2.1 Local stability analysis

The Eckhaus instability in finite domains was analysed in [Reference Tuckerman and Barkley57], and we briefly review the basic linear analysis here. Separating (2.1) into real and imaginary parts $B = u + iv$ and linearising about $(u,v) = (B_*(j), 0)$ , we find

\begin{align*} u_t &= u_{xx} - 2 j v_x - 2 (\mu - j^2) u \\[2pt] v_t &= v_{xx} + 2 j u_x. \end{align*}

The linearisation is block diagonal in Fourier modes $ u(x) = \sum _{\ell \in \mathbb{Z}} u_\ell \mathrm{e}^{\mathrm{i} \ell x},\ v(x) = \sum _{\ell \in \mathbb{Z}} v_\ell \mathrm{e}^{\mathrm{i} \ell x}$ ,

(2.2) \begin{align} \begin{pmatrix} \dot{u_\ell } \\[5pt] \dot{v_\ell } \end{pmatrix} = \begin{pmatrix} -\ell ^2 - 2( \mu -j^2) & - 2 \mathrm{i} j \ell \\[5pt] 2 \mathrm{i} j \ell & -\ell ^2 \end{pmatrix} \begin{pmatrix} u_\ell \\[5pt] v_\ell \end{pmatrix}. \end{align}

This matrix is self-adjoint, a reminder that the Ginzburg–Landau equation is a gradient flow. It has a negative trace, so that it always has at least one negative real eigenvalue. From a short calculation, one finds that the other eigenvalue is positive when

(2.3) \begin{align} \mu \lt \mu _{\ell,\mathrm{c}} \,:\!=\, 3j^2 - \frac{\ell ^2}{2} \end{align}

for $\ell \neq 0$ , while the $\ell = 0$ block always has one zero and one negative eigenvalue. As $\mu$ decreases, the instability is therefore always first seen in the $\ell = \pm 1$ mode. The eigenvalue that becomes unstable past $\mu _{\ell,\mathrm{c}}$ is given by

(2.4) \begin{align} \lambda _{\ell,+}(\mu ) = - \ell ^2 - (\mu -j^2) + \sqrt{4\ell ^2 j^2 + (\mu -j^2)^2}. \end{align}

For notational simplicity, we usually suppress the dependence of $\lambda _{\ell,+}(\mu )$ and $\mu _{\ell,\mathrm{c}}$ on the base wavenumber $j$ . To further simplify notation, we write $\lambda _\ell$ instead of $\lambda _{\ell,+}$ since the eigenvalue $\lambda _{\ell,-}$ will be irrelevant for the discussion in this work. We also write simply $\mu _{\mathrm{c}}\,:\!=\,\mu _{\mathrm{1,c}}$ for the boundary of stability, noting that modes with $|\ell | \neq 1$ destabilise for smaller values of $\mu$ , reflecting the side-band nature of the Eckhaus instability. Eigenvectors are

\begin{equation*} (u_{\ell },v_{\ell })(\mu )=\left (j^2-\mu +\sqrt {4\ell ^2 j+(\mu -j^2)^2},2\mathrm {i}\ell j\right ). \end{equation*}

Note that, since the equilibrium is fixed under translations $T_{x_0}$ and the reflection $T_-S_-: B(x)\mapsto \bar{B}({-}x)$ , the linearisation commutes with these symmetries which henceforth act on eigenspaces (and on the flows on invariant manifolds tangent to those eigenspaces). Since the algebraic expression for eigenvalues and eigenvectors are somewhat cumbersome, we will later verify assumptions explicitly in the limit of large $j$ , where we set $\mu =\mu _{\mathrm{c}}+\hat{\mu }$ , and find

(2.5) \begin{align} \lambda _{\ell }(\mu )=\frac{-\ell ^2}{2}\left (\frac{\ell ^2}{2}-\frac{1}{2}+\hat{\mu }\right )j^{-2}+\mathrm{O}\left (j^{-4}\right ),\qquad (u_{\ell },v_{\ell })(\mu )=\left (\frac{\ell }{2}+\mathrm{O}\left (j^{-2}\right ),\mathrm{i} j\right ), \end{align}

also writing $e_{\ell }(\mu )$ short for the normalised eigenvector; see Figure 4 for an illustration of the dependence of eigenvalues on $\mu$ . The stability threshold $\hat{\mu }=0$ corresponds to the Eckhaus boundary $\mu = 3 j^2$ in the infinite domain, but with the finite size correction $-\frac{1}{2}$ . Note that since $u$ and $v$ are real, we have $u_\ell = \overline{u_{-\ell }}, v_\ell = \overline{v_{-\ell }}$ , and in particular $\lambda _{\ell }(\mu ) = \lambda _{-\ell }(\mu ).$

Figure 4.

Eigenvalues of the linearisation at $E_j$ with $j=8$ (top) and $j=4$ (bottom), for parameter values $\mu =j^2$ at onset to $\mu _{\mathrm{c}}+10$ . Enlargement near criticality (right) reveals the intricate crossovers that are explicit in the limit of large $j$ . Eigenvalues are shown from $\ell =1$ in dark blue to $\ell =7$ (top) and $\ell =5$ (bottom), respectively, in light blue.

