6.1 Fold-Neimark–Sacker dynamics

The fold-Neimark–Sacker bifurcation is sometimes called the Hopf saddle node for maps bifurcation. It has been studied by Broer, Simó & Vitolo (Reference Broer, Simó and Vitolo2008) in a low-dimensional model problem that was designed to be as ‘generic as possible’. The model was constructed by perturbing the Poincaré map of the truncated normal form for the Hopf-saddle-node bifurcation. In essence, the Hopf-saddle-node bifurcation is treated as being the stroboscopic map of the fold-Neimark–Sacker bifurcation. As such, steady states and limit cycles of the Hopf-saddle-node bifurcation correspond to limit cycles and 2-tori of the fold-Neimark–Sacker bifurcation, and the strobes of those limit cycles and 2-tori are fixed points and cycles in the approximating map. But complications set in due to possible resonances when the frequencies of the 2-tori are in low-order rational ratios. Broer et al. (Reference Broer, Simó and Vitolo2008) provide several conjectures based on numerical studies of their model problem that involve various bifurcations of limit cycles and 2-tori, yielding a cascade of bifurcations of 2-tori, and heteroclinic and homoclinic behaviour. The Hopf-saddle-node bifurcation scenario on which their model was based is the one that includes the complex dynamics associated with the breaking of a heteroclinic connection. The nonlinear dynamics in the neighbourhood of such a Hopf-saddle-node bifurcation has been studied numerically and experimentally in short aspect-ratio Taylor–Couette flows (Lopez, Marques & Shen Reference Lopez, Marques and Shen2004; Abshagen et al. Reference Abshagen, Lopez, Marques and Pfister2005). Here, we find that much of the simpler dynamics in the neighbourhood of the fold-Neimark–Sacker bifurcation FN indicated in figure 5 can be diagnosed along these same lines, and as in all Hopf-saddle-node and fold-Neimark–Sacker bifurcations that have been studied, the complex dynamics is very much case dependent, but intrinsically associated with the heteroclinic region. In the following, we describe in some detail such dynamics.

Figure 6.

Schematic of the Hopf-saddle-node bifurcation, which is used to discuss the fold-Neimark–Sacker (FN) bifurcation. The central plot is the bifurcation diagram, and the surrounding plots are typical phase portraits in the various sections of the bifurcation diagram, labelled 1–7. In these phase portraits, the symbols on the horizontal axis are stable (blue) or unstable (yellow) equilibria, which correspond to the upper and lower branch L $_{1:1}$ limit cycles in the full problem.

Figure 6 is a schematic of a Hopf-saddle-node bifurcation which we adopt to discuss the dynamics in the neighbourhood of the fold-Neimark–Sacker point FN in figure 5. The FN point is at the origin of the bifurcation diagram, $(\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2})=(0,0)$ , where the coordinates $\unicode[STIX]{x1D707}_{1}$ and $\unicode[STIX]{x1D707}_{2}$ are the variables in the codimension-two normal form. These are related to the two variables our full problem, $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x1D6FC}$ , by linear transformations near the point FN. The bifurcation diagram has a number of bifurcation curves that divide the $(\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2})$ space into various regions, labelled 1–7. Typical phase portraits in these regions are provided in the figure. In these phase portraits, the symbols on the horizontal axis are stable (blue) or unstable (yellow) equilibria, which correspond to the upper and lower branch L $_{1:1}$ limit cycles in the full problem. In region 1 ( $\unicode[STIX]{x1D707}_{1}<0$ ), there are no solutions nearby in phase space and all initial conditions evolve away from this region of phase space. In the full problem the evolutions end up at the stable basic state, which is far removed from the neighbourhood of the fold-Neimark–Sacker bifurcation in phase space.

Crossing the fold bifurcation curve from region 1 into region 2, the fold bifurcation leads to the creation of the upper and lower branch L $_{1:1}$ . Crossing the Neimark–Sacker curve NS from region 2 to region 3, the stable upper branch L $_{1:1}$ loses stability and spawns a stable 2-torus $\text{T}_{2}$ . In the phase portraits for regions 3a and 3b, this 2-torus is indicated as a (blue) equilibrium off the axis. The way to interpret the nature of the equilibria in the phase portraits is such that equilibria on the axis have one frequency (corresponding to $\unicode[STIX]{x1D714}/2$ since the L $_{1:1}$ are subharmonic); equilibria off the axis correspond to 2-tori and have an additional frequency (associated with rotations of the equilibria around the axis). The second frequency varies along the Neimark–Sacker curve NS, and may be in a rational ratio with the first frequency ( $\unicode[STIX]{x1D714}/2$ ) at various points along the curve. From such points emerge Arnold tongues into region 3 in which the flow is phase locked.

Figure 7.

Schematic of the transverse across the green wedge-shaped region 5 in figure 6. The blue and green curves are the stable and unstable manifolds of the two saddle states (yellow symbols), that intersect transversely.

Crossing from region 3 into region 4, the curve labelled ‘Torus’ is crossed, at which the 2-torus loses stability (it goes from being blue to yellow) and a stable (blue) invariant circle is spawned in the phase portrait for region 4. This invariant circle has three frequencies associated with it: two are inherited from the 2-torus and the third is new corresponding to cycling around the blue circle. This indicates the birth of a 3-torus $\text{T}_{3}$ . Below we give descriptions of numerical solutions exhibiting 3-torus dynamics in the corresponding parameter region in figure 5. The dynamics in region 5 is not smooth due to various resonances, as suggested by the model case study in Broer et al. (Reference Broer, Simó and Vitolo2008). In the Hopf-saddle-node normal form, the invariant circle becomes heteroclinic to the two unstable equilibria on the axis (the green curve in the region 5 phase portrait). Even for the Hopf-saddle-node bifurcation, this heteroclinic connection does not persist if higher-order terms in the normal form are included. Typically, the higher-order terms result in the stable and unstable manifolds emanating from the unstable equilibria on the axis intersecting transversely instead of coinciding. This is illustrated in figure 7. The wedge-shaped region 5 in green demarcates the region where the transverse intersections occur. It is in this region that very complicated dynamics occurs, some of which we describe below. Continuing from region 5 into regions 6 and 7, all states in the phase space neighbourhood of the fold-Neimark–Sacker bifurcation are unstable, and so all initial conditions eventually evolve away from this part of phase space.

