3=t_1+T/2$ (closest approach to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ ), corresponding to the labelled bullets in figure 4. The period of the solution is $T=15.321$ . Styles as for figure 3.

In addition to $\mathit{E}_{\textit{A}}^{\textit{f}}$ and ${\mathit{M}_{\backslash }}{\mathit{E}_{\textit{A}}^{\textit{f}}}$ , $\mathit{R}_{\pi }$ -restricted time evolution at $ \textit{Re}=500$ occasionally converges on a third instance of stable invariant solution in the form of a space–time symmetric cycle $\mathit{C}_{\textit{S}}$ , four snapshots of which are shown in figure 5 (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2026.11516). Unlike $\mathit{E}_{\textit{S}}$ and $\mathit{E}_{\textit{A}}$ , which were already known (Chen et al. Reference Chen, Tsai and Luo2013), the cycle $\mathit{C}_{\textit{S}}$ is reported here for the first time. The cycle can be traced back in $ \textit{Re}$ to a Hopf bifurcation $\mathit{H}_{\textit{S}}$ of $\mathit{E}_{\textit{S}}$ that breaks both the $\mathit{M}_{\backslash }$ and $\mathit{M}_{\textit{/}}$ symmetries at $ \textit{Re}_{{\mathit{H}_{\textit{S}}}}\simeq 453.7$ , and also forward until its annihilation in a heteroclinic bifurcation $\mathit{Het}_{\textit{S}}$ at $ \textit{Re}_{{\mathit{Het}_{\textit{S}}}}\simeq 551.7$ involving $\mathit{E}_{\textit{A}}$ states. In addition to the time invariance $\textit{TP}$ of periodic orbits to evolution over a full period $T$ ,

(4.1) \begin{align} \textit{TP} : [u,v](x,y;t) = [u,v](x,y;t+T), \end{align}

$\mathit{C}_{\textit{S}}$ possesses also a space–time symmetry ST that makes it also invariant under evolution over a half-period $T/2$ followed by reflection with respect to any of the two diagonals:

(4.2) \begin{align} \textit{ST} : [u,v](x,y;t) = [v,u](y,x;t+T/2) = [-v,-u](-y,-x;t+T/2). \end{align}

The ST symmetry is apparent from comparing pairs of snapshots exactly $T/2$ apart. Figure 5(c) at time $t_0+T/2$ and figure 5(d) at $t_1+T/2$ are the $\mathit{M}_{\backslash }$ (and $\mathit{M}_{\textit{/}}$ ) mirror images of figure 5(a) at $t_0$ and figure 5(b) at $t_1=t_0+3.32$ , respectively. As for $\mathit{E}_{\textit{A}}$ , the instantaneous flow topology of $\mathit{C}_{\textit{S}}$ consists of two impermeable $\mathit{R}_{\pi }$ -related vortices attached to two facing walls and a large region connecting the other two walls across the centre of the cavity. The dynamics, however, generate a dislocation every half-period at the very centre of the cavity that merges the extant two separate vortices into a new large connected region and, by so doing, splits the old domain-crossing region into two separate vortices. Unlike $\mathit{E}_{\textit{A}}$ , which avoided the collision of the jets at the cavity centre permanently, $\mathit{C}_{\textit{S}}$ accomplishes the same end dynamically by having the jets oscillate about the diagonal in phase opposition (see Appendix A). One bends up and the other down in an alternated fashion – or, equivalently, both clockwise, then anticlockwise. In the phase map projection of figure 4, the stable (stable in the $\mathit{R}_{\pi }$ invariant subspace, that is, hence the superscript n) cycle $\mathit{C}_{\textit{S}}^{\textit{n}}$ shows as a $\pi$ -rotational symmetric closed loop (solid pink line) around $\mathit{E}_{\textit{S}}^{\textit{uf}}$ , from which it has bifurcated. Indeed, the unstable manifold of the latter state drives the dynamics directly to the former. Incidentally, figures 5(a) and 5(c) are the mutually symmetric closest approaches to the vicinity of $E_{\textit{S}}$ , occurring shortly after the back and forth vortex rearrangements, while figures 5(b) and 5(d) are closest visits to the two mutually symmetric branches of an unstable asymmetric steady state that we shall discuss presently.

The coexistence of more than one local attractor at $ \textit{Re}=500$ suggests that a phase space threshold must be in place that separates the respective basins of attraction. A useful tool to study the basin boundary with nothing but a time stepping code is the edge tracking algorithm (Itano & Toh Reference Itano and Toh2001; An et al. Reference An, Bergada and Mellibovsky2019). In our case, the shoot-and-refine process has been applied to initial conditions generated by linear interpolation between the flow fields of $\mathit{E}_{\textit{A}}^{\textit{f}}$ and those of an arbitrary snapshot of $\mathit{C}_{\textit{S}}$ . Successive refinement iterations have the transient dynamics approach, for increasingly longer lapses, a saddle steady state ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ , whose flow field is shown in figure 3(c). While the topology of the solution coincides with that of $\mathit{E}_{\textit{A}}^{\textit{f}}$ , the actual flow field is half-way between those of $\mathit{E}_{\textit{S}}^{\textit{uf}}$ and $\mathit{E}_{\textit{A}}^{\textit{f}}$ . As all $\mathit{R}_{\pi }$ -invariant non-mirror-symmetric solutions, ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ comes also in symmetry-conjugate pairs, mutually related by both the $\mathit{M}_{\backslash }$ and $\mathit{M}_{\textit{/}}$ mirror symmetries. The new branch can be traced back in $ \textit{Re}$ , duly switching to edge tracking between $\mathit{E}_{\textit{S}}^{\textit{n}}$ and $\mathit{E}_{\textit{A}}^{\textit{f}}$ below $\mathit{H}_{\textit{S}}$ , to a subcritical symmetry-breaking pitchfork bifurcation $\mathit{P}_{\textit{2S}}$ of $\mathit{E}_{\textit{S}}$ that breaks both $\mathit{M}_{\backslash }$ and $\mathit{M}_{\textit{/}}$ at $ \textit{Re}_{{\mathit{P}_{\textit{2S}}}}\simeq 355.1$ . Forward continuation via edge tracking beyond $\mathit{Het}_{\textit{S}}$ requires trading the $\mathit{C}_{\textit{S}}$ snapshot for $\mathit{E}_{\textit{S}}^{\textit{uf}}$ and bisecting between trajectories going back to either $\mathit{E}_{\textit{A}}^{\textit{f}}$ or its mirror image.

According to Chen et al. (Reference Chen, Tsai and Luo2013) and Meseguer et al. (Reference Meseguer, Alonso, Batiste, An and Mellibovsky2024), the saddle branch ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ coalesces at even higher $ \textit{Re}$ in a saddle-node bifurcation with the unstable nodal branch $\mathit{E}_{\textit{A}}^{\textit{uf}}$ . Signs of the approach to the saddle-node bifurcation $\mathit{SN}_{\textit{A}}$ are apparent from the increasing slope of the branch in figure 2(a), but the divergingly slow dynamics in the vicinity of saddle-node bifurcations and the inability of time evolution to converge the asymmetric unstable focus hinder the precise determination of its occurrence. A steady-state solver would be required to analyse $\mathit{SN}_{\textit{A}}$ properly.

Figure 6.

Asymmetric periodic orbit $\mathit{C}_{\textit{A}}^{\textit{n}}$ at $ \textit{Re}=744$ . Snapshots of the solution at (a) $t_0$ , (b) $t_1=t_0+5.08$ , (c) $t_2=t_0+15.63$ and (d) $t_3=t_0+26.17$ (closest approach to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ ), corresponding to the labelled bullets in figure 7. The period of the solution is $T=37.432$ . Styles as for figure 3.

The stable asymmetric focus $\mathit{E}_{\textit{A}}^{\textit{f}}$ may be continued in $ \textit{Re}$ with the time stepper up to a supercritical Hopf bifurcation $\mathit{H}_{\textit{A}}$ , beyond which point it becomes an unstable focus $\mathit{E}_{\textit{A}}^{\textit{uf}}$ and is inevitably lost to time evolution codes (shown in figure 2 as the prolongation of the $\mathit{E}_{\textit{A}}^{\textit{f}}$ branch beyond the $\mathit{H}_{\textit{A}}$ point, approximated by extrapolation). Two $\mathit{R}_{\pi }$ -invariant, $\mathit{M}_{\backslash }$ - and $\mathit{M}_{\textit{/}}$ -breaking (but symmetry-conjugate) stable limit cycles $\mathit{C}_{\textit{A}}^{\textit{n}}$ and ${\mathit{M}_{\backslash }}{\mathit{C}_{\textit{A}}^{\textit{n}}}$ , replace $\mathit{E}_{\textit{A}}^{\textit{f}}$ and ${\mathit{M}_{\backslash }}{\mathit{E}_{\textit{A}}^{\textit{f}}}$ in their role as attractors. Neither the $\mathit{H}_{\textit{A}}$ point nor the resulting pair of cycles has been reported elsewhere. Four snapshots of $\mathit{C}_{\textit{A}}^{\textit{n}}$ are shown in figure 6 (see supplementary movie 2) at $ \textit{Re}=744$ . The flow topology is the same as for the two $\mathit{E}_{\textit{A}}$ branches, nodal and saddle, at all times, but while the flow field resembles E $_{\textit{A}}^{\textit{n/f}}$ (i.e. the stable $\mathit{E}_{\textit{A}}$ , be it node or focus) over the best part of a period (compare figures 6 a– 6 c with figure 8 h), the solution conspicuously approaches ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ once every period (see figure 6 c for the closest approach, to be compared with figure 8 i). The jets configuration is similar to that for $\mathit{E}_{\textit{A}}^{\textit{uf}}$ , but the impingement point on the walls oscillates around the equilibrium position (see Appendix A).