2.2 Connecting orbits

To understand the evolution of periodic patterns in the dynamic problem, we are interested not only in stability boundaries in the static problem but also in connecting orbits. That is, we ask the following question: if we start with an initial condition $B_0$ near the equilibrium $E_j$ which is Eckhaus unstable, where does the solution $B(t)$ go as $t \to \infty$ ?

The complex Ginzburg–Landau equation is in fact a gradient flow in $L^2([0,2\pi ),\mathbb{C})$ ,

(2.6) \begin{equation} B_t=-\nabla _{L^2}\mathcal{E}[B],\qquad \text{ with energy }\quad \mathcal{E}[B] = \frac{1}{2 \pi }\int _0^{2 \pi } \left (\frac{1}{2} |B_x+\mathrm{i} jB|^2 - \frac{\mu }{2} |B|^2 + \frac{1}{4} |B|^4 \right )\, \mathrm{d} x. \end{equation}

As a consequence, the solution $B(t)$ necessarily converges to an equilibrium with energy lower than $E_j$ . A direct calculation shows that from this perspective, out of all equilibria $E_k$ , only wavenumbers $|k| \lt j$ are eligible, and one may suspect that solutions $B(t)$ originating near $E_j$ converge to a specific $E_k$ . Unfortunately, it seems difficult to establish analytically which equilibrium $E_k$ is selected in this sense. In fact, there are also spatially patterned equilibria other than the $E_k$ , believed to all be unstable as they limit, in a large-domain limit, onto unstable phase defect solutions such as

(2.7) \begin{equation} A_{\mathrm{d}}(x;\,k)=\left (\sqrt{2}k+\mathrm{i} \sqrt{1-3k^2}\tanh (\sqrt{1-3k^2}x/\sqrt{2})\right )\mathrm{e}^{\mathrm{i} k x}; \end{equation}

see [Reference Morrissey and Scheel42]. Some, for our purposes rather restrictive, results have been obtained in [Reference Eckmann, Gallay and Wayne18], where heteroclinic orbits between two distinct equilibrium sets $E_j$ and $E_{j-k}$ have been constructed. The arguments there are based on the variational structure (2.6), roughly demonstrating that there is in fact a unique candidate for connecting orbits with given co-periodicity. In our contexts, the results only give conclusive information in the case $\frac{1}{2} \lt j \lt 1$ , in which case a generic perturbation of $E_j$ will converge to $E_{j-1}$ .

On the other hand, the fate of perturbations is rather straightforward to establish numerically. We distill our findings in the following conceptual assumption about the static problem, which we will later rely on to draw conclusions in the problem with slowly varying parameters.

First note that for a generic $\mu$ , the eigenvalues $\lambda _{\ell } (\mu )$ will be distinct up to the symmetry $\lambda _{\ell } (\mu ) = \lambda _{-\ell } (\mu )$ . The special crossover points at which this condition is violated, that is, at which eigenvalues for different wavenumbers $\ell$ collide, will play a role later but are excluded in the following discussion. We denote the crossover point, at which $\lambda _{\ell }(\mu ) = \lambda _{k}(\mu )$ , by $\mu = \mu _{\ell \,:\, k}$ . Away from these crossover points, we let $\mathrm{e}^{\mathrm{u}}_\ell$ denote the real two-dimensional eigenspace associated with the unstable eigenvalues $\lambda _{\pm \ell }(\mu )$ .

The following hypothesis is sometimes colloquially referred to as node conservation or linear mode selection.

To explain the basic intuition, notice that perturbations of the constant equilibrium in the direction of the eigenvector $\lambda _{\ell }$ , that is perturbations where $\delta _1=0$ , have a sinusoidal shape and winding number $\ell$ relative to $E_j$ . The hypothesis states that as the amplitude grows, this ‘modal’ shape is preserved and terminates in the equilibrium $E_{j-\ell }$ , with winding number $-\ell$ relative to the origin.

Figure 5.

Left: existence and stability boundaries, as well as spatio-temporal resonances responsible for cross-overs in the drop number changes, together with mode number of a sample trajectory. Resonance curves $\mu _{1^2:2}$ , $\mu _{1,2:3}$ , etc. correspond to parameter values where $\lambda _1+\lambda _1=\lambda _2$ , $\lambda _1+\lambda _2=\lambda _3$ , and so forth. In the region between solid and dashed part of the curves, $\lambda _{j+1}\gt \lambda _1+\lambda _j$ ; see Section 3.1 for details on resonances and their relevance. Right: space-time plot of $\mathrm{Re}(A)$ of the same sample trajectory, with time axis represented in terms of the slowly varying $\mu$ , varying up until $\mu \lt 0$ and no nontrivial equilibria exist.

Figure 6.