Figure 8.

Time series of the temperature at $(x_{p},z_{p})=(1/\sqrt{8},1/\sqrt{8})$ , $T_{p}$ (a–c) and the normalized thermal ${\mathcal{L}}^{2}$ measure $E_{T}$ (d–f) for (a,d) L $_{1:1}$ at $\unicode[STIX]{x1D714}=1.367$ , (b,e) T $_{2}$ at $\unicode[STIX]{x1D714}=1.359$ and (c,f) T $_{3}$ at $\unicode[STIX]{x1D714}=1.356$ , all with $\unicode[STIX]{x1D6FC}=0.08$ , corresponding to the ‘ $+$ ’ marks in figure 5.

Figure 9.

Frequency spectra of the time series shown in figure 8. The angular response frequency $\unicode[STIX]{x1D714}_{R}$ has been normalized by the forcing frequency $\unicode[STIX]{x1D714}$ .

6.2 Torus bifurcation to $\text{T}_{3}$

Figure 5 includes ‘ $+$ ’ markers, these indicate the $(\unicode[STIX]{x1D714},\unicode[STIX]{x1D6FC})$ locations of some sample states whose time series are shown in figure 8. Time is shown in units of $n_{\unicode[STIX]{x1D70F}}$ , the number of forcing periods $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}$ . All are shown at $\unicode[STIX]{x1D6FC}=0.08$ and the variation in $\unicode[STIX]{x1D714}$ corresponds to a traverse through the NS and Torus bifurcation curves. For each case, the time series on the left is of $T_{p}$ , the temperature at the point $(x_{p},z_{p})=(1/\sqrt{8},1/\sqrt{8})$ (a local measure). The one on the right is of the normalized thermal ${\mathcal{L}}^{2}$ measure $E_{T}$ (a global measure). The first case at $\unicode[STIX]{x1D714}=1.367$ is the upper branch limit cycle L $_{1:1}$ . As described earlier, it is a subharmonic response so that the local measure is $2\unicode[STIX]{x1D70F}$ -periodic, but it is setwise ${\mathcal{G}}$ symmetric so its global measure is $\unicode[STIX]{x1D70F}$ -periodic. The power spectral densities (PSD) of these time series are shown in figure 9(a). The spectrum of $E_{T}$ shows a single peak at $\unicode[STIX]{x1D714}_{R}/\unicode[STIX]{x1D714}=f_{1}=1.0$ , plus harmonics, whereas that of $T_{p}$ also shows a (stronger) peak at $\unicode[STIX]{x1D714}_{R}/\unicode[STIX]{x1D714}=f_{1}/2$ , plus harmonics. The next state at $\unicode[STIX]{x1D714}=1.359$ is to the low- $\unicode[STIX]{x1D714}$ side of the NS curve. It is a 2-torus $\text{T}_{2}$ that was spawned at the Neimark–Sacker bifurcation where L $_{1:1}$ loses stability. The time series of $\text{T}_{2}$ are very similar to those of L $_{1:1}$ with a small amplitude modulation. Together with the corresponding spectra, we see that this modulation frequency is $f_{2}\approx 0.223f_{1}$ . The $E_{T}$ spectrum of $\text{T}_{2}$ consists of peaks at these two frequencies plus their linear combinations. The $T_{p}$ spectrum has many more peaks, corresponding to $f_{1}/2$ , $f_{2}/2$ and all their linear combinations. This is evidence that $\text{T}_{2}$ is also setwise ${\mathcal{G}}$ symmetric. The third case at $\unicode[STIX]{x1D714}=1.356$ is to the low- $\unicode[STIX]{x1D714}$ side of the Torus curve. On crossing the Torus curve, $\text{T}_{2}$ loses stability and a 3-torus $\text{T}_{3}$ is spawned. The $T_{p}$ time series shows multiple frequencies, but the $E_{T}$ time series of $\text{T}_{3}$ is essentially that of $\text{T}_{2}$ with a very large amplitude (approximately a factor of 10) low-frequency modulation. The corresponding $E_{T}$ spectrum of $\text{T}_{3}$ consists of peaks at $f_{1}$ and $f_{2}$ (just like that of $\text{T}_{2}$ ), together with a very low-frequency peak at $f_{3}\approx 0.0075f_{1}$ (corresponding to a period of approximately 133 $\unicode[STIX]{x1D70F}$ ), and all their linear combinations. The spectrum of the local measure $T_{p}$ has many more peaks as it consists of all the linear combinations of $f_{1}/2$ , $f_{2}/2$ and $f_{3}/2$ .

Figure 10.

Local and global strobe maps of the flows with $\unicode[STIX]{x1D6FC}=0.08$ at the specified forcing frequency $\unicode[STIX]{x1D714}$ , showing the strobed behaviour of 1-tori, 2-tori and 3-tori as $\unicode[STIX]{x1D714}$ is decreased. The states shown in (a,c) correspond to the $+$ markers in figure 5 and those in (b,d) to the $\times$ markers.