If the $\mathit{C}_{\textit{A}}^{\textit{n}}$ branch is bounded from below by the Hopf bifurcation $\mathit{H}_{\textit{A}}$ from which it springs at $ \textit{Re}_{{\mathit{H}_{\textit{A}}}}\simeq 737.6$ , neither can it be continued beyond a fold of cycles $\mathit{FC}_{\textit{A}}$ occuring very shortly after at $ \textit{Re}_{{\mathit{FC}_{\textit{A}}}}\simeq 744.2$ .

Figure 7.

Phase map projection at $ \textit{Re}=744$ . Styles as for figure 4. The location of $\mathit{E}_{\textit{A}}^{\textit{uf}}$ has been quadratically extrapolated from the stable focus $\mathit{E}_{\textit{A}}^{\textit{f}}$ at $ \textit{Re}=735$ , 736 and 737. Labelled bullets indicate snapshots in figures 6, 8 and 9. The region within the box has been enlarged in the inset.

Figure 7 depicts the phase map portrait of the 4LDSSC problem at $ \textit{Re}=744$ . All steady states of figure 4 are still present, but while $\mathit{E}_{\textit{S}}^{\textit{uf}}$ (red blank circle) and ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (blue half-filled circle) are still an unstable focus and an unstable saddle, respectively, $\mathit{E}_{\textit{A}}^{\textit{f}}$ has turned into an unstable focus $\mathit{E}_{\textit{A}}^{\textit{uf}}$ (blue blank circle). As an asymmetric unstable focus, $\mathit{E}_{\textit{A}}^{\textit{uf}}$ can neither be reached through time evolution nor by edge tracking, so its location has been approximated through quadratic extrapolation of the stable focus $\mathit{E}_{\textit{A}}^{\textit{f}}$ at neighbouring values of the Reynolds number, i.e. $ \textit{Re}=735$ , 736 and 737. The pair of $\mathit{C}_{\textit{A}}^{\textit{n}}$ cycles (solid cyan lines) show as two closed loops related by a $180^\circ$ rotation about the origin. Each revolves around one of the mutually symmetric $\mathit{E}_{\textit{A}}^{\textit{uf}}$ states and seemingly approaches, once every period, the nearmost ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ state.

Figure 8.

(a–f) Space–time symmetric periodic orbit $C_{\mathit{S^\prime }}^{\mathit{n}}$ at $ \textit{Re}=744$ , together with steady states (g) $E_{\textit{S}}^{\textit{uf}}$ , (h) $E_{\textit{A}}^{\textit{uf}}$ (quadratically extrapolated from $E_{\textit{A}}^{\textit{f}}$ at $ \textit{Re}=735$ , 736 and 737) and (i) $E_{\textit{A}}^{\textit{s}}$ . Snapshots of C $_{\mathit{S^\prime }}^{\mathit{n}}$ are taken at (a) $t_0$ , (b) $t_1=t_0+7.42$ , (c) $t_2=t_0+26.5$ (closest approach to $M_{\backslash }$ $E_{\textit{ A}}^{\textit{s}}$ ), (d) $t_3=t_0+T/2$ , (e) $t_4=t_1+T/2$ , (f), $t_5=t_2+T/2$ (closest approach to $E_{\textit{A}}^{\textit{s}}$ ), corresponding to the labelled bullets in figure 7. $T=66.338$ is the period of the solution. Styles as for figure 3.

Even though $\mathit{C}_{\textit{A}}^{\textit{n}}$ and its mirror-symmetric counterpart are stable at $ \textit{Re}=744$ , time evolution initialised from the symmetric base state $\mathit{E}_{\textit{S}}^{\textit{uf}}$ is eventually seized by yet another stable limit cycle $\mathit{C}_{\textit{S}^\prime}$ . This periodic orbit, a few snapshots of which are shown in figure 8(a–f) (see supplementary movie 3), has the same flow topology and space–time symmetry $ST$ of $\mathit{C}_{\textit{S}}$ , but with significantly longer period. Its amplitude is also substantially larger. Also shown, for reference, are the flow fields of $\mathit{E}_{\textit{S}}^{\textit{uf}}$ (figure 8 g), computed through time stepping by enforcing all problem symmetries, an approximation to $\mathit{E}_{\textit{A}}^{\textit{uf}}$ (figure 8 h), quadratically extrapolated from $E_{\textit{A}}^{\textit{f}}$ at the neighbouring $ \textit{Re}=735$ , 736 and 737, and ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (figure 8 i), obtained via edge tracking bewteen arbitrary snapshots of $\mathit{C}_{\textit{A}}^{\textit{n}}$ and $\mathit{C}_{\textit{S}^\prime}$ , bisecting between trajectories following two different paths to $\mathit{C}_{\textit{S}^\prime}$ . While $\mathit{C}_{\textit{S}}$ orbited in the vicinity of $\mathit{E}_{\textit{S}}$ with alternate periodic approaches to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ and its mirror image, $\mathit{C}_{\textit{S}^\prime}$ reaches further and revolves in turns around $\mathit{E}_{\textit{A}}^{\textit{uf}}$ and its symmetric (compare the flow fields of figures 8 b,e with figure 8 h). At other times, the trajectory evolves in the close neighbourhood of $\mathit{E}_{\textit{S}}$ (compare the flyby in figures 8 a and 8 d, just after the merging/splitting of vortex pairs across the cavity centre, with figure 8 g) and twice every period approaches within a very short distance of ${\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}$ and ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (see figures 8 c and 8 f, to be compared with figure 8 i). The cycle $\mathit{C}_{\textit{S}^\prime}$ can be traced back to a heteroclinic bifurcation $\mathit{Het}_{\textit{S'}}$ at $ \textit{Re}_{{\mathit{Het}_{\textit{S'}}}}\simeq 727.0$ and forward to Reynolds numbers in excess of $ \textit{Re}\gt 1000$ , still retaining its stability and having become the sole attractor of the 4LDSSC problem from $\mathit{FC}_{\textit{A}}$ onwards, and the sole non-trivial solution beyond the surmised $\mathit{SN}_{\textit{A}}$ . This orbit, whose dynamics reconnect the two $\mathit{M}_{\backslash }$ -symmetry related halves of phase space, is the one responsible for the onset of chaos at larger $ \textit{Re}$ .

Figure 9.

Asymmetric periodic orbit $\mathit{C}_{\textit{A}}^{\textit{s}}$ at $ \textit{Re}=744$ . Snapshots of the solution at (a) $t_0$ , (b) $t_0+5.27$ , (c) $t_0+15.23$ , (d) $t_0+29.18$ (closest approach to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ ), corresponding to the labelled bullets in figure 7. The period of the solution is $T=42.070$ . Styles as for figure 3.

The seeming occurrence of a fold of cycles suggests at once that the stable branch $\mathit{C}_{\textit{A}}$ must be turning towards lower $ \textit{Re}$ and, therefore, that the nodal cycle $\mathit{C}_{\textit{A}}^{\textit{n}}$ must be shielded by a saddle cycle $\mathit{C}_{\textit{A}}^{\textit{s}}$ . Such an unstable periodic orbit should, in principle, attract the dynamics on the basin boundary separating the two coexisting stable solutions $\mathit{C}_{\textit{A}}^{\textit{n}}$ and $\mathit{C}_{\textit{S}^\prime}$ and would, therefore, be accessible by edge tracking between any two instantaneous flow fields, each taken from one of the two attractors. Indeed, the edge tracking procedure at $ \textit{Re}=744$ produces an unstable limit cycle $\mathit{C}_{\textit{A}}^{\textit{s}}$ , portrayed in figure 9 (see supplementary movie 4). The proximity in phase space makes this saddle periodic orbit very similar to the nodal cycle $\mathit{C}_{\textit{A}}^{\textit{n}}$ , but the flow field differences can still be detected with the naked eye, particularly when comparing well-defined analogous instants, such as their respective closest approaches to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (see figures 9 d and 6 d). The $\mathit{C}_{\textit{A}}^{\textit{s}}$ branch, issued backwards from $\mathit{FC}_{\textit{A}}$ , is short and dies in a homoclinic bifurcation $\mathit{Hom}_{\textit{A}}$ at $ \textit{Re}_{{\mathit{Hom}_{\textit{A}}}}\simeq 742.9$ . Needless to say, the branch and bounding bifurcations have all their mirror-symmetric counterparts.