Schematic of the picture implied by Hypothesis 2.1: the unstable manifold of $E_j$ , here 2-dimensional, contains open sets of orbits that connect to $E_{j-1}$ and $E_{j-2}$ , respectively. The hypothesis guarantees that the connecting orbits to $E_{j-1}$ and $E_{j-2}$ , respectively, contain the caps of conical regions around the respective eigenspaces. In particular, trajectories in the boundary between connecting orbits to $E_{j-1}$ and $E_{j-2}$ are not arbitrarily close to the eigenspaces. It seems plausible that the boundary of validity of the hypothesis is related to trajectories on a codimension-one stable manifold of saddle equilibria, here referred to as ‘defect’, since equilibria of the form (2.7) are natural candidates for such a role in the setting of an unbounded domain.

Figure 7.

Testing the linear prediction hypothesis in the fixed- $\mu$ problem, we computed trajectories with small perturbations of an unstable equilibrium $E_j$ in the modes $\ell =1$ (left) and $\ell =2$ (right). We found that the trajectories converged to the equilibrium $E_{j-\ell }$ in a range (shaded grey) far exceeding the possible drop ranges investigated below. In the left figure, $\ell =1$ , drops to $E_{j-1}$ were confirmed up for all $\mu$ larger than and up to a critical value $\mu _{1,\mathrm{bdy}}$ , which includes a region where $\lambda _2\gt \lambda _1$ (shown as $\mu \gt \mu _{1:2}$ ) and a region where $\lambda _2\gt 2\lambda _1$ (shown as $\mu \gt \mu _{1^2:2}$ ). However, it does not encompass the entire parameter region where $E_{j-\ell }$ is stable (shown as $\mu \gt \mu _{\mathrm{stab}}^{-1}$ ). Remarkably, higher drops appear to occur past a somewhat fixed distance to the instability boundary $\mu _{\mathrm{stab}}$ . On the right, the region with consistent drop-by-2 (shaded grey) also encompasses well the region where we predict a local drop-by-two. We refer to (3.4) for precise definitions of the various resonance curves shown and the main prediction in Section 3.1 relying on these resonances to predict drop numbers and drop times.

Hypothesis 2.1 is reasonably well supported by numerical experiments; see Figure 7 for experiments in the cases $\ell =1$ and $\ell =2$ . In the numerical experiments, we perturbed the unstable equilibrium in the eigenvector to $\lambda _{\ell }$ , only, thus checking the milder version of the hypothesis with $\delta _1=0$ . We found consistent drops by $\ell$ in a large region (shaded grey in the figure). We superimposed this region with eigenvalue resonance curves that we computed using numerical continuation. In the left panel, we show the transition from drop-by-one to drop-by-two as predicted in the next section at $\mu _{1^2:2}$ . This resonance occurs well inside the shaded grey region. Note that for smaller $\mu$ the resonance is not present, the numerical continuation of the resonance condition undergoes a saddle-node with a second resonance curve $\mu ^*_{1^2:2}$ . In the right panel, we show in particular the resonance $\mu _{1,2:3}$ , which indicates the transition to a drop-by-three, again well inside the region where the hypothesis holds for $\ell =2$ ; for details, compare the definition of the log amplitudes $b_j$ (3.5)–(3.8) and the main prediction (3.10).

In the spirit of this assumption, we fix $\delta \gt 0$ sufficiently small and consider a trajectory that starts in a neighbourhood of the unstable equilibrium $E_j$ . We define the local drop number (depending on $\delta$ ) at the time when the distance to the $E_j$ reaches $\delta$ by finding the index of the unstable eigenspace with maximal amplitude in the sense of Hypothesis 2.1. In the results presented in the next section, it then turns out that the distance to the eigenspace is in fact very small, that is, $\delta _1\ll \delta _0$ .

We say that the global drop number of such a trajectory is $\ell$ if $E_{j-\ell }$ is the first equilibrium so that the trajectory reaches its neighbourhood of size $\delta$ . Hypothesis 2.1 is geared to imply that local drop numbers and global drop numbers are equal in the static ( $\mu$ fixed) problem. In fact, local and global drop numbers agree well also in the dynamical problem, where $\mu$ varies slowly; compare Figure 11.

We note that, thinking about proving a version of Hypothesis 2.1, detailed information on connecting orbits usually requires information in addition to energy levels of equilibria or more general topological information [Reference Mischaikow41]. One such property are comparison principles with the associated refined Sturm oscillation properties in the real scalar version of (1.1), known as the Allen-Cahn equation, $u_t = u_{xx} + \mu u - u^3$ . Here, in fact the structure of connecting orbits is completely determined; see for instance [Reference Henry28, Chapter 5.3], the review [Reference Fiedler and Scheel22], or [Reference Fiedler and Rocha21] for a more recent perspective. Some extensions towards systems of the type considered here were investigated in [Reference Büger6].