Figure 10 shows strobe maps using both local (panel $a$ ) and global (panel $c$ ) quantities of the three states just described above (corresponding to the $+$ markers in figure 5). The local map uses the horizontal and vertical components of velocity at the point $(x_{p},z_{p})=(1/\sqrt{8},1/\sqrt{8})$ every two forcing periods at forcing phase $\unicode[STIX]{x03C0}$ . The global map strobes the kinetic and thermal global measures, given in (2.3), also every two forcing periods and at a half-period phase shift. The first state is L $_{1:1}$ at $\unicode[STIX]{x1D714}=1.367$ . Its strobe map is a point (yellow circular symbol). The next state is $\text{T}_{2}$ at $\unicode[STIX]{x1D714}=1.359$ . Its strobe map is a cycle (closed red loop), which is clearly seen in the local map but requires a significant zoom in to be clearly seen in the global map. The third state is $\text{T}_{3}$ at $\unicode[STIX]{x1D714}=1.356$ . The global strobe map of $\text{T}_{3}$ is the black doughnut-shaped structure, typical of a 2-torus; its structure in the local strobe map is also a 2-torus. Figure 10(b,d) shows the local and global strobe maps for two other $\text{T}_{3}$ states. These are indicated by the ‘ $\times$ ’ markers in figure 5 at slightly smaller $\unicode[STIX]{x1D714}=1.355$ and $1.354$ . Their strobe maps show how they rapidly expand in phase space with very small decreases in $\unicode[STIX]{x1D714}$ . Also, the $\unicode[STIX]{x1D714}=1.355$ case is nearly phase locked; the strobe map is a convoluted cycle on a 2-torus.

Figure 11.

Bifurcation diagram for $\unicode[STIX]{x1D714}=1.35$ .

6.3 Homoclinic-doubling cascade on the $T_{3}$

We now explore some of the dynamics inside the region enclosed by the Torus curve (shaded purple in figure 5). We begin by fixing $\unicode[STIX]{x1D714}=1.35$ . Figure 11 is a bifurcation diagram at this $\unicode[STIX]{x1D714}$ , using the temporal standard deviation in the kinetic energy, $\unicode[STIX]{x1D6F4}_{K}$ , as a measure of the state. The basic state has $\unicode[STIX]{x1D6F4}_{K}=0$ . It loses stability at $\unicode[STIX]{x1D6FC}\approx 0.181$ in a subcritical flip bifurcation, spawning an unstable limit cycle L $_{1:1}$ to lower $\unicode[STIX]{x1D6FC}$ . For larger $\unicode[STIX]{x1D6FC}$ , the flow evolves to the quasi-periodic $\text{T}_{2}$ , which we have continued down to smaller $\unicode[STIX]{x1D6FC}$ . At $\unicode[STIX]{x1D6FC}\approx 0.089$ , $\text{T}_{2}$ loses stability at the Torus bifurcation, and 3-torus $\text{T}_{3}$ is spawned. Using the stable $\text{T}_{2}$ and $\text{T}_{3}$ , together with the stable basic state, we have used the edge state technique to capture the unstable L $_{1:1}$ . At $\unicode[STIX]{x1D6FC}\approx 0.074426$ , the $\text{T}_{3}$ begins a cascade of bifurcations which are described below in some detail. This essentially corresponds to entering region 5 marked in the schematic of the Hopf-saddle-node bifurcation in figure 6.

Figure 12 illustrates some characteristics of a state in the above-mentioned region 5. It shows the local phase portrait (figure 12 a), using the two velocity components at the point $(x_{p},z_{p})=(1/\sqrt{8},1/\sqrt{8})$ and the global phase portrait (figure 12 c), using $E_{T}$ and $E_{K}$ , of the basic state (green dot in the local phase portrait), the unstable L $_{1:1}$ (yellow cycle in the local and global phase portraits; note however that $E_{T}$ of L $_{1:1}$ is essentially time invariant), and a complicated state that resides in region 5 (black cloud of dots). Further insight is gained by examining the strobed phase portraits, shown in figure 12(b,d). In the strobe maps, the limit cycle L $_{1:1}$ is simply a point (yellow dot), and the complicated state in the global strobe map is a thin 2-torus with five loops and in the local strobe map it appears as a wide 2-torus structure with five scrolls. Furthermore, this $\text{T}_{3}$ is very close to forming a homoclinic connection to the unstable L $_{1:1}$ .

Figure 12.

(a) Local phase portraits of the basic state (green dot), the unstable lower branch L $_{1:1}$ (yellow cycle computed using the edge state technique) and the stable attractor (black cloud), and (b) the corresponding strobe map (strobed every two forcing periods at forcing phase $\unicode[STIX]{x03C0}/2$ ), in which L $_{1:1}$ is a single (yellow) dot, and the stable attractor is in black, all at $(\unicode[STIX]{x1D714},\unicode[STIX]{x1D6FC})=(1.35,0.0743178)$ . (c,d) Illustrate the global data associated with (a,b). Note that the basic state is not shown in (c); it is located at $(E_{K},E_{T})=(0,1)$ .

Figure 13.

Local and global strobe maps (strobed every two forcing periods at forcing phase $\unicode[STIX]{x03C0}/2$ ) for $\unicode[STIX]{x1D714}=1.35$ and $\unicode[STIX]{x1D6FC}$ as indicated, illustrating states in the period-doubling cascade. The yellow marker corresponds to the strobed unstable L $_{1:1}$ , determined using the edge state technique.

The $\text{T}_{3}$ for $\unicode[STIX]{x1D714}=1.35$ and $\unicode[STIX]{x1D6FC}\in [0.07443,0.089]$ exhibits behaviour which we describe as having an $\ell =1$ loop. Reducing the forcing amplitude just below $\unicode[STIX]{x1D6FC}\approx 0.07442$ , the global strobe map shows that $\text{T}_{3}$ has $\ell =2$ loops. In a sense, there has been a period doubling of the $\text{T}_{3}$ . Further reducing $\unicode[STIX]{x1D6FC}$ below approximately $0.074374$ , there is another period doubling resulting in an $\ell =4$ $\text{T}_{3}$ , and an $\ell =8$ $\text{T}_{3}$ at $\unicode[STIX]{x1D6FC}\approx 0.074365$ . Further doublings were not resolved in this neighbourhood. Examples of $\text{T}_{3}$ with $\ell =1$ , 2, 4 and 8 are shown in figure 13(a–d), which gives both the local and global strobe maps of the $\text{T}_{3}$ (they also include the unstable L $_{1:1}$ obtained using the edge state technique, indicated as a yellow dot). The strobe maps for smaller $\unicode[STIX]{x1D6FC}$ (figure 13 e–h) generally did not show distinct loops or scrolls, except for some very narrow windows in $\unicode[STIX]{x1D6FC}$ where an $\ell =6$ $\text{T}_{3}$ at $\unicode[STIX]{x1D6FC}=0.074359$ and an $\ell =5$ $\text{T}_{3}$ at $\unicode[STIX]{x1D6FC}=0.074350$ were detected. The start of a second period-doubling cascade with $\ell =3\rightarrow \ell =6$ at $\unicode[STIX]{x1D6FC}\approx 0.07434$ was also found. This behaviour of the strobe maps is very reminiscent of the behaviour of the logistic map (May Reference May1976).