5. Phase portraits

All exact $\mathit{R}_{\pi }$ -invariant solutions of the 4LDSCC problem are contained between the phase portraits at $ \textit{Re}=500$ (figure 4) and $ \textit{Re}=744$ (figure 7). We now turn our attention to the various phase map topologies that succeed one another as the Reynolds number is increased from very low up to $ \textit{Re}\leq 1000$ . Figure 10 shows all structurally stable phase portraits, in addition to those presented earlier in figures 4 and 7 for illustration purposes, in order of increasing $ \textit{Re}$ . At $ \textit{Re}=100$ (figure 10 a), the symmetric state $\mathit{E}_{\textit{S}}^{\textit{n}}$ is the sole exact solution and, as such, acts as a global attractor. At $ \textit{Re}=300$ (figure 10 b), the symmetric state $\mathit{E}_{\textit{S}}^{\textit{s}}$ has become a saddle and is flanked at either side of its stable manifold by a pair of $\mathit{M}_{\backslash }$ -conjugate $\mathit{E}_{\textit{A}}^{\textit{f}}$ , now the only attractors of the problem. Incidentally, the states had been stable nodes $\mathit{E}_{\textit{A}}^{\textit{n}}$ for some $ \textit{Re}$ -range after their bifurcation, but they have already become focuses in a non-bifurcation node–focus transition by the time $ \textit{Re}=300$ is reached. The unstable manifold (grey lines) of $\mathit{E}_{\textit{S}}^{\textit{s}}$ points directly towards $\mathit{E}_{\textit{A}}^{\textit{f}}$ , the final approach following a spiralling trend that is barely noticeable at this value of $ \textit{Re}$ . At $ \textit{Re}=360$ (figure 10 c), the symmetric state has recovered its former stability (as long as the $\mathit{R}_{\pi }$ invariance is enforced) and a second pair ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ of asymmetric states, now saddles, have appeared. The unstable manifolds of these new states point on one side towards the symmetric state $\mathit{E}_{\textit{S}}^{\textit{n}}$ and on the other to the closest asymmetric stable state $\mathit{E}_{\textit{A}}^{\textit{f}}$ , now more clearly a focus. At $ \textit{Re}=425$ (figure 10 d), the symmetric state $\mathit{E}_{\textit{S}}^{\textit{f}}$ has also undergone a node-focus transition (compare its stable manifold in the insets). At $ \textit{Re}=500$ (see figure 4), the symmetric state has destabilised anew and become an unstable focus $\mathit{E}_{\textit{S}}^{\textit{uf}}$ , while a new attractor, in the form of a space–time symmetric cycle $\mathit{C}_{\textit{S}}$ collects the dynamics issued from $\mathit{E}_{\textit{S}}^{\textit{uf}}$ or from the side of the unstable manifold of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ that pointed towards $\mathit{E}_{\textit{S}}^{\textit{f}}$ at lower $ \textit{Re}$ . At $ \textit{Re}=600$ (figure 10 e), the cycle $\mathit{C}_{\textit{S}}$ has vanished into thin air, the phase space region it enclosed has opened and trajectories issued from the neighbourhood of $\mathit{E}_{\textit{S}}^{\textit{uf}}$ are now free to reach the only surviving attractors, the pair $\mathit{E}_{\textit{A}}^{\textit{f}}$ . At $ \textit{Re}=732$ (figure 10 f), a new stable space–time symmetric cycle $\mathit{C}_{\textit{S}^\prime}$ orbiting around all other existing solutions has come into existence. All trajectories issued from the vicinity of $\mathit{E}_{\textit{S}}^{\textit{uf}}$ , and from most of the phase space, are driven towards $\mathit{C}_{\textit{S}^\prime}$ . The stable and unstable manifolds of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ divide its neighbourhood into four quadrants, three of which belong to the basin of attraction of $\mathit{C}_{\textit{S}^\prime}$ , while the fourth pushes trajectories towards $\mathit{E}_{\textit{A}}^{\textit{f}}$ . At $ \textit{Re}=740$ (figure 10 g), this asymmetric pair of solutions have become unstable focuses $\mathit{E}_{\textit{A}}^{\textit{uf}}$ and trajectories starting from their neighbourhood converge onto the nearest of the two stable asymmetric limit cycles $\mathit{C}_{\textit{A}}^{\textit{n}}$ , mutually related by the $\mathit{M}_{\backslash }$ -symmetry. At $ \textit{Re}=744$ (see figure 7), a pair of saddle cycles $\mathit{C}_{\textit{A}}^{\textit{s}}$ have emerged that surround either cycle of the pair $\mathit{C}_{\textit{A}}^{\textit{n}}$ . The stable cycles are effectively shielded by the saddle cycles, so that the former are no longer reachable from the vicinity of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ and the full phase space, with exception of the region enclosed within $\mathit{C}_{\textit{A}}^{\textit{s}}$ , which remains drawn towards $\mathit{C}_{\textit{A}}^{\textit{n}}$ , becomes the basin of attraction of $\mathit{C}_{\textit{S}^\prime}$ . At $ \textit{Re}=800$ (figure 10 h), the cycles $\mathit{C}_{\textit{A}}^{\textit{n}}$ and $\mathit{C}_{\textit{A}}^{\textit{s}}$ have collided in pairs and disappeared altogether, leaving $\mathit{C}_{\textit{S}^\prime}$ as the sole and, therefore, global attractor and the full phase space as its basin of attraction. Finally, at $ \textit{Re}=1000$ (figure 10 i), the asymmetric saddle ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ and nodal $\mathit{E}_{\textit{A}}^{\textit{un}}$ steady states, the latter having, at some point, transitioned into unstable nodes, have collided in pairs and vanished. The cycle $\mathit{C}_{\textit{S}^\prime}$ is not only a global attractor, but also the only remaining non-trivial invariant solution, exception made, of course, of the unstable base state $\mathit{E}_{\textit{S}}^{\textit{uf}}$ . Shortly after, at slightly higher $ \textit{Re}$ , $\mathit{C}_{\textit{S}^\prime}$ is succeeded by a chaotic attractor (not shown).

Figure 10.

Phase portraits of the 4LDSSC problem at (a) $ \textit{Re}=100$ , (b) 300, (c) 360, (d) 425, (e) 600, (f) 732, (g) 740, (h) 800 and (i) 1000. Styles as for figure 4.

6. Analysis of bifurcations

Except for the (non-bifurcation) node-focus transition occurring between figures 10(c) and 10(d), all other $ \textit{Re}$ -consecutive phase portraits of figures 4, 7 and 10 are mediated by bifurcations, either local or global. In this section, we investigate the nature of each of these bifurcations and analyse them to the extent that is feasible with time integration only. Details of the stability analysis methodology can be found in Appendix B. All bifurcations and the phase space topology alterations they induce will be revisited in § 7 with the aid of phase portrait sketches to place them in context.

6.1. Local bifurcations

From $ \textit{Re}=100$ to 300 (figures 10 a and 10 b), the base state $\mathit{E}_{\textit{S}}$ becomes unstable and the pair of $\mathit{M}_{\backslash }$ -related stable equilibria $\mathit{E}_{\textit{A}}^{\textit{n}}$ emerge. The scenario is suggestive of a pitchfork bifurcation and figure 11(a) confirms it, while revealing, at the same time, its supercritical nature. The quantity $u_{\backslash }$ has been used to monitor the degree of disymmetry of the steady states $\mathit{E}_{\textit{A}}$ as a function of $ \textit{Re}$ . A square root fit of the form

(6.1) \begin{align} u_{{\backslash }}^{\textit{fit}}=u_{{\backslash }}^1\sqrt {\textit{Re}-\textit{Re}_{{\mathit{P}_{\textit{1S}}}}} \end{align}

to the leftmost $\mathit{E}_{\textit{A}}^{\textit{n}}$ points substantiates the supercritical nature of the pitchfork bifurcation $\mathit{P}_{\textit{1S}}$ and establishes its occurrence at $ \textit{Re}_{{\mathit{P}_{\textit{1S}}}}=132.974\pm 0.002$ , in fair agreement with previously reported values (Wahba 2009; Cadou et al. Reference Cadou, Guevel and Girault2012; Chen et al. Reference Chen, Tsai and Luo2013). The leading eigenvalue $\lambda$ has been estimated, at each value of $ \textit{Re}$ , from exponential fits to the linear regime of the $v_{\backslash }$ time series in their final approach to (initial departure from) the stable (unstable) state, having previously discarded, for unstable solutions, the very first initial transients on account of multimodality (see Appendix B). The unstable $\mathit{E}_{\textit{S}}^{\textit{s}}$ has been converged by enforcing $\mathit{M}_{\backslash }$ in the time stepper. As expected for a supercritical pitchfork bifurcation, the leading eigenvalue $\lambda$ of the bifurcating solution $\mathit{E}_{\textit{S}}$ is real and crosses the imaginary axis precisely at the bifurcation point, while that of the bifurcated solution $\mathit{E}_{\textit{A}}^{\textit{n}}$ debuts with zero real part and evolves towards negative values.

Figure 11.

Pitchfork bifurcations of the $\mathit{R}_{\pi }$ , $\mathit{M}_{\backslash }$ - and $\mathit{M}_{\textit{/}}$ -symmetric base state $\mathit{E}_{\textit{S}}$ , breaking the $\mathit{M}_{\backslash }$ and $\mathit{M}_{\textit{/}}$ invariance. The symmetry parameter $u_{\backslash }$ and the bifurcating eigenvalue $\lambda = \lambda _r+\textrm {i}\lambda _i$ are monitored as a function of $ \textit{Re}$ . (a) Supercritical pitchfork bifurcation $\mathit{P}_{\textit{1S}}$ (here, $\lambda _i=0$ so that $\lambda =\lambda _r$ over the full $ \textit{Re}$ -range considered). (b) Subcritical pitchfork bifurcation $\mathit{P}_{\textit{2S}}$ . Square-root fits to locate the pitchfork points are indicated with dash-dotted lines.