3.1 Main result – spatio-temporal resonances and drop criteria

We summarise here the results of the analysis in the subsequent sections in the form of an algorithm that predicts, at leading order, local drop parameter values $\mu _{\mathrm{out}}$ and local drop number $m$ given an initial parameter value $\mu _{\mathrm{in}}$ and initial wavenumber $j$ in (3.2). Recall the eigenvalues of the linearisation $\lambda _{\ell }(\mu )$ and define resonant parameter values $\mu =\mu _{k_1,\ldots,k_s\,:\,k_0}$ as the values of $\mu$ where a resonance condition in the eigenvalues holds:

(3.3) \begin{align} \lambda _{k_0}(\mu )&=\sum _{\sigma =1}^s \lambda _{k_\sigma }(\mu ) \quad \text{ at } \quad \mu =\mu _{k_1,\ldots,k_s\,:\,k_0}, \end{align}

(3.4) \begin{align} k_0&=\sum _{\sigma =1}^s k_\sigma. \end{align}

For instance,

\begin{equation*} \lambda _{3}(\mu )=\lambda _{1}(\mu )+\lambda _{2}(\mu ),\quad \text { at } \quad \mu =\mu _{1,2:3}, \end{equation*}

and introducing short-hand notation,

\begin{equation*} \lambda _{3}(\mu )=3\lambda _{1}(\mu )\quad \text { at } \quad \mu =\mu _{1,1,1:3}\,=\!:\,\mu _{1^3:3}. \end{equation*}

The parameter values $\mu _{k_1,\ldots,k_s\,:\,k_0}$ represent spatio-temporal resonances in the sense that at these parameter values linear solutions $\mathrm{e}^{\mathrm{i} kx+\lambda t}$ exist that are resonant both in time (3.3) and space (3.4).

Next, we recursively define $\log$ -amplitudes of modes $b_\ell$ given $\mu _{\mathrm{in}}$ as follows:

(3.5) \begin{align} b_1(\mu )&=\int _{\mu }^{\mu _{\mathrm{in}}}\lambda _1(\hat \mu )\mathrm{d}\hat \mu, \end{align}

(3.6) \begin{align} b_2(\mu )&=\int _{\mu }^{\mu _{1^2:2}}\lambda _2(\hat \mu )\mathrm{d}\hat \mu + 2 b_1(\mu _{1^2:2}), \qquad \mu \lt \mu _{1^2:2} \end{align}

(3.7) \begin{align} b_3(\mu )&=\int _{\mu }^{\mu _{1,2:3}}\lambda _3(\hat \mu )\mathrm{d}\hat \mu + b_1(\mu _{1,2:3})+b_2(\mu _{1,2:3}), \qquad \mu \lt \mu _{1,2:3} \end{align}

(3.8) \begin{align} b_4(\mu )&=\int _{\mu }^{\mu _{2,2:4}}\lambda _4(\hat \mu )\mathrm{d}\hat \mu + 2b_2(\mu _{2^2:4}), \qquad \mu \lt \mu _{2,2:4}. \end{align}

Main prediction – drops $m\leqslant 4$ . Fix $\ell$ and an initial parameter value $\mu _{\mathrm{in}}\gt \mu _{\mathrm{c}}$ and consider dynamics with slowly decreasing $\mu =\mu _{\mathrm{in}}-\varepsilon t$ . The drop parameter value $\mu _{\mathrm{out}}$ is given to zeroth order in $\varepsilon$ by

(3.9) \begin{equation} \mu _{\mathrm{out}}=\mathrm{argmax}_{\mu \lt \mu _{\mathrm{c}}} \mathrm{max}_{1\leqslant \ell \leqslant 4 } b_\ell (\mu )=0, \end{equation}

that is, the largest (dynamically the first) $\mu$ -value at which any of the log amplitudes return to 0. The local drop number is

(3.10) \begin{equation} m= \mathrm{argmax}_{1\leqslant \ell }\, b_\ell (\mu _{\mathrm{out}}), \end{equation}

that is, the index $\ell$ that achieves $b_\ell (\mu _{\mathrm{out}})=0.$ The maxima are taken only over log amplitudes that are defined for the given parameter value $\mu$ . As a function of $\mu _{\mathrm{in}}$ , $\mu _{\mathrm{out}}\lt \mu _{\mathrm{c}}$ is strictly decreasing and $m$ is strictly increasing.

The $\log$ -amplitudes before the resonances can be defined as well, but they are always much smaller than the $\log$ -amplitudes with smaller index.