Figure 14.

(a) Variation with $\unicode[STIX]{x1D6FC}$ of $E_{T}$ strobed every two forcing periods at forcing phase $\unicode[STIX]{x03C0}/2$ for $\unicode[STIX]{x1D714}=1.35$ , (b) response period relative to the forcing period, $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{R}$ , in the periodic windows in (a), with the integer $\ell$ related to the logistic map like period-doubling cascade, and (c) $\unicode[STIX]{x1D714}/\ell \unicode[STIX]{x1D714}_{R}$ . See supplementary movie 2, which shows how the local and global strobed phase portraits (shown in figure 13) vary with $\unicode[STIX]{x1D6FC}$ .

We further explore the logistic map-like behaviour of the strobe maps by taking a Poincaré-like section of the global strobe map. Of course, the strobe map is a discrete-time map and so we cannot simply take a Poincaré section of it. Instead, we define a transverse section, give it a little thickness and record $E_{T}$ whenever the strobe map is within this thin transverse rectangular section. The strobe map was iterated (i.e. the flow was simulated) for up to $10^{6}$ forcing periods for each value of $\unicode[STIX]{x1D6FC}$ (124 values of $\unicode[STIX]{x1D6FC}\in [0.0743178,0.074520]$ ). The results (discarding the early transients) are shown in figure 14(a). The similarity to the logistic map is striking. Supplementary movie 2 animates the 124 cases in this small range in $\unicode[STIX]{x1D6FC}$ , with each frame corresponding to a different value of $\unicode[STIX]{x1D6FC}$ , and consisting of the local and global strobe maps at that $\unicode[STIX]{x1D6FC}$ , and a vertical line at the $\unicode[STIX]{x1D6FC}$ value of the plot in figure 14(a). At several of the $\unicode[STIX]{x1D6FC}$ values where the strobe maps have distinct $\ell$ loops/scrolls, we plot in figure 14(b) the period corresponding to the very low frequency in the $\text{T}_{3}$ , and in figure 14(c) this period is divided by $\ell$ , giving essentially the averaged period $1/f_{3}$ over the $\ell$ loops. This averaged period $1/f_{3}$ grows linearly with decreasing $\unicode[STIX]{x1D6FC}$ .

The logistic map-like dynamics just described also has much in common with what is theoretically expected in a so-called homoclinic-doubling cascade (Oldeman, Krauskopf & Champneys Reference Oldeman, Krauskopf and Champneys2000), in which an entire period-doubling cascade collides with a saddle equilibrium. In period-doubling cascades that do not involve homoclinic collisions, the parameter values at which the period-doubling bifurcations occur follow a self-similar scaling given by the Feigenbaum constant (Feigenbaum Reference Feigenbaum1978). In contrast, for the homoclinic-doubling cascade, the scaling constants depend on the eigenvalues of the saddle equilibrium. While Oldeman et al. (Reference Oldeman, Krauskopf and Champneys2000) studied a system of three ordinary differential equations with quadratic nonlinearity that was constructed to satisfy conditions for the existence of a homoclinic-doubling cascade, they concluded that such a cascade is a generic mechanism that should be found in many physical systems. We believe that the cascade we have found is the first to be reported in a hydrodynamic system.

7.1 Dynamics restricted to the ${\mathcal{G}}$ and ${\mathcal{K}}_{z}$ subspaces

L $_{2:2}$ is pointwise ${\mathcal{G}}$ invariant. Fixing $\unicode[STIX]{x1D714}=1.41$ and computing in the ${\mathcal{G}}$ subspace, L $_{2:2}$ is found to bifurcate subharmonically (so it is $2\unicode[STIX]{x1D70F}$ periodic) from the base state by $\unicode[STIX]{x1D6FC}\approx 0.084$ , which is very close to the forcing amplitude predicted by Floquet analysis, $\unicode[STIX]{x1D6FC}\approx 0.0831$ (Yalim et al. Reference Yalim, Lopez and Welfert2018). It remains stable up to $\unicode[STIX]{x1D6FC}\approx 0.155$ . However, if the simulations are only restricted to the ${\mathcal{K}}_{z}$ subspace, it loses stability at $\unicode[STIX]{x1D6FC}\approx 0.087$ via a ${\mathcal{K}}_{x}$ -symmetry-breaking Neimark–Sacker bifurcation. The new frequency $f_{2}$ is approximately an order of magnitude smaller than the forcing frequency, $1/f_{2}\approx 7$ , and corresponds to an internal sloshing state that modulates the L $_{2:2}$ oscillation. Figure 16 shows a few snapshots of the temperature and vorticity of the sloshing state, S $_{2}$ , at $\unicode[STIX]{x1D6FC}=0.111$ (supplementary movie 4 shows the temporal evolution over six forcing periods). S $_{2}$ loses stability at $\unicode[STIX]{x1D6FC}\approx 0.114$ at a secondary Neimark–Sacker bifurcation that spawns a 3-torus, S $_{3}$ , which inherits the two frequencies of S $_{2}$ and has a new much lower third frequency, approximately another order of magnitude smaller. Figure 17 shows the variations with $\unicode[STIX]{x1D6FC}$ of the variance of the horizontal velocity $\unicode[STIX]{x1D6F4}_{u}^{2}$ at the point $(x_{p},z_{p})=(1/\sqrt{8},1/\sqrt{8})$ and the number of forcing periods $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{R}$ associated with the primary $E_{T}$ response frequencies of L $_{2:2}$ , S $_{2}$ and S $_{3}$ for $\unicode[STIX]{x1D714}=1.41$ .