A second pitchfork bifurcation $\mathit{P}_{\textit{2S}}$ may be conjectured between $ \textit{Re}=300$ and 360 (figures 10 b and 10 c). The symmetric base state $\mathit{E}_{\textit{S}}$ recovers stability to $\mathit{M}_{\backslash }$ perturbations, while a second pair of $\mathit{M}_{\backslash }$ -related equilibria ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ , this time unstable, pop into existence. Incidentally, the base state is only really stable if the $\mathit{R}_{\pi }$ symmetry is specifically enforced. Otherwise, phase map trajectories are apt to escape it by temporarily violating the $\mathit{R}_{\pi }$ invariance, although the symmetry is eventually recovered later on upon their final approach to either of the stable $\mathit{E}_{\textit{A}}^{\textit{f}}$ evolved from the $\mathit{E}_{\textit{A}}^{\textit{n}}$ pair bifurcated at $\mathit{P}_{\textit{1S}}$ . The pair $\mathit{E}_{\textit{A}}^{\textit{f}}$ is, therefore, globally stable when considered in the full phase space. The added instability is the result of a third pitchfork bifurcation P $_{\textit{3S}}$ , almost coincidental with $\mathit{P}_{\textit{2S}}$ , this time breaking the $\mathit{R}_{\pi }$ symmetry (Meseguer et al. Reference Meseguer, Alonso, Batiste, An and Mellibovsky2024). We will not be concerned here with instabilities disrupting the $\mathit{R}_{\pi }$ invariance of the 4LDSSC problem. The analysis of $\mathit{P}_{\textit{2S}}$ , analogous to that of $\mathit{P}_{\textit{1S}}$ , is illustrated in figure 11(b). While the unstable $\mathit{E}_{\textit{S}}^{\textit{s}}$ have been obtained by enforcing the $\mathit{M}_{\backslash }$ symmetry in the time evolution, the saddle ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ solutions have required $\mathit{R}_{\pi }$ -restricted edge tracking between the base state $\mathit{E}_{\textit{S}}^{\textit{n/f}}$ and one of the stable $\mathit{E}_{\textit{A}}^{\textit{f}}$ , now already stable focuses. Because of the $\mathit{R}_{\pi }$ -breaking pitchfork bifurcation P $_{\textit{3S}}$ and a closely related $\mathit{R}_{\pi }$ -breaking pitchfork P $_{\textit{A}}$ on the ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ branch, the whole bifurcation analysis for $\mathit{P}_{\textit{2S}}$ must be done on the $\mathit{R}_{\pi }$ -invariant subspace. Imposing the symmetry stabilises $\mathit{E}_{\textit{S}}^{\textit{n/f}}$ and turns ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ into an edge state by removing the unstable direction associated with R $_\pi$ -disrupting perturbations. A square root fit analogous to that of (6.1) to the few leftmost points on the ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ branch fixes the occurrence of $\mathit{P}_{\textit{2S}}$ at $ \textit{Re}_{{\mathit{P}_{\textit{2S}}}}=355.05\pm 0.02$ , very close to the value reported by Chen et al. (Reference Chen, Tsai and Luo2013), and attests to the subcritical nature of the pitchfork bifurcation. The evolution of the leading eigenvalue of $\mathit{E}_{\textit{S}}$ across the critical point is also consistent with a pitchfork bifurcation. The eigenvalue is real and crosses the imaginary line, from positive to negative, while that of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ starts from the origin of the complex plane and moves along the positive real axis away from the bifurcation point. Somewhere in between $ \textit{Re}=365$ and 370, the leading eigenvalue of $\mathit{E}_{\textit{S}}$ , already in the stable half of the complex plane, collides with a subleading eigenvalue moving towards instability, and the couple merges and splits into a complex-conjugate pair in a non-bifurcation node-focus transition (see the evolution of $\lambda _i$ in figure 11 b). The count and stability of the solutions does not change, but the stable manifold of $\mathit{E}_{\textit{S}}$ is no longer governed by two distinct eigenmodes, but a complex pair that induces a spiralling recovery of the equilibrium. The real part of the leading complex pair of eigenvalues of $\mathit{E}_{\textit{S}}$ is seen to evolve in the direction of becoming unstable already in figure 11(b). Indeed, the pair crosses the imaginary axis as $ \textit{Re}$ is increased further. Complex eigenvalues have been estimated from oscillatory exponential fits to the linear regime of the $v_{\backslash }$ time series (see Appendix B).

Figure 12.

Hopf bifurcations of the steady states $\mathit{E}_{\textit{S}}$ and $\mathit{E}_{\textit{A}}$ and fold of cycles of $\mathit{C}_{\textit{A}}$ . (a) Supercritical Hopf bifurcation $\mathit{H}_{\textit{S}}$ of the base state $\mathit{E}_{\textit{S}}$ , breaking the $\mathit{M}_{\backslash }$ symmetry and producing the ST symmetric cycle $\mathit{C}_{\textit{S}}$ . (b) Supercritical Hopf bifurcation $\mathit{H}_{\textit{A}}$ of the steady state $\mathit{E}_{\textit{A}}$ , producing the cycle $\mathit{C}_{\textit{A}}$ , and fold bifurcation $\mathit{FC}_{\textit{A}}$ of the latter. The oscillation amplitude $A_{{u_{\backslash }}}=u_{\backslash }^{\textit{max}}-u_{\backslash }^{\textit{ min}}$ (nil for steady states), the bifurcating eigenvalue $\lambda = \lambda _r+\textrm {i}\lambda _i$ of the steady states, and the period $T$ and multiplier $\mu$ of the bifurcated cycles are monitored as a function of $ \textit{Re}$ . Dash-dotted and dotted lines denote square root fits to pinpoint Hopf and cyclic fold bifurcations, respectively.

Figure 12(a) illustrates the analsysis of the Hopf bifurcation undergone by the focus $\mathit{E}_{\textit{S}}$ , which occurs somewhere in between $ \textit{Re}=425$ and 500 (see figures 10 d and 4). In this case, solutions are characterised by the oscillation amplitude $A_{{u_{\backslash }}}=u_{\backslash }^{\textit{max}}-u_{\backslash }^{\textit{min}}$ to detect time dependence. The symmetric equilibrium $\mathit{E}_{\textit{S}}$ has zero amplitude all along, but evolves from stable to unstable across the Hopf bifurcation. Meanwhile, the ST symmetric cycle $\mathit{C}_{\textit{S}}$ starts with zero amplitude that grows with increasing $ \textit{Re}$ . A square root fit following

(6.2) \begin{align} A_{u_{\backslash }}^{\textit{fit}} = A_{u_{\backslash }}^1\sqrt {Re-Re_{{\mathit{H}_{\textit{S}}}}} \end{align}

to the few points closest to the critical point, establishes the occurrence of a supercritical Hopf bifurcation at $ \textit{Re}_{{\mathit{H}_{\textit{S}}}}=453.6766\pm 0.0004$ . This value is very close to the $452.8$ reported by Meseguer et al. (Reference Meseguer, Alonso, Batiste, An and Mellibovsky2024), despite their smooth (or rather only moderately sharp) regularisation of corner singularities. The real part of the leading eigenpair of $\mathit{E}_{\textit{S}}$ crosses the imaginary axis at precisely this point and the dominant multiplier of the bifurcated cycle $\mathit{C}_{\textit{S}}$ is real, starts on the unit circle ( $\mu =1$ ) and decreases thereafter. The bottom subpanel depicts the quantity $2\pi /\lambda _i$ for the leading eigenpair of $\mathit{E}_{\textit{S}}$ instead of the imaginary part $\lambda _i$ to allow comparison with the period $T$ of $\mathit{C}_{\textit{S}}$ . As expected, the period of the cycle is inherited from the angular frequency associated with the bifurcating eigenpair of the steady state at the bifurcation point, but evolves independently away from it. See Appendix B for a detailed account of the stability analysis of periodic solutions.

Because of its involving a global bifurcation, we shall postpone the discussion on the fate of C $_{\textit{S}}$ to the next section and deal instead now with the remaining local bifurcations that befall the stable $\mathit{E}_{\textit{A}}$ branch and its offshoots. Recall that $\mathit{E}_{\textit{A}}^n \rightarrow \mathit{E}_{\textit{A}}^f$ has already transitioned from node to focus half-way between $\mathit{P}_{\textit{1S}}$ and $ \textit{Re}=300$ (see figure 10 b). Nothing much happens along the branch across a wide Reynolds number range until some place in between $ \textit{Re}=732$ and 740 (see figures 10 f and 10 g), where the focus becomes unstable and a $\mathit{R}_{\pi }$ -invariant but $\mathit{M}_{\backslash }$ -asymmetric limit cycle materialises. If there are two branches of $\mathit{E}_{\textit{A}}^{\textit{f}}$ mutually related by the $\mathit{M}_{\backslash }$ symmetry, the same holds for $\mathit{C}_{\textit{A}}$ and the Hopf bifurcation $\mathit{H}_{\textit{A}}$ at its origin. The analysis of H $_{\textit{A}}$ is shown in figure 12(b). A fit like that of (6.2) to the leftmost amplitudes of the converged $\mathit{C}_{\textit{A}}$ cycles sets the supercritical Hopf bifurcation at $ \textit{Re}_{{\mathit{H}_{\textit{A}}}}=737.554\pm 0.002$ . In this case, the comparison with the regularised 4LDSSC flow is not so favourable (see Meseguer et al. Reference Meseguer, Alonso, Batiste, An and Mellibovsky2024, where a value of 698.4 was reported), but this was expected, since their $ \textit{Re}_{{\mathit{SN}_{\textit{A}}}}$ was also off with respect to the non-regularised problem (Chen et al. Reference Chen, Tsai and Luo2013). The evolution of the real and imaginary parts of the leading eigenpair is again consistent with the occurrence of a Hopf bifurcation, and the same can be said about the period and the dominant multiplier of the bifurcated cycle. Unlike what happened for $\mathit{H}_{\textit{S}}$ , where we had access to all solutions involved in the bifurcation, for $\mathit{H}_{\textit{A}}$ , we have to do without the unstable focus $\mathit{E}_{\textit{A}}^{\textit{uf}}$ into which the steady state evolves. Having a 2-D unstable manifold that can neither be stabilised by symmetry enforcement nor by edge tracking places it beyond reach of time evolution. Its depiction in phase maps beyond $ \textit{Re}_{{\mathit{H}_{\textit{A}}}}$ has been approximated with a quadratic extrapolation from the latest available stable instances at $ \textit{Re}=737$ , 738 and 739.