The reader will notice that only specific resonances occur in the definition of the $b_\ell$ . We found these resonances to be relevant in the sense that their effects dominate the effects of all other resonances in the examples we studied, in particular in the limit of large $j$ . We do not know of a general rule to predict which resonances dominate for larger potential drop numbers, so the algorithm for drops by $5$ or higher is more complex. For the mode $b_5$ , one needs to find $\log$ -amplitudes depending on different spatio-temporal resonances and maximise over all of those, for instance defining

\begin{align*} b_5^{2,3:5}(\mu )&=\int _{\mu }^{\mu _{2,3:5}}\lambda _5(\hat \mu )\mathrm{d}\hat \mu + b_2(\mu _{2,3:5}) + b_3(\mu _{2,3:5}) \qquad \mu \lt \mu _{2,3:5},\\[5pt] b_5^{1,4:5}(\mu )&=\int _{\mu }^{\mu _{1,4:5}}\lambda _5(\hat \mu )\mathrm{d}\hat \mu + b_1(\mu _{1,4:5}) + b_4(\mu _{1,4:5}) \qquad \mu \lt \mu _{1,4:5},\\[5pt] b_5^{1,2^2:5}(\mu )&=\int _{\mu }^{\mu _{1,2^2:5}}\lambda _5(\hat \mu )\mathrm{d}\hat \mu + b_1(\mu _{1,2^2:5}) + 2b_2(\mu _{1,2^2:5}) \qquad \mu \lt \mu _{1,2^2:5},\qquad \text{etc.} \end{align*}

With this information, we can define critical parameter values where changes in drop numbers occur. At these parameter values, the argmax in (3.10) is realised simultaneously at two different values of $\ell$ . In all situations we encountered, those two values of $\ell$ were adjacent, $\ell \in \{m,m+1\}$ and the drop number increases by one as $\mu _{\mathrm{in}}$ increases through this critical parameter value. We then denote this critical parameter value by $\mu _{\mathrm{in},m\to m+1}$ , with associated local drop parameter values $\mu _{\mathrm{out},m\to m+1}$ .

It turns out that $\mu _{\mathrm{out},1\to 2}=\mu _{1^2:2}$ , but for $m\geqslant 2$ , $\mu _{\mathrm{out},m\to m+1}$ is strictly less than the resonances associated with the mode $m+1$ , for instance $\mu _{1,m:m+1}$ , $\mu _{1^2,m-1:m+1}$ , etc. We will see that, as a consequence, corrections to the transition points are $\mathrm{O}(\sqrt{\varepsilon })$ for $\mu _{\mathrm{out},1\to 2}$ but $\mathrm{O}(\varepsilon )$ for other transitions.

We refer to Figures 8, 10, and 11 for numerical demonstration of the validity of the results presented here. The remainder of this section presents the rationale behind these predictions, explaining to what extent one should expect rigorous results, and some details on $\varepsilon$ -corrections as well as asymptotics for $j\to \infty$ . The numerical results also show that global drop numbers agree well with local drop number predictions. This is also confirmed by the direct testing of Hypothesis 2.1, Figure 7, which shows that the hypothesis holds well beyond the region where drop number changes are expected, at least for drop numbers of 1 and 2.

3.2 Drop-by-1 transitions

First, we consider initial conditions for which $B_0$ is close to an equilibrium $E_j$ , and the initial parameter value $\mu _{\mathrm{in}}$ is just above the stability boundary $\mu _{\mathrm{c}}=\mu _{1,\mathrm{c}}$ at which $\lambda _{1}(\mu )$ becomes positive. Starting very close to the stability boundary, we expect the first drop in the winding number to occur while $\lambda _{1}(\mu )$ is the only unstable eigenvalue, and so we study the reduced problem on the associated centre manifold. In this case, no further assumptions are necessary to make precise and rigorous statements about the local drop time.

When $\varepsilon = 0$ and $\mu =\mu _{1,\mathrm{c}}$ , the static problem has a complex one-dimensional, real two-dimensional centre manifold. A parameter-dependent centre manifold reduction [Reference Tuckerman and Barkley57] gives the leading order equation for the complex amplitude $a_1$ near $\mu = \mu _{1,\mathrm{c}}$ ,

(3.11) \begin{align} \dot{a}_1 = \lambda _{1} (\mu ) a_1 + c_1(\mu ) a_1 |a_1|^2, \quad c_1(\mu ) \gt 0, \end{align}

which is the normal form for a subcritical pitchfork bifurcation with rotational symmetry. The cubic coefficient, obtained through a centre-manifold expansion was shown to be positive in [Reference Tuckerman and Barkley57]. Solutions to (2.1) on this centre manifold are then recovered using the eigenvectors $e_1(\mu )$ (2.5) through

(3.12) \begin{align} B(x,t) = B_*(j) + u + \mathrm{i} v,\qquad u + \mathrm{i} v = a_1(t) e_{1}(\mu ) \mathrm{e}^{\mathrm{i} x} + \bar{a}_1(t) \bar{e}_{1}(\mu ) \mathrm{e}^{-\mathrm{i} x} + \mathrm{O}\left (|a_1|^2\right ). \end{align}

Considering these centre manifolds in the system (3.1)–(3.2) for $\varepsilon = 0$ , we find an invariant manifold, of real dimension three, given by the union over $\mu$ of the centre manifolds constructed above. This manifold is normally hyperbolic from the explicit splitting in the linearisation at the equilibria and hence persists for $\varepsilon$ small [Reference Bates, Lu and Zeng4, Reference Henry28], with dynamics given by

(3.13) \begin{align} \dot{a}_1 &= \lambda _{1} (\mu ) a_1 + c_1 (\mu ) a_1 |a_1|^2+\mathrm{O}\left (|a_1|^5\right ), \end{align}

(3.14) \begin{align} \dot{\mu } &= - \varepsilon. \end{align}

As noted above, the flow commutes with the action of translations $a_1\mapsto \mathrm{e}^{\mathrm{i}\varphi }a_1$ and reflections $a_1\mapsto \bar{a}_1$ , so that the real subspace is invariant and dynamics for arbitrary initial conditions are conjugate to the dynamics in the real space by complex rotation. Alternatively, one could also use a time-dependent centre-manifold reduction [Reference Chicone and Latushkin11] after an appropriate modification of the $\mu$ -dynamics outside of the parameter regime considered and arrive at the same reduced equation.