Figure 16.

Snapshots of (a) the vorticity and (b) the temperature of S $_{2}$ at $\unicode[STIX]{x1D714}=1.41$ and $\unicode[STIX]{x1D6FC}=0.111$ shown at six consecutive forcing periods at phase $\unicode[STIX]{x03C0}$ . Supplementary movie 4 shows the corresponding animations over the six forcing periods.

Figure 17.

Variations with $\unicode[STIX]{x1D6FC}$ at $\unicode[STIX]{x1D714}=1.41$ of (a) the variance of the horizontal velocity $\unicode[STIX]{x1D6F4}_{u}^{2}$ at the point $(x_{p},z_{p})=(1/\sqrt{8},1/\sqrt{8})$ , computed in the ${\mathcal{K}}_{z}$ symmetric subspace, and (b) the number of forcing periods corresponding to the $E_{T}$ response frequencies of L $_{2:2}$ ( $f_{1}$ ), S $_{2}$ ( $f_{1}$ , $f_{2}$ ) and S $_{3}$ ( $f_{1}$ , $f_{2}$ , $f_{3}$ ). The part of the L $_{2:2}$ branch with open circles is unstable in the ${\mathcal{K}}_{z}$ , but is stable in the ${\mathcal{G}}$ subspace.

7.2 Dynamics in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace

We now consider what happens in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ invariant subspace. L $_{L}$ and L $_{R}$ which bifurcated from L $_{1:1}$ at the pitchfork bifurcation at $\unicode[STIX]{x1D6FC}\approx 0.084$ are ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ invariant, as is L $_{1:1}$ . In the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace they remain stable until $\unicode[STIX]{x1D6FC}\approx 0.1208$ , where they lose stability at a Neimark–Sacker bifurcation NS $_{s}$ . A 2-torus pair, Q $_{L}$ and Q $_{R}$ , are spawned and are ${\mathcal{G}}$ symmetry conjugates. The period associated with the second frequency acquired at NS $_{s}$ is approximately 400 forcing periods near onset and suddenly becomes unbounded as $\unicode[STIX]{x1D6FC}$ is increased. This second frequency corresponds to a slow drift toward and away from the ${\mathcal{G}}$ symmetry subspace, and at $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{G1}\approx 0.125536$ , Q $_{L}$ and Q $_{R}$ each become homoclinic to the saddle L $_{1:1}$ which resides in the ${\mathcal{G}}$ subspace. For $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{G1}$ , a single setwise ${\mathcal{G}}$ invariant 2-torus, Q, results from the homoclinic gluing (Glendinning Reference Glendinning1984). The homoclinic gluing bifurcation occurs when two symmetrically related states simultaneously become homoclinic to a saddle and ‘glue’ together to form a single symmetric state. Figure 18 shows the velocity variances of the states involved in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace as $\unicode[STIX]{x1D6FC}$ is increased, along with the second frequency of the 2-tori states. The critical $\unicode[STIX]{x1D6FC}_{G1}$ was determined by using a logarithmic fit $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{R}=c_{1}\ln (1/|\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FC}_{G1}|)+c_{2}$ (Gaspard Reference Gaspard1990; Lopez & Marques Reference Lopez and Marques2000; Marques, Lopez & Shen Reference Marques, Lopez and Shen2001; Lopez et al. Reference Lopez, Marques and Shen2004). Also, the variance $\unicode[STIX]{x1D6F4}_{u}^{2}$ of the unstable L $_{1:1}$ is included in figure 18 as open circular symbols. These were obtained by using the unstable basic state as initial condition; the unstable L $_{1:1}$ appears as a long-lived transient (of the order of hundreds of forcing periods). The flow evolves quickly from the unstable basic state, spirals in toward the saddle focus L $_{1:1}$ and eventually evolves away from it to a nearby stable trajectory.

Figure 18.

Variations with $\unicode[STIX]{x1D6FC}$ at $\unicode[STIX]{x1D714}=1.41$ of (a) variance of the horizontal velocity $\unicode[STIX]{x1D6F4}_{u}^{2}$ at the point $(x_{p},z_{p})=(1/\sqrt{8},1/\sqrt{8})$ for various observed states in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetric subspace, as indicated, and (b) the number of forcing periods of the slow response versus forcing amplitude for 2-tori states in (a). The data are fitted to logarithmic fits $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{R}=c_{1}\ln (1/|\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FC}_{c}|)+c_{2}$ , with $\unicode[STIX]{x1D6FC}_{G1}=0.125536$ , $\unicode[STIX]{x1D6FC}_{G2}=0.135994$ and $\unicode[STIX]{x1D6FC}_{G3}=0.146631$ . The plotted slow periods were computed by finding the highest-amplitude frequency of the $E_{T}$ power spectral density over 20 000 forcing periods. Supplementary movie 5 animates the strobe maps of the states in steps of $0.01$ in $\unicode[STIX]{x1D6FC}$ .