This same figure 12(b) contains the necessary elements for a rough analysis of the fold-of-cycles to which the limit cycle branch $\mathit{C}_{\textit{A}}$ is subject at slightly higher values of the parameter, between $ \textit{Re}=744$ and 800 (see figures 7 and 10 h). Since the $\mathit{C}_{\textit{A}}$ branch had not been reported in the literature, neither had the eventual occurrence of a fold-of-cycles. Two closeby versions of the cycle, one stable ( $\mathit{C}_{\textit{A}}^{\textit{n}}$ ) and the other unstable (the saddle $\mathit{C}_{\textit{A}}^{\textit{s}}$ ), coexist at the lower $ \textit{Re}$ but have disappeared at the higher. The unstable cycle $\mathit{C}_{\textit{A}}^{\textit{s}}$ can be converged to reasonable accuracy by deploying edge tracking between some velocity field on the stable cycle $\mathit{C}_{\textit{A}}^{\textit{n}}$ and either the symmetric equilibrium $\mathit{E}_{\textit{S}}^{\textit{uf}}$ or another velocity field on the stable ST-symmetric cycle $\mathit{C}_{\textit{S}^\prime}$ , and bisecting between the last trajectory falling back on $\mathit{C}_{\textit{A}}^{\textit{n}}$ and the first to leave towards $\mathit{C}_{\textit{S}^\prime}$ . Be as it might, the amplitudes of $\mathit{C}_{\textit{A}}^{\textit{n}}$ and $\mathit{C}_{\textit{A}}^{\textit{s}}$ approach each other as $ \textit{Re}$ is increased towards a critical point. A square root fit following

(6.3) \begin{align} A_{u_{\backslash }}^{\textit{fit}} = A_{u_{\backslash }}^{0}+A_{u_{\backslash }}^1\sqrt {Re_{{\mathit{FC}_{\textit{A}}}}-Re} \end{align}

to the rightmost points on the stable focus branch, this time allowing for a vertical shift $A_{u_{\backslash }}^{0}$ , provides an estimate for the cyclic fold as taking place at $ \textit{Re}_{{\mathit{FC}_{\textit{A}}}}=744.193\pm 0.002$ . A parabolic fit of the form $ \textit{Re}^{\textit{fit}}=Re_{{\mathit{FC}_{\textit{A}}}}-((A_{u_{\backslash }}-A_{\backslash }^0)/A_{\backslash }^1)^2$ using points on both the stable and unstable sides of the branch would have served the same purpose. The dominant multiplier $\mu$ is real throughout, but exits the unit circle precisely as the branch bends back in the fold of cycles. The period $T$ also evolves continuously across the bifurcation and its rapid increase as $ \textit{Re}$ is lowered along the saddle branch is an indicator that a boundary crisis is close at hand. In a boundary crisis, a time dependent solution (usually a strange attractor; periodic orbits in our case) comes into contact with a saddle solution (always ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ for the 4LDSSC flow) and/or with its stable manifold and disappears catastrophically. As a result, the phase space topology is reconfigured and phase space regions are reconnected differently, meaning that initial conditions that evolved into one attractor may lead to another after the crisis. We will investigate the crisis of $\mathit{C}_{\textit{A}}^{\textit{s}}$ , along with two others involving the other two limit cycles, in the next section.

The full set of local bifurcations of the $\mathit{R}_{\pi }$ -invariant 4LDSSC problem in the studied range $ \textit{Re}\in [0,1000]$ is completed by a saddle-node point at which the nodal and saddle branches of $\mathit{E}_{\textit{A}}$ collide and disappear. This has been reported to happen at $ \textit{Re}\simeq 876.6$ for the singular corners cavity (Chen et al. Reference Chen, Tsai and Luo2013) and at $ \textit{Re}\simeq 708.4$ for the regularised cavity (Meseguer et al. Reference Meseguer, Alonso, Batiste, An and Mellibovsky2024). Our time-stepping approach loses track of the nodal branch $\mathit{E}_{\textit{A}}^{\textit{uf}}$ beyond the Hopf bifurcation $\mathit{H}_{\textit{A}}$ and the accuracy of the saddle branch ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ , although in principle accessible through edge tracking, is compromised by the divergingly slow dynamics that are typical of a saddle-node point. Consequently, it has not been possible to study this final bifurcation in any detail, other than by noting the disappearance of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ , as illustrated by the phase portraits at $ \textit{Re}=800$ and $ \textit{Re}=1000$ of figures 10(h) and 10(i), respectively.

6.2. Global bifurcations

We now turn our attention to the global bifurcations, none of which had been previously detected, let alone discussed in the literature, despite the fundamental part they play in the emergence of $\mathit{C}_{\textit{S}^\prime}$ .

Figure 13.

Heteroclinic bifurcation of $\mathit{C}_{\textit{S}}$ . (a) Time series $v_{\backslash }(t)$ of several $\mathit{C}_{\textit{S}}$ (pink solid lines with labels indicating $ \textit{Re}$ ) approaching the bifurcation. The steady states $\mathit{E}_{\textit{S}}$ (dotted red), $\mathit{E}_{\textit{A}}^{\textit{f}}$ (solid blue) and ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (dashed blue) are indicated with horizontal lines. (b) Phase portrait at $ \textit{Re}=551.7$ , very close to the crisis at $ \textit{Re}_{\mathit{Het}_{\textit{S}}}\simeq 551.73$ . Styles as for figure 4. The inset shows a magnification around ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ . (c) Distances time series of $\mathit{C}_{\textit{S}}$ with respect to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ ( $d_{{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ , solid) and ${\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}$ ( $d_{{\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ , dash-dotted) at $ \textit{Re}=551.7$ . The time axis is shared with panel (a). (d) Snapshot of $\mathit{C}_{\textit{S}}$ at the closest approach to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ at $ \textit{Re}=551.7$ , indicated with pink bullets in the time series of panels (a) and (c). (e) The ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ state at the same $ \textit{Re}$ .

The ST symmetric limit cycle $\mathit{C}_{\textit{S}}$ created at $\mathit{H}_{\textit{S}}$ disappears at some point between $ \textit{Re}=500$ and 600 (see figures 4 and 10 e). The snapshot of figure 5(d) at $ \textit{Re}=500$ is clearly suggestive of a close approach to the saddle steady-state ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ originated at $\mathit{P}_{\textit{2S}}$ , thus pointing at the possible occurrence of a boundary crisis in the form of a saddle-loop bifurcation. Such a crisis involves the collision of a limit cycle with a saddle steady solution, followed by its catastrophic disappearance. In the absence of symmetries, the limit cycle merges with the stable and unstable manifolds of the saddle state forming a non-robust homoclinic orbit. The symmetries of the 4LDSSC problem and of the solutions involved in the boundary crisis require that the collision occurs simultaneously and symmetrically with ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ and ${\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}$ , and that the manifolds connection is not homoclinic, i.e. from each of the saddles to itself, but heteroclinic, with the stable and unstable manifolds of both states connecting them back and forth in an heteroclinic loop. The analysis of the heteroclinic bifurcation $\mathit{Het}_{\textit{S}}$ of $\mathit{C}_{\textit{S}}$ is presented in figure 13. In particular, figure 13(a) shows a full period of the $v_{\backslash }(t)$ time series of several $\mathit{C}_{\textit{S}}$ at $ \textit{Re}$ approaching the crisis (increasingly darker shades of pink, labels indicating $ \textit{Re}$ value). As expected for a boundary crisis of a limit cycle, the period grows fast as the global bifurcation is approached, and the dynamics spend longer and longer times in the vicinity of the saddle state involved in the collision. The time spent in the neighbourhood of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ and ${\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}$ (dashed blue horizontal lines), symmetrically in turns, is rapidly protracted as $ \textit{Re}$ is increased. The approach, which could be a mere projection artefact is, to some extent, confirmed by the phase portrait of figure 13(b), particularly the inset, where $\mathit{C}_{\textit{S}}$ is seen to evolve increasingly closer to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ , to the point of becoming confounded with one side of the stable and unstable manifolds of the saddle solution (grey lines with arrows) at $ \textit{Re}=551.7$ , very near the crisis. Definite confirmation of the approaching collision is provided by the distance time series $d_{{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ (solid pink) and $d_{{\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ (dash-dotted pink) of $\mathit{C}_{\textit{S}}$ with respect to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ and ${\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}$ , respectively, shown in figure 13(c). The encounter between cycle and saddles is seen to occur at very close quarters at $ \textit{Re}=551.7$ , and both the approach and departure follow the exponential behaviour that is characteristic of the linear regime of manifolds in the proximity of a saddle equilibrium. Indeed, the instantaneous flow field of $\mathit{C}_{\textit{S}}$ at the time that minimises the distance $d_{{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ (figure 13 d and supplementary movie 5) is indistinguishable from that of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (figure 13 e) for all practical purposes.