We then consider a boundary-value problem where we prescribe the initial amplitude $a_{1,\mathrm{in}}=\delta \ll 1$ and initial parameter value $\mu _{\mathrm{in}}$ at a time $-T_{\mathrm{in}}$ , and seek time $T_{\mathrm{out}}$ and associated parameter value $\mu _{\mathrm{out}}$ where the amplitude $a_1$ is of size $\delta$ again,

(3.15) \begin{align} a_1({-}T_{\mathrm{in}}) &= a_{1,\mathrm{in}}=\delta,\qquad \mu ({-}T_{\mathrm{in}}) = \mu _{\mathrm{in}} \nonumber \\[5pt] a_1(T_{\mathrm{out}}) &= a_{1,\mathrm{out}} = \delta,\qquad \mu (T_{\mathrm{out}}) = \mu _{\mathrm{out}}. \end{align}

For convenience, we set

(3.16) \begin{equation} \mu _{\mathrm{in}}-\mu _{1,\mathrm{c}} = \varepsilon T_{\mathrm{in}}, \text{ so that } \mu _{1,\mathrm{c}}-\mu _{\mathrm{out}} = \varepsilon T_{\mathrm{out}},\quad \mu (0) =\mu _{1,\mathrm{c}}. \end{equation}

Because $\lambda _{1}(\mu _{\mathrm{in}})$ is stable, we expect that $a_1$ will initially decay exponentially, and then grow once $\mu$ decreases past $\mu _{1,\mathrm{c}}$ , so that the time $T_{\mathrm{out}}$ and thereby the value of $\mu _{\mathrm{out}}$ at which $a_1$ returns to its initial size, $a_1(T_{\mathrm{out}}) = \delta$ , are well defined.

Since we are only considering a time interval in which $a_1$ remains small, as a first approximation we analyse (3.13)–(3.14) by neglecting the nonlinear terms and solving the resulting linear equation explicitly for initial conditions $(a_{1,\mathrm{in}},\mu _{\mathrm{in}})$ , with solution

(3.17) \begin{align} a_1(t) = \exp \left ( \int _{-T_{\mathrm{in}}}^t \lambda _{1}(\mu (\varepsilon s)) \, \mathrm{d} s \right ) a_{1,\mathrm{in}}. \end{align}

Since $\lambda _{1}(\mu _{\mathrm{in}}) \lt 0$ , the integral is initially negative and begins increasing as $\mu$ decreases past $\mu _{1,\mathrm{c}}$ . Then in this linear prediction, the time $T_{\mathrm{out}}$ is the first time after $T_{\mathrm{in}}$ at which the equal area condition

(3.18) \begin{equation} \int _{-T_{\mathrm{in}}}^{T_{\mathrm{out}}} \lambda _{1}(\mu (\varepsilon s)) \, \mathrm{d} s = 0, \qquad \text{ or, equivalently, }\quad \int _{\mu _{\mathrm{in}}}^{\mu _{\mathrm{out}}} \lambda _{1}(\mu ) \, \mathrm{d}\mu = 0, \end{equation}

holds. Note that the latter integral describes precisely the vanishing $\log$ -amplitude $b_1=0$ from Section 3.1.

Lemma 3.1 (Equal-area prediction). The system (3.12) together with the boundary conditions (3.16) has a unique solution for $0\lt \delta, \mu _{1,\mathrm{in}}-\mu _{1,\mathrm{c}} \ll 1$ , sufficiently small, and we have the expansion

(3.19) \begin{equation} \mu _{\mathrm{out}}=\mu _{\mathrm{out}}^0+\mathrm{O}(\varepsilon ), \end{equation}

where $\mu _{\mathrm{out}}^0\lt \mu _{\mathrm{in}}$ is the largest solution to the equal-area condition (3.18). In particular, $\mu _{1,\mathrm{c}}-\mu _{\mathrm{out}}=\mu _{\mathrm{in}}-\mu _{1,\mathrm{c}}+\mathrm{O}(|\delta |+|\varepsilon |+|\mu _{\mathrm{in}}-\mu _{1,\mathrm{c}}|^2)$ .