Figures 19 and 20 illustrate the dynamics just before and after the gluing bifurcation at $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{G1}$ . At $\unicode[STIX]{x1D6FC}=0.125$ , the strobe map of Q $_{L}$ shows a slow–fast trajectory composed of saddle-focus behaviour with the slow behaviour being at the focus, followed by a rapid excursion out from the focus. The snapshots in figure 20 show the focus to be L $_{1:1}$ , and the trajectory away from the focus resembles L $_{L}$ , then L $_{2:2}$ briefly as it loops back, again resembling L $_{L}$ , and spirals in toward the focus again. The numbers labelled on the various strobe maps in figure 19 corresponds to the snapshots in figure 20. For Q $_{R}$ , there is also a spiral into L $_{1:1}$ , but the ejection out is via L $_{R}$ and the L $_{2:2}$ -like part of the trajectory at the top of the loop has more of a ‘right-hand’ bias than the L $_{2:2}$ -like part in Q $_{L}$ . After the gluing, the result is Q which consists of alternating Q $_{L}$ and Q $_{R}$ loops, as illustrated in figure 20(c) at $\unicode[STIX]{x1D6FC}=0.126$ . Supplementary movie 6 animates the strobes of Q $_{L}$ , Q $_{R}$ and Q shown in the figures over several hundred forcing periods, mapping out the near-homoclinic cycles involved in the gluing. Everything just described for Q $_{L}$ , Q $_{R}$ and Q also takes place for ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{L})$ , ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{R})$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q})$ , as illustrated by the thin curves in figure 19.

Q unglues at $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{G2}\approx 0.135994$ . This ungluing is essentially like the first gluing at $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{G1}$ in reverse as $\unicode[STIX]{x1D6FC}$ is increased. The essential difference between the first and second gluings is that the flows traverse a larger part of phase space at the larger $\unicode[STIX]{x1D6FC}$ . In both the first and second gluing bifurcations, for $\unicode[STIX]{x1D6FC}$ close to $\unicode[STIX]{x1D6FC}_{G1}$ and $\unicode[STIX]{x1D6FC}_{G2}$ , the period $1/f_{2}$ becomes unbounded smoothly and monotonically.

Figure 19.

Strobe maps (every two forcing periods at forcing phase $\unicode[STIX]{x03C0}$ ) for (a) Q $_{L}$ and Q $_{R}$ (thick blue and red curves) as well as ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{L})$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{R})$ (thin green and grey curves) at $\unicode[STIX]{x1D6FC}=0.125$ , before the gluing bifurcation, and (b) Q (thick magenta curve) and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q})$ (thin black curve) at $\unicode[STIX]{x1D6FC}=0.126$ , after the gluing of Q $_{L}$ and Q $_{R}$ (and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{L})$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{R})$ ). The simulations were restricted to the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetric subspace.

Figure 20.

Snapshots of the vorticity at forcing phase $\unicode[STIX]{x03C0}$ with $\unicode[STIX]{x1D714}=1.41$ at the markers indicated in figure 19 (the relative times of the markers are indicated in terms of forcing periods): (a,b) Q $_{L}$ and Q $_{R}$ at $\unicode[STIX]{x1D6FC}=0.125$ and (c) Q at $\unicode[STIX]{x1D6FC}=0.126$ . Supplementary movie 6 animates the strobe maps and vorticity of the Q $_{L}$ , Q $_{R}$ and Q states at the respective parameters.

Figure 21.

Strobe maps (every two forcing periods at forcing phase $\unicode[STIX]{x03C0}$ ) for (a) Q $_{L}$ and Q $_{R}$ (thick blue and thin green curves) as well as ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{L})$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{R})$ (thick blue and thin grey curves) at $\unicode[STIX]{x1D6FC}=0.146$ , before the gluing bifurcation, and (b) Q $_{B}$ (thick magenta curve) and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{B})$ (thin black curve) at $\unicode[STIX]{x1D6FC}=0.147$ , after the gluing of Q $_{L}$ and Q $_{R}$ (and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{L})$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{R})$ ). The simulations were restricted to the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetric subspace.

Figure 22.

Snapshots of the vorticity at forcing phase $\unicode[STIX]{x03C0}$ with $\unicode[STIX]{x1D714}=1.41$ at the markers indicated in figure 21 (the relative times of the markers are indicated in terms of forcing periods): (a,b) Q $_{L}$ and Q $_{R}$ at $\unicode[STIX]{x1D6FC}=0.146$ and (c) Q $_{B}$ at $\unicode[STIX]{x1D6FC}=0.147$ . Supplementary movie 7 animates the strobe maps and vorticity of the Q $_{L}$ , Q $_{R}$ and Q $_{B}$ states at the respective parameters.

In figure 18(a), at $\unicode[STIX]{x1D6FC}_{G3}\approx 0.146631$ the period $1/f_{2}$ again becomes unbounded, but now this process is neither smooth nor monotonic. This gluing event is different to the two gluing bifurcations described above. The most obvious difference is that now Q $_{R}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{L})$ glue, i.e. the trajectories that glue have opposing phases at the L $_{1:1}$ saddle-focus. This is seen both on the strobe maps in figure 21 and in the snap shots in figure 22. There is also the conjugate gluing between Q $_{L}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{R})$ .

Figure 23.

Strobed time series of the local velocity $u$ sampled at $(x_{p},z_{p})=(1/\sqrt{8},1/\sqrt{8})$ and corresponding strobe map sampled every two periods at forcing phase $\unicode[STIX]{x03C0}$ for (a) ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{L})$ and Q $_{R}$ (blue and red curves) at $\unicode[STIX]{x1D6FC}=0.14663$ before the gluing bifurcation, and (b) Q $_{B}$ (magenta curve) at $\unicode[STIX]{x1D6FC}=0.14664$ , post gluing bifurcation. The simulations were restricted to the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetric subspace, and only the last 10 000 forcing periods are shown of simulations with 200 000 forcing periods.