The global bifurcation we are envisaging involves the simultaneous collision of a limit cycle with two hyperbolic saddle equilibria. For the collision of a stable limit cycle with a single hyperbolic saddle equilibrium (the homoclinic or saddle-loop bifurcation), the period of the former is known to diverge following a logarithmic law of the form

(6.4) \begin{align} T \sim -\frac {1}{\lambda }\log {\lvert \alpha _c-\alpha \rvert }, \end{align}

where $\alpha$ is the parameter ( $ \textit{Re}$ in our case), $\alpha _c$ is its value at the bifurcation point and $\lambda \gt 0$ is the eigenvalue of the saddle in the direction of its unstable manifold, which, together with the stable manifold merge to form a non-robust homoclinic orbit (Gaspard Reference Gaspard1990). If the cycle is unstable, $\lambda$ is the negative of the least stable eigenvalue of the saddle equilibrium. Following the same arguments as Gaspard (Reference Gaspard1990), but for a stable space–time symmetric cycle colliding simultaneously with a pair of saddles related by the Z $_2$ symmetry of which the space-time symmetry is a remnant, simply modifies (6.4) to

(6.5) \begin{align} T \sim -\frac {2}{\lambda }\log {\lvert \alpha _c-\alpha \rvert }, \end{align}

with the unstable eigenvalue contributing twice to the slowing of the dynamics on account of the two saddles involved. The limit cycle, rather than becoming a non-robust homoclinic orbit, transforms instead into a pair of heteroclinic connections between the two mutually symmetric saddles. The protraction of the period as the bifurcation is approached, which is dictated by the unstable eigenvalue, occurs at two mutually symmetric stages of the cycle, such that the period diverges exactly twice as fast as for the ordinary saddle-loop in the absence of symmetry.

Figure 14.

Heteroclinic bifurcations of the ST-symmetric limit cycles (a) C $_{\textit{S}}$ and (b) C $_{\textit{S'}}$ . Period $T$ as a function of $ \textit{Re}$ . The dash-dotted lines are logarithmic fits to a few data points closest to the crisis, indicated with the vertical asymptote. The inset presents the same data in log–log scale.

Figure 14(a) depicts the period $T$ of $\mathit{C}_{\textit{S}}$ as a funtion of $ \textit{Re}$ . The divergence of the period is evident and a fit of the form $T = -2\lambda ^{-1}\log (Re_{{\mathit{Het}_{\textit{S}}}}-Re)+T_1$ , following (6.5), places the heteroclinic bifurcation $\mathit{Het}_{\textit{S}}$ at $ \textit{Re}_{{\mathit{Het}_{\textit{S}}}}=551.73366\pm 0.00006$ with $\lambda =0.3977\pm 0.0004$ , in fair agreement with the leading eigenvalue of the saddle E $_{\textit{A}}^{\textit{s}}$ , which has been estimated as ${\lambda }_{{\mathit{E}}_{\textit{A}}^{\textit{s}}}=0.39136\pm 1\times 10^{-5}$ at $ \textit{Re}_{{\mathit{Het}_{\textit{S}}}}$ .

At some place in bewteen the phase portraits at $ \textit{Re}=600$ and 732 (figures 10 e and 10 f), the $\mathit{C}_{\textit{S}^\prime}$ periodic orbit comes to exist without the mediation of a Hopf or a cyclic fold bifurcation. Parametric continuation of the $\mathit{C}_{\textit{S}^\prime}$ branch in the direction of decreasing $ \textit{Re}$ has the periodic orbit disappear slightly below $ \textit{Re}\lesssim 727$ . The time series $v_{\backslash }(t)$ of $\mathit{C}_{\textit{S}^\prime}$ , shown in figure 15(a) for decreasing values of $ \textit{Re}$ , preserves the same temporal structure, with alternated overshoot of $\mathit{E}_{\textit{A}}^{\textit{f}}$ and ${\mathit{M}_{\backslash }}{\mathit{E}_{\textit{A}}^{\textit{f}}}$ (solid blue), followed each by increasingly protracted relaxation onto ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ and ${\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}$ (dashed blue), respectively, that results in the overall elongation of the period. The phase portrait (figure 15 b) reveals a close approach to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (see the inset) to the point that, at $ \textit{Re}=727$ , the cycle $\mathit{C}_{\textit{S}^\prime}$ becomes indistinguishable form the stable and unstable manifolds of the saddle (grey arrows). The flow field distance time series $d_{{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ and $d_{{\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ of figure 15(c) confirm that the visits of the cycle to the saddles are indeed legitimate and not an artefact of the phase map projection. The similarity of the flow field colourmaps of $\mathit{C}_{\textit{S}^\prime}$ at the closest approach to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (figure 15 d, and supplementary movie 6) and of the saddle itself (figure 15 e) attest to the proximity of a boundary crisis. Thorough examinination of the divergence of the period $T$ as a function of $ \textit{Re}$ , as done in figure 14(b), and a logarithmic fit following $T = -2\lambda ^{-1}\log (Re-Re_{{\mathit{Het}_{\textit{S'}}}})+T_1$ , pinpoints the occurrence of the heteroclinic bifurcation $\mathit{Het}_{\textit{S'}}$ at $ \textit{Re}_{{\mathit{Het}_{\textit{S'}}}}=726.9916\pm 0.0004$ with $\lambda =0.330\pm 0.004$ , in excellent agreement with the leading eigenvalue $\lambda _{{\mathit{E}}_{\textit{A}}^{\textit{s}}}=0.33110\pm 3\times 10^{-5}$ of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ at the bifurcation.

Figure 15.

Heteroclinic bifurcation of $\mathit{C}_{\textit{S}^\prime}$ . (a) Time series $v_{\backslash }(t)$ of several $\mathit{C}_{\textit{S}^\prime}$ (orange solid lines with labels indicating $ \textit{Re}$ ) approaching the bifurcation. The steady states $\mathit{E}_{\textit{S}}$ (dotted red), $\mathit{E}_{\textit{A}}^{\textit{f}}$ (solid blue) and ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (dashed blue) are indicated with horizontal lines. (b) Phase portrait at $ \textit{Re}=727$ , very close to the crisis at $ \textit{Re}_{\mathit{Het}_{\textit{S'}}}\simeq 726.99$ . Styles as for figure 7. The inset shows a magnification around ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ . (c) Distances time series of $\mathit{C}_{\textit{S}}^\prime$ with respect to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ ( $d_{{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ , solid) and ${\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}$ ( $d_{{\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ , dash-dotted) at $ \textit{Re}=727$ . The time axis is shared with panel (a). (d) Snapshot of $\mathit{C}_{\textit{S}^\prime}$ at the closest approach to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ at $ \textit{Re}=727$ , indicated with orange bullets in the time series of panels (a) and (c). (e) The ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ state at the same $ \textit{Re}$ .

The phase portrait at $ \textit{Re}=744$ (figure 7) features an unstable asymmetric limit cycle $\mathit{C}_{\textit{A}}^{\textit{s}}$ that was not present at $ \textit{Re}=740$ (figure 10 g). We have already shown that this cycle results from a cyclic fold with the stable nodal cycle $\mathit{C}_{\textit{A}}^{\textit{n}}$ at the slightly higher $ \textit{Re}_{{\mathit{FC}_{\textit{A}}}}$ , the analysis of which was presented in figure 12(b). This same figure holds the key for interpreting the fate of $\mathit{C}_{\textit{A}}^{\textit{s}}$ as the parameter is lowered from the fold of cycles. The branch of saddle cycles has larger amplitude than the nodal branch and increases away from the fold, as also does the period, which seems to eventually diverge. This is a manifest sign that an infinite period bifurcation may be in play.

Figure 16.

Homoclinic bifurcation $\mathit{C}_{\textit{A}}^{\textit{s}}$ . (a) Time series $v_{\backslash }(t)$ of several $\mathit{C}_{\textit{A}}$ (solid blue-to-cyan lines for the nodal cycle $\mathit{C}_{\textit{A}}^{\textit{n}}$ and dashed cyan with labels indicating $ \textit{Re}$ for the saddle cycle $\mathit{C}_{\textit{A}}^{\textit{s}}$ ; same colour denotes same $ \textit{Re}$ ) approaching the bifurcation. The steady states $\mathit{E}_{\textit{A}}^{\textit{uf}}$ (dotted blue, extrapolated) and ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (dashed blue) are indicated with horizontal lines. (b) Phase portrait at $ \textit{Re}=742.87$ accompanied by solutions on the asymmetric branch at several other $ \textit{Re}$ leading to the bifurcation. The steady state $\mathit{E}_{\textit{A}}^{\textit{f}}$ is shown from $ \textit{Re}=550$ on in steps $\Delta Re=50$ (filled circles of increasingly darker shades of blue) up to the Hopf point (empty blue circle). The nodal cycle $\mathit{C}_{\textit{A}}^{\textit{n}}$ is depicted at $ \textit{Re}=737.7, 739$ and $741$ (solid lines transitioning from blue to cyan) up to the homoclinic bifurcation and at several $ \textit{Re}$ beyond (increasingly light shades of cyan up to the cyclic fold at $ \textit{Re}_{{\textit {FC}}_{\textit{A}}}$ ). The saddle cycle $\mathit{C}_{\textit{A}}^{\textit{s}}$ is shown at the same $ \textit{Re}$ as the $\mathit{C}_{\textit{A}}^{\textit{n}}$ (increasingly dark shades of cyan from the cyclic fold to the homoclinic point, with labels indicating $ \textit{Re}$ ). (c) Distance $d_{{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ time series of $\mathit{C}_{\textit{A}}^{\textit{s}}$ with respect to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ at $ \textit{Re}=742.87$ . The time axis is shared with panel (a). (d) Snapshot of $\mathit{C}_{\textit{A}}^{\textit{s}}$ at the closest approach to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ at $ \textit{Re}=742.87$ , indicated with cyan bullets in the time series of panels (a) and (c). (e) The ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ at the same $ \textit{Re}$ .