From Corollary 3.2, we conclude that for $\mu _{\mathrm{in}}$ sufficiently close to $\mu _{1,\mathrm{c}}$ , we will always observe a drop by one and can easily characterise the drop time. Naturally, we wish to see how far this result extends. The key limiting factor in the result is the presence of a leading eigenvalue with $\lambda _{1}(\mu )\gt \lambda (\mu )$ for all other $\lambda (\mu )$ in the spectrum of the linearisation. Technically, centre manifold reductions in the literature also require a uniform splitting, $\lambda _{1}(\mu )\gt \eta _0\gt \lambda (\mu )$ with $\eta _0$ independent of $\mu$ , possibly with sufficiently large gaps to ensure smoothness. We believe that a simple splitting of the leading eigenvalue $\lambda _{1}(\mu )\gt \lambda (\mu )$ for all $\mu \in (\mu _{\mathrm{in}},\mu _{\mathrm{out}})$ should be sufficient to establish Lemma 3.1 but will not attempt to do so here. We observed excellent validity of our predictions even when the uniform gap condition does not hold.

The procedure thus far predicts how and when a first transition will lead to a drop by one in wavenumber or possibly a more severe drop. At the end of this drop, from say $E_j$ to $E_{j-1}$ near $\mu =\mu _{\mathrm{out}}$ , we have followed a global heteroclinic to a neighbourhood of $E_{j-1}$ , where we typically expect that upon entering the neighbourhood, we have a generic perturbation of this stable equilibrium and can now repeat the analysis with the new $\mu _{\mathrm{in}}$ given by the previous $\mu _{\mathrm{out}}$ . A short calculation using our main prediction (3.10) and expressions for eigenvalues in (2.5) shows that for $j\gg 1$ , the distance between $\mu _{1,\mathrm{c}}$ for modes $j$ and $j+1$ is $6j+1$ , while the distance between $\mu _{1,\mathrm{c}}$ and $\mu _{2,\mathrm{c}}$ is only $\frac{3}{2}$ . Hence, one expects that, for large $j$ , a drop-by-1 is followed by a large distance to criticality in parameter space and a potentially subsequent higher drop; compare also Figure 3. Therefore, we now analyse such higher drops, when more than one eigenvalue is unstable at the predicted drop time.

3.3 The drop-by-2 transition

We first consider the static problem with $\mu _{3,\mathrm{c}} \lt \mu \lt \mu _{\mathrm{2,c}}$ , so that the equilibrium $E_j$ has two unstable eigenvalues $\lambda _{1}(\mu )$ and $\lambda _{2}(\mu )$ . There is an associated complex two-dimensional, real four-dimensional centre-unstable manifold. The dynamics on this manifold are governed by equations for the complex amplitudes $a_1, a_2$ in the associated eigenspaces,

(3.20) \begin{align} \dot{a}_1 &= \lambda _{1}(\mu ) a_1 + c_{1,2} (\mu ) \bar{a}_1 a_2 +\mathrm{O}(|a_1|(|a_1|^2+|a_2|^2)),\nonumber \\[5pt] \dot{a}_2 &= \lambda _{2} (\mu ) a_2 + c_{1^2} (\mu ) a_1^2 +\mathrm{O}(|a_1|(|a_1|^2+|a_2|^2)), \end{align}

with coefficients $c_{1,2}(\mu ), c_{1^2}(\mu ) \in \mathbb{C}$ that can be readily computed by evaluating the nonlinearity on the associated eigenspace and projecting. Again, the vector field commutes with the isotropy of the equilibrium, that is, translations $(a_1,a_2)\mapsto (\mathrm{e}^{\mathrm{i}\varphi } a_1,\mathrm{e}^{2\mathrm{i}\varphi } a_2),$ and reflections $(a_1,a_2)\mapsto (\bar{a}_1,\bar{a}_2)$ . As consequence, $c_{1^2}$ and $c_{1,2}$ are real and the real subspace $(a_1,a_2)\in \mathbb{R}^2$ is invariant. Note that the nonlinear terms correspond to the spatial resonances referred to in (3.4): the $a_j$ are amplitudes of Fourier modes $\mathrm{e}^{\mathrm{i} j x}$ , and the complex rotation is induced by the spatial shift $x\mapsto x+\varphi$ in the full equation. The associated manifold is normally hyperbolic in parameter regimes where $\lambda _{1}(\mu )$ and $\lambda _{2}(\mu )$ are separated from other eigenvalues; its smoothness (making in particular the cubic terms meaningful) is determined by spectral gaps; see for instance [Reference Henry28, Reference Hirsch, Pugh and Shub30]. These gaps can be easily evaluated in the large- $j$ limit (2.5) where $|\lambda _{2}(\mu )|/|\lambda _{3}(\mu )|\gt 2$ for all $\hat{\mu }\gt 0$ , providing the desired smoothness of the manifold and validity of expansions.