The trajectories at this third gluing are not homoclinic to L $_{1:1}$ (although they do get very close to L $_{1:1}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{L}_{1:1})$ ). Instead, as the trajectories are ejected away from the saddle foci L $_{1:1}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{L}_{1:1})$ they approach the stable manifold of the saddle L $_{2:2}$ and become homoclinic to it at $\unicode[STIX]{x1D6FC}_{G3}\approx 0.146631$ . For $\unicode[STIX]{x1D6FC}\gtrsim \unicode[STIX]{x1D6FC}_{G3}$ , the glued trajectory Q $_{B}$ spirals around L $_{1:1}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{L}_{1:1})$ , but spends a long time near L $_{2:2}$ (near points labelled 4 and 8 in figure 21 b). This suggests the existence of heteroclinic connections between the unstable manifolds of L $_{1:1}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{L}_{1:1})$ and the stable manifold of L $_{2:2}$ , but this connection is structurally unstable and instead of coinciding, the stable and unstable manifolds intersect transversely (much like that shown in the schematic in figure 7). This implies the existence of a horse-shoe map and chaos (Smale Reference Smale1967). Indeed, we have found such chaotic dynamics for $\unicode[STIX]{x1D6FC}$ very close to $\unicode[STIX]{x1D6FC}_{G3}$ , primarily manifesting itself as trajectories with irregular numbers of orbits around L $_{1:1}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{L}_{1:1})$ . Figure 23(a) shows a strobed time series of $u_{p}=u(1/\sqrt{8},1/\sqrt{8},t)$ at $\unicode[STIX]{x1D6FC}=0.14663$ and $\unicode[STIX]{x1D6FC}=0.14664$ , either side of the gluing at $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{G3}$ . For the lower $\unicode[STIX]{x1D6FC}$ case, the responses are very regular Q $_{L}$ and Q $_{R}$ states, where as for $\unicode[STIX]{x1D6FC}$ slightly beyond the gluing, the Q $_{B}$ response shows an irregular number of left and right loops, and the accompanying strobe map in panel (b) demonstrates that the loops are also variable especially as they are ejected along the unstable manifolds of L $_{2:2}$ .

As $\unicode[STIX]{x1D6FC}$ is increased beyond $\unicode[STIX]{x1D6FC}_{G3}$ , the trajectory is farther away from the heteroclinic tangle, and instead follows a smooth path where it executes a single loop around L $_{1:1}$ followed by a single loop around ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{L}_{1:1})$ ; such a trajectory is illustrated in figure 21(b) for $\unicode[STIX]{x1D6FC}=0.147$ . Supplementary movie 7 animates Q $_{R}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D70F}}(\text{Q}_{L})$ at $\unicode[STIX]{x1D6FC}=0.146$ and Q $_{B}$ at $\unicode[STIX]{x1D6FC}=0.147$ .

Supplementary movie 5 summarizes the dynamics in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace at $\unicode[STIX]{x1D714}=1.41$ . It animates the strobe maps in steps of $0.01$ in $\unicode[STIX]{x1D6FC}$ over the $\unicode[STIX]{x1D6FC}$ range in figure 18.

7.3 Dynamics in the full space

We now return to the dynamics in the full space. L $_{L}$ and L $_{R}$ lose stability at $\unicode[STIX]{x1D6FC}\approx 0.11$ via a Neimark–Sacker bifurcation which breaks the pointwise ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetry (in comparison, L $_{L}$ and L $_{R}$ lost stability in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace at $\unicode[STIX]{x1D6FC}\approx 0.1208$ ). This Neimark–Sacker bifurcation is designated NS $_{a}$ . A pair of ${\mathcal{G}}$ related 2-tori, T $_{2L}$ and T $_{2R}$ , are spawned, which are setwise ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ invariant. They inherit the subharmonic frequency $\unicode[STIX]{x1D714}/2$ and gain a new frequency corresponding to between six and seven forcing periods (in contrast, the 2-tori Q $_{L}$ and Q $_{R}$ that were spawned at NS $_{s}$ in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace have a second frequency which is two orders of magnitude smaller). This new frequency corresponds to small oscillations in the degree to which the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetry is broken. The ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ asymmetry is quantified by

(7.2) $$\begin{eqnarray}A_{\unicode[STIX]{x03C0}}({\mathcal{S}})=\Vert T_{{\mathcal{R}}_{\unicode[STIX]{x03C0}}({\mathcal{S}})}-T_{{\mathcal{S}}}\Vert /\Vert T_{{\mathcal{S}}}\Vert ,\end{eqnarray}$$

where ${\mathcal{S}}$ is the state, $T_{{\mathcal{S}}}$ is its temperature, and ${\mathcal{R}}_{\unicode[STIX]{x03C0}}({\mathcal{S}})$ is the result of the action of ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ on ${\mathcal{S}}$ . For T $_{2L}$ and T $_{2R}$ , $A_{\unicode[STIX]{x03C0}}\sim 10^{-3}$ ; in essence they are weakly modulated L $_{L}$ and L $_{R}$ .

Figure 24.

Variance of the horizontal velocity $\unicode[STIX]{x1D6F4}_{u}^{2}$ at the point $(x_{p},z_{p})=(1/\sqrt{8},1/\sqrt{8})$ for $\unicode[STIX]{x1D714}=1.41$ for various observed states, as indicated. In panel (a), the L $_{1:1}$ is unstable for $\unicode[STIX]{x1D6FC}>0.102$ and is obtained through a transient. Supplementary movie 8 animates the strobe maps of the states in steps of 0.01 in $\unicode[STIX]{x1D6FC}$ .

Figure 24 shows the velocity variances of the states involved in the full space as $\unicode[STIX]{x1D6FC}$ is increased for fixed $\unicode[STIX]{x1D714}=1.41$ , along with the long periods of some of the quasi-periodic states. It is very similar to figure 18, which shows the corresponding measures in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace. On the scale of the figure, T $_{2L}$ and T $_{2R}$ are stable only at the symbol shown (in yellow), and they lose stability with a very slight increase in $\unicode[STIX]{x1D6FC}$ . A pair of ${\mathcal{G}}$ -related 3-tori, T $_{3L}$ and T $_{3R}$ are spawned. They inherit the two frequencies from T $_{2L}$ and T $_{2R}$ , and the third new frequency is very low, approximately two orders of magnitude smaller, and corresponds to slow drifts towards and away from the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetric subspace. During these drifts, $A_{\unicode[STIX]{x03C0}}$ varies between $O(10^{-3}{-}10^{-4})$ and $O(10^{-1})$ .