Figure 16 clarifies the situation and reveals the advent of a homoclinic bifurcation. For completeness, we have included in the figure solutions across both the Hopf ( $\mathit{H}_{\textit{A}}$ ) and cyclic fold ( $\mathit{FC}_{\textit{A}}$ ) bifurcations. Figure 16(b) essentially depicts the phase portrait at the homoclinic bifurcation point ( $ \textit{Re}_{{\mathit{Hom}_{\textit{A}}}}$ ), but includes also the stable equilibrium $\mathit{E}_{\textit{A}}^{\textit{f}}$ (circles of increasingly darker blue) from $ \textit{Re}=550$ in steps $\Delta Re=50$ up to $\mathit{H}_{\textit{A}}$ (blue empty circle) and the stable nodal cycle $\mathit{C}_{\textit{A}}^{\textit{n}}$ at several $ \textit{Re}$ from $\mathit{H}_{\textit{A}}$ to the $\mathit{FC}_{\textit{A}}$ (lines transitioning from blue to cyan up to $\mathit{Hom}_{\textit{A}}$ , followed by increasingly lighter shades of cyan thereafter). The four $\mathit{C}_{\textit{A}}^{\textit{n}}$ cycles in the range $ \textit{Re}_{\textit{Hom}_{\textit{A}}}$ to $ \textit{Re}_{\textit{FC}_{\textit{A}}}$ ( $ \textit{Re}=743,743.5,744$ and 744.2) are in one to one correspondence with coexisting saddle cycles $\mathit{C}_{\textit{A}}^{\textit{s}}$ , drawn in the same shade but with dashed lines. The $\mathit{C}_{\textit{A}}^{\textit{n}}$ grows fast after its appearance in the Hopf bifurcation, then the growth rate relaxes for a while before speeding up again as the fold is approached. The larger $\mathit{C}_{\textit{A}}^{\textit{n}}$ and the smaller $\mathit{C}_{\textit{A}}^{\textit{s}}$ , corresponding to the same $ \textit{Re}=744.2$ , are already on the verge of colliding and disappearing in the cyclic fold at $ \textit{Re}_{{\mathit{FC}_{\textit{A}}}}$ . From the fold backwards, but now on the saddle branch, the $\mathit{C}_{\textit{A}}^{\textit{s}}$ keep growing and conspicuously approach the saddle equilibrium ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ .

The $v_{\backslash }$ time series of figure 16(a), drawn in the same colours and shadings as figure 16(b), illustrate the full evolution of $\mathit{C}_{\textit{A}}$ along its branch. The nodal cycle $\mathit{C}_{\textit{A}}^{\textit{n}}$ starts with vanishing amplitude, nearly sinusoidal time dependence and short yet finite period following $\mathit{H}_{\textit{A}}$ . The amplitude increases fast thereafter and the period slowly, and both keep increasing up to $\mathit{FC}_{\textit{A}}$ . The cycle becomes unstable at the fold and the branch bends back towards lower $ \textit{Re}$ , now transformed into the saddle $\mathit{C}_{\textit{A}}^{\textit{s}}$ . As $ \textit{Re}$ is decreased towards the crisis, the amplitude keeps growing, albeit slowly, while the period increases fast as the dynamics spend increasingly longer times in the vicinity of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ (horizontal blue dashed line). Definite evidence that the slowing dynamics are associated with a forthcoming collision with ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ is provided by figure 16(c), where the instantaneous distance $d_{{\mathit{E}}_{\textit{A}}^{\textit{s}}}$ from $\mathit{C}_{\textit{A}}^{\textit{s}}$ to ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ is shown to decrease exponentially fast as the saddle equilibrium is approached and then depart again exponentially fast shadowing its unstable manifold at $ \textit{Re}=742.87$ , close to the bifurcation. To fathom the proximity of the fly-by, figures 16(d) and 16(e) (and supplementary movie 7) compare flow fields of $\mathit{C}_{\textit{A}}^{\textit{s}}$ and ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ , respectively, at the time instant corresponding to the closest approach of the cycle to the saddle equilibrium, indicated with cyan bullets in figures 16(a) and 16(c). As expected, there is no telling them apart.

Unlike the heteroclinic bifurcations $\mathit{Het}_{\textit{S}}$ and $\mathit{Het}_{\textit{S'}}$ , which involved ST-symmetric cycles and mutually symmetric saddle equilibria, the boundary crisis we have now at hand concerns only an asymmetric cycle and a single equilibrium. The symmetries of the LDSSCF problem imply that all solutions breaking the $\mathit{M}_{\backslash }$ symmetry must occur in pairs and the same applies to all bifurcations undergone by these solutions. The boundary crisis of $\mathit{C}_{\textit{A}}^{\textit{s}}$ upon hitting ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ is not an exception, so that ${\mathit{M}_{\backslash }}{\mathit{C}_{\textit{A}}^{\textit{s}}}$ must be colliding with ${\mathit{M}_{\backslash }}{\mathit{E}}_{\textit{A}}^{\textit{s}}$ at the same exact $ \textit{Re}$ . Since ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ is an hyperbolic saddle equilibrium, the boundary crisis is a homoclinic (or saddle-loop) bifurcation, whereby the cycle and the stable and unstable manifolds of the equilibrium merge in a non-robust homoclinic orbit connecting instantly the equilibirum with itself and the periodic orbit disappears thereafter. In this type of infinite period bifurcation, the period $T$ is known to diverge following (6.4), so figure 17 explores the evolution of $T$ as a function of $ \textit{Re}$ . A logarithmic fit according to $T = -\lambda ^{-1}\log (Re-Re_{{\mathit{Hom}_{\textit{A}}}})+T_1$ , locates the homoclinic bifurcation Hom $_{\textit{A}}$ as occurring at $ \textit{Re}_{{\mathit{Hom}_{\textit{A}}}}=742.8643\pm 0.0002$ with $\lambda =0.187\pm 0.004$ . The $\mathit{C}_{\textit{A}}^{\textit{s}}$ being a saddle cycle, the fitting parameter $\lambda$ is to be compared with the stable, rather than the unstable, eigenvalue of the saddle equilibrium. The least stable of the stable eigenvalues of ${\mathit{E}}_{\textit{A}}^{\textit{s}}$ at $ \textit{Re}_{{\mathit{Hom}_{\textit{A}}}}$ has been estimated, from exponential fits to the final stages of the edge tracking trajectories leading to the saddle equilibrium, as $\lambda _{{\mathit{E}}_{\textit{A}}^{\textit{s}}}=-0.195\pm 0.008$ , which is compatible with the divergence rate of the period of $\mathit{C}_{\textit{A}}^{\textit{s}}$ .

Figure 17.

Homoclinic bifurcation of $\mathit{C}_{\textit{A}}$ . Period $T$ as a function of $ \textit{Re}$ . The dash-dotted line is a logarithmic fit to a few data points closest to the crisis, indicated with the vertical asymptote. The inset presents the same data in log–log scale.

7. Double-zero bifurcation with Z $_{\boldsymbol{2}}$ symmetry and degeneration of third-order coefficients

We will now argue that the solutions, bifurcations and phase space topologies reported in the preceding sections are compatible with a 1-D parameter sweep in the vicinity of a codimension-4 point. The one-dimensionality of our exploration does not allow for the direct determination of the change of variables that would transform the 4LDSSC flow problem into the corresponding normal form, which would not be of any particular use anyway. Instead, we will draw the correspondence from phenomenological observation.

The double zero bifurcation with Z $_2$ symmetry was first studied by Takens (Reference Takens1974) as an approximating ordinary differential equations system to the 1 : 2 strong resonance in the context of codimension-2 bifurcations of maps. Provided non-degeneracy of the third order terms is granted, a complete unfolding of the normal form is given by

(7.1) \begin{align} \left \{ \begin{array}{l} \dot {x} = y, \\ \dot {y} = \alpha x + \beta y + s x^3 - x^2y, \end{array} \right . \end{align}

with $s=\pm 1$ , of which only $s=-1$ is structurally stable. The associated bifurcation diagrams feature states akin to $\mathit{E}_{\textit{S}}$ , $\mathit{E}_{\textit{A}}$ , $\mathit{C}_{\textit{S}}$ , $\mathit{C}_{\textit{A}}$ and $\mathit{C}_{\textit{S}^\prime}$ , but lack several of the characteristic features observed in the 4LDSSC problem. Knobloch (Reference Knobloch1986) presented the normal forms for the double zero bifurcation subject to failure of the non-degeneracy conditions, both for the cases with and without reflectional symmetry. In the case of mirror symmetry, up to four nonlinear terms of up to fifth order need be retained. If the degeneracy affects the two extant third-order terms (Knobloch Reference Knobloch1986, case 4B), a complete unfolding of the structurally stable normal form of the reflectionally symmetric double zero bifurcation is given by

(7.2) \begin{align} \left \{ \begin{array}{l} \dot {x} = y, \\ \dot {y} = \alpha x + \beta y + \gamma x^3 + \delta x^2y - x^5 - x^4y, \end{array} \right . \end{align}

Figure 18.

Structurally stable phase portrait sketches of the double zero bifurcation with Z $_2$ symmetry and degeneracy of the cubic terms that are relevant to the 4LDSSC flow problem.

Table 2.

Correspondence of phase portraits and intervening bifurcations with those for the double zero bifurcation with Z $_2$ symmetry and degeneracy of cubic terms. Each row corresponds to a structurally stable phase portrait (first four columns) and the bifurcation or transition (last seven columns) that leads to the next. For each phase portrait and bifurcation/transition, the columns indicate the figures correspondence between our sketches (Sketch) and the actual flow problem (4LDSSC), the sketch of Dangelmayr et al. (Reference Dangelmayr, Neveling and Armbruster1986) (D86) and the sketch of Dangelmayr et al. (Reference Dangelmayr, Armbruster and Neveling1985) (D85). The correspondence between our naming of the bifurcations/transitions and those in D86 and D85 is also given in the columns labelled as Bif. Note that the correspondence fails for figures 18(i)/10(g) and 18(j)/7 because $\mathit{H}_{\textit{A}}$ and $\mathit{Hom}_{\textit{A}}$ are reversed (see shaded cells). Phase portraits and bifurcations/transitions not shown are indicated with a hyphen (–).

and the full collection of bifurcation diagrams was produced by Dangelmayr et al. (Reference Dangelmayr, Neveling and Armbruster1986), henceforth abreviatted as D86. Notice that we have reversed the sign of all four parameters $(\alpha ,\beta ,\gamma ,\delta )$ with respect to D86 for consistency with Knobloch (Reference Knobloch1986). These bifurcation diagrams contain almost all the phenomenology we have observed in the 4LDSSC flow problem. Our sweep in terms of $ \textit{Re}$ can be interpreted as a 1-D path through the four-dimensional parameter space $(\alpha ,\beta ,\gamma ,\delta )$ . In particular, we start with $\alpha ,\beta ,\gamma ,\delta \lt 0$ in region J1 of parameter space (see figures 2 and 3J in D86) and, in particular, with sketch 5.1, to be compared with our figure 10(a). To ease comparison with D86, we reproduce in figure 18 the phase portrait sketches that are relevant to the 4LDSSC problem, adapting colours and styles. Thus, figure 18(a) is a schematic representation of the case shown in figure 10(a) and corresponds to sketch 5.1 in D86. This and all correspondences discussed hereafter between the sketches in figure 18 and the phase portraits shown in previous sections or the sketches in D86 have been tabulated in table 2, to assist the reader. When $ \textit{Re}$ is increased from its starting value, eventually $\alpha$ becomes positive, a pitchfork bifurcation occurs (P in D86, $\mathit{P}_{\textit{1S}}$ here), and the path crosses into region J5; compare sketch 5.5 of D86 with our figure 18(b), corresponding to figure 10(b), or, strictly speaking, figure 18(c). The difference between figures 18(b) and 18(c) illustrates a node-focus ( $\mathit{NF_{1A}}$ ) transition of $\mathit{E}_{\textit{A}}$ . At this point, $\gamma$ becomes positive and J5 turns into A5 with no intervening bifurcation (see figures 2 and 3A in D86). With $\gamma$ already positive, $\alpha$ becomes negative again, the system undergoes a second pitchfork bifurcation (again P in D86, our $\mathit{P}_{\textit{2S}}$ ), and the path crosses into region A16; see sketch 5.16 in D86 and our figures 18(d) and 18(e), corresponding to figures 10(c) and 10(d), respectively, which illustrate, incidentally, the node-focus transition of $\mathit{E}_{\textit{S}}$ . From there, $\beta$ becomes positive, the system undergoes a Hopf bifurcation (H $_0$ in D86, $\mathit{H}_{\textit{S}}$ here) and the system lands in region A18 (sketch 5.18 in D86 versusfigure 18 f here, corresponding to figure 4). The path then traverses a heteroclinic bifurcation (SL $_{\textit{i}}$ in figure 6 of D86, $\mathit{Het}_{\textit{S}}$ of figure 13 here) into region A17 (figure 5.17 in D86, figure 18 g here, akin to figure 10 e). Sketches of the three global bifurcations observed in the 4LDSSC problem are shown in figure 19, reproducing the relevant sketches of figure 6 in D86, but with colours and styles matching those we have used throughout. The Heteroclinic point $\mathit{Het}_{\textit{S}}$ of figure 13 is schematically represented as figure 19(a). A second heteroclinic bifurcation (SL $_+$ of figure 6 in D86, $\mathit{Het}_{\textit{S'}}$ of figure 19 b here, corresponding to figure 15) takes the system into region A20 (figure 5.20 in D86, figure 18 h, akin to figure 10 f). At this point occurs the only discrepancy we have observed between our scenario and the unfolding of the reflectionally symmetric double zero bifurcation. As we have seen, the ensuing sequence takes the 4LDSSC problem through the Hopf ( $\mathit{H}_{\textit{A}}$ ), homoclinic ( $\mathit{Hom}_{\textit{A}}$ ) and cyclic fold ( $\mathit{FC}_{\textit{A}}$ ) bifurcations, while the double zero bifurcation concatenates the homoclinic (SL $_{\textit{a}}$ ) and Hopf (H $_{\textit{a}}$ ) bifurcations in reverse order and dispenses with the cyclic fold. Presumably, this is the effect of higher order terms that induce a crossing of the SL $_{\textit{a}}$ and H $_{\textit{a}}$ bifurcation lines, rendering the Hopf subcritical at a Bautin (or degenerate Hopf) bifurcation, whence a cyclic fold bifurcation line is issued (Kuznetsov Reference Kuznetsov2023). The unique pair of asymmetric limit cycles observed by D86 is unstable and appears in the homoclinic bifurcation SL $_{\textit{a}}$ the same way our unstable $\mathit{C}_{\textit{A}}^{\textit{s}}$ appears in $\mathit{Hom}_{\textit{A}}$ (compare sketches 6 and 5.22 in D86 with figures 19 c and 18 j here, corresponding to figures 16 and 7). In contrast, our Hopf $\mathit{H}_{\textit{A}}$ is supercritical and precedes $\mathit{Hom}_{\textit{A}}$ , while their Hopf H $_{\textit{a}}$ is subcritical and must necessarily come after SL $_{\textit{a}}$ . To restore the equivalence between $\mathit{Hom}_{\textit{A}}$ and SL $_{\textit{a}}$ , the limit cycle resulting from $\mathit{H}_{\textit{A}}$ must turn unstable, and hence the presence of the cyclic fold $\mathit{FC}_{\textit{A}}$ . A sketch of $\mathit{FC}_{\textit{A}}$ is depicted in figure 20(a), where the non-hyperbolic $\mathit{C}_{\textit{A}}$ cycle has been styled as dash-dotted. In a way, our figures 18(i) and 18(j), which are schematic representations of figures 10(g) and 7, replace the sketch 5.22 in D86. Either subsequence of bifurcations, that of the degenerate mirror symmetric double zero bifurcation or the one we have found for the 4LDSSC flow problem, leads to the same state of affairs, region A19 (sketch 5.19 in D86 or figure 18 k here, corresponding to figure 10 h). The last step, and the only one we have not been able to study in any depth, takes the system across a saddle-node bifurcation ( $\mathit{SN}_{\textit{A}}$ ) into region A2 (sketch 5.2 in D86 and figure 18 m, corresponding to figure 10 i). For the occurrence of $\mathit{SN}_{\textit{A}}$ to be possible (see figure 20 b, where the non-hyperbolic $\mathit{E}_{\textit{A}}$ is depicted with triangles), the focus $\mathit{E}_{\textit{A}}^{\textit{uf}}$ must first become a node $\mathit{E}_{\textit{A}}^{\textit{un}}$ in a reverse node–focus transition ( $\mathit{NF_{2A}}$ ), as illustrated by the differences from figures 18(k) and 18(l).

Figure 19.

Phase portrait sketches of the global bifurcations occurring in the double zero bifurcation with Z $_2$ symmetry and degeneracy of the cubic terms that are relevant to the 4LDSSC flow problem.

Figure 20.

Phase portrait sketches of (a) the $\mathit{FC}_{\textit{A}}$ and (b) the $\mathit{SN}_{\textit{A}}$ bifurcations of the 4LDSSC flow problem. The non-hyperbolic $\mathit{C}_{\textit{A}}$ and $\mathit{E}_{\textit{A}}$ solutions are denoted with dash-dotted lines and triangles, respectively.

In this parallel, the parameter $\delta$ remains negative all along. This suggests that the 4LDSSC problem may be understood in the framework of a codimension-3 bifurcation, rather than having to resort to the fully degenerate reflectionally symmetric double zero bifurcation, which is codimension-4. Actually, case 3B of Knobloch (Reference Knobloch1986), which has the degeneracy only in the term $x^3$ , was unfolded by Dangelmayr et al. (Reference Dangelmayr, Armbruster and Neveling1985) and already contains all the necessary ingredients. For reference, the parameter space regions 1, 5, 16, 18, 17, 20, 22, 19 and 2 of D86 correspond to 1, 6, 7, 8, 9, 10, 12, 13 and 2 of Dangelmayr et al. (Reference Dangelmayr, Armbruster and Neveling1985), respectively.

Table 2 summarises, for completeness, the correspondence of phase portraits and intervening bifurcations of the 4LDSSC flow problem with those for the double zero bifurcation with Z $_2$ symmetry and degeneracy of cubic terms.