At $\unicode[STIX]{x1D6FC}\approx 0.121$ two new stable 3-tori states are observed. They are very similar to of Q $_{L}$ and Q $_{R}$ in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace, with the only essential difference being that in the full space they have a very weak drift away from and back to the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace. As such, we shall simply call them Q $_{L}$ and Q $_{R}$ . In contrast, T $_{3L}$ and T $_{3R}$ are never ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetric even though they also drift toward and away from the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace in a regular fashion. The T $_{3L}$ and T $_{3R}$ also drift toward and away from the ${\mathcal{K}}_{z}$ subspace during the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace drifts. This ${\mathcal{K}}_{z}$ subspace excursion gains intensity as the forcing amplitude $\unicode[STIX]{x1D6FC}$ is increased, and is associated with the 2-torus S $_{2}$ and 3-torus S $_{3}$ sloshing flows identified in the ${\mathcal{K}}_{z}$ symmetric subspace.

Figure 25.

Strobe maps for the flow taken every two forcing periods at forcing phase $\unicode[STIX]{x03C0}$ at $\unicode[STIX]{x1D6FC}=0.138$ for the stable Q $_{R}$ (blue), stable T $_{3R}$ (black) and unstable L $_{1:1}$ (yellow). See supplementary movie 9 for animated strobes of Q $_{R}$ and T $_{3R}$ .

Q $_{L}$ and Q $_{R}$ undergo the first and second gluing bifurcations with essentially no difference from what we observed earlier in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace. Over the same interval in $\unicode[STIX]{x1D6FC}$ , T $_{3L}$ and T $_{3R}$ follow similar trajectories in phase space with two important differences. They do not become homoclinic to $L_{1:1}$ (although they do spiral in close to it), and they have a pronounced excursion to the sloshing S $_{2}$ or S $_{3}$ (depending on the value of $\unicode[STIX]{x1D6FC}$ ). Q $_{L}$ and Q $_{R}$ are completely oblivious to S $_{2}$ and S $_{3}$ as S $_{2}$ and S $_{3}$ do not exist in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace. Strobes of Q $_{R}$ and T $_{3R}$ are shown in supplementary movie 9 at $\unicode[STIX]{x1D6FC}=0.138$ ; the erratic behaviour in T $_{3R}$ is due to the trajectory passing very close by S $_{3}$ . The corresponding strobe maps are shown in figure 25 with the unstable L $_{1:1}$ shown as a yellow marker. Q $_{L}$ and Q $_{R}$ are no longer stable for $\unicode[STIX]{x1D6FC}\gtrsim 0.139$ , and T $_{3L}$ or T $_{3R}$ are the only observed states until $\unicode[STIX]{x1D6FC}\gtrapprox 0.146$ , the flow then consists of intermittent bursts, displaying behaviour associated with the previously described states for irregular time intervals. This intermittent bursting state IB roughly coincides with the appearance of the Q $_{B}$ state in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace.

Supplementary movie 8 summarizes the dynamics as $\unicode[STIX]{x1D6FC}$ is increased with $\unicode[STIX]{x1D714}=1.41$ in the full space. The left panel has figure 24 for reference and the $(u_{p},E_{T})$ strobe maps are in the right panel.

7.4 The big picture of the L $_{1:1}$ tip

Now, we consider how the various dynamic responses in the different parameter regimes of the tip of the $1:1$ horn described earlier are interrelated. We have traced out both the pitchfork bifurcation and the Neimark–Sacker bifurcation NS $_{a}$ in $(\unicode[STIX]{x1D714},\unicode[STIX]{x1D6FC})$ space, and their loci are shown in figure 26. These curves appear to meet the Neimark–Sacker curve NS (described in § 6) at a point PN (green circular symbol in the figure). For $\unicode[STIX]{x1D714}\lesssim 1.415$ , T $_{2L}$ and T $_{2R}$ are only stable in a very thin wedge region emanating from PN (too thin to see on the scale of the figure). For $\unicode[STIX]{x1D714}\gtrsim 1.415$ , the Neimark–Sacker bifurcation does not break the pointwise ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetry. The Neimark–Sacker bifurcation curve is the continuation of NS $_{s}$ which was identified in the ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ subspace (see § 7.2). It also meets the point PN (it is shown as a dashed red curve for $\unicode[STIX]{x1D714}\lesssim 1.415$ ). The 2-tori that are spawned at NS $_{s}$ , Q $_{L}$ and Q $_{R}$ , are stable over a larger range in $\unicode[STIX]{x1D6FC}$ than those spawned at NS $_{a}$ . Above the NS $_{a}$ and NS $_{s}$ there are multiple stable states, some of which undergo homoclinic bifurcations. The region between these Neimark–Sacker curves and the dashed curve denoted ‘breakup’ has very complicated dynamics, some of which we describe above for $\unicode[STIX]{x1D714}=1.41$ (see § 7.3). In the middle of the complex region is a pair of homoclinic gluing bifurcation curves along which ${\mathcal{G}}$ -related states simultaneously become homoclinic to a setwise ${\mathcal{G}}$ invariant saddle state (the unstable L $_{1:1}$ ). The green region of figure 26 demarcates where pointwise ${\mathcal{R}}_{\unicode[STIX]{x03C0}}$ symmetry is broken. In between the second gluing bifurcation curve and the breakup curve, there are multiple states manifesting slow drifts towards and away from various saddle solutions that are related not just to L $_{1:1}$ , but also the secondary instability of the basic state (L $_{2:2}$ ), and various secondary instabilities of these that we have been able to identify.

Figure 26.

Bifurcation curves in the $1:1$ resonance tongue.