For $\varepsilon \neq 0$ , this manifold persists as a slow manifold with dynamics at quadratic order given by

(3.21) \begin{align} \dot{a}_1 &= \lambda _{1}(\mu ) a_1 + c_{1,2} (\mu ) \bar{a}_1 a_2, \end{align}

(3.22) \begin{align} \dot{a}_2 &= \lambda _{2} (\mu ) a_2 + c_{1^2} (\mu ) a_1^2, \end{align}

(3.23) \begin{align} \dot{\mu } &= - \varepsilon. \end{align}

To predict drop times and drop numbers, we will try to determine when $(a_1,a_2)$ leave a small neighbourhood of the origin, and whether $|a_1|\gg |a_2|$ or $|a_2|\gg |a_1|$ at that time, thus concluding that the local drop number is $1$ or $2$ , respectively. Hypothesis 2.1 then allows us to conclude global drop numbers. We therefore consider the system (3.21)–(3.23) with initial conditions $a_1({-}T_{\mathrm{in}}) = a_2 ({-}T_{\mathrm{in}}) = \delta$ , $\mu ({-}T_{\mathrm{in}}) = \mu _{\mathrm{in}}$ , with $\mu _{\mathrm{in}} \gt \mu _{1,\mathrm{c}} \gt \mu _{2,\mathrm{c}}$ so that both eigenvalues are initially stable. As we shall quickly see, the initial value of $a_2$ is irrelevant so that this particular choice is not restrictive. The solution will therefore initially contract, and we look for the first time $T_{\mathrm{out}}$ at which

(3.24) \begin{align} \max \left \{ | a_1(T_{\mathrm{out}} )|, |a_2 (T_{\mathrm{out}} )| \right \} = \delta. \end{align}

Asymptotics from variation-of-constant formula. We neglect higher order terms and study (3.21)–(3.23) setting $c_{1,2}\equiv 0$ , noticing that $a_2\ll 1$ so that $|c_{1,2}\bar{a_1}a_2|\ll |a_1|$ ; see also (3.35) and the discussion there. We also use $\mu$ as an equivalent time variable and find amplitudes

(3.25) \begin{align} a_1(\mu )&=\mathrm{e}^{\frac{1}{\varepsilon }\int _{\mu }^{\mu _{\mathrm{in}}}\lambda _1(\nu )\mathrm{d}\nu }\delta,\nonumber \\[5pt] a_2(\mu )&=a_2^0(\mu )+a_2^{1^2}(\mu ),\nonumber \\[5pt] a_2^0(\mu )&=\mathrm{e}^{\frac{1}{\varepsilon }\int _{\mu }^{\mu _{\mathrm{in}}}\lambda _2(\nu )\mathrm{d}\nu }\delta,\nonumber \\[5pt] a_2^{1^2}(\mu )&=\int _{\mu }^{\mu _{\mathrm{in}}}c_{1^2}(\nu )\mathrm{e}^{\frac{1}{\varepsilon }\Lambda (\nu )}\mathrm{d}\nu \,\delta ^2,\nonumber \\[5pt] \Lambda (\nu ;\,\mu )&=\int _\mu ^\nu \lambda _2(\rho )\mathrm{d}\rho + \int _\nu ^{\mu _{\mathrm{in}}}2\lambda _1(\rho )\mathrm{d}\rho. \end{align}

Here, we suppress the dependence on $\mu _{\mathrm{in}}$ in $\Lambda (\nu ;\,\mu )$ and notice that $\Lambda (\nu ;\,\mu )$ is smooth and maximal when $\nu =\mu _{1^2:2}$ so that we can expand the integrand near $\nu =\mu _{1^2:2}$ to find

(3.26) \begin{align} a_2^{1^2}(\mu )&=c_{1^2}\mathrm{e}^{\frac{1}{\varepsilon }\Lambda }\delta ^2 \int _{\mu }^{\mu _{\mathrm{in}}}\mathrm{e}^{\frac{1}{2\varepsilon } \Lambda _{\nu \nu }(\nu -\mu _{1^2:2})^2}\mathrm{d}\nu (1+\mathrm{O}(\sqrt \varepsilon ))\nonumber \\[5pt] &=c_{1^2}\mathrm{e}^{\frac{1}{\varepsilon }\Lambda }\delta ^2\sqrt{\frac{-2\varepsilon }{\Lambda _{\nu \nu }}}\mathrm{err}\,\left (\frac{\mu _{1^2:2}-\mu }{\sqrt{\varepsilon }}\right )\left (1+\mathrm{O}(\sqrt \varepsilon )\right ), \end{align}

where $\Lambda _{\nu \nu }=(\lambda _2-2\lambda _1)'\lt 0$ is the derivative in $\mu$ of the resonance condition, and $c_{1^2},\Lambda$ , and $\Lambda _{\nu \nu }$ are evaluated at $\nu =\mu _{1^2:2}$ , and $\mathrm{err}\,(z)=\int _{-\infty }^z \exp ({-}z^2)\mathrm{d} z$ .

First, one sees that $a_2^0\ll a_2^{1^2}$ . In order to identify the largest amplitude upon exiting a neighbourhood, thus determining if the local drop number is 1 or 2, we therefore need to compare $a_1$ and $a_2\sim a_2^{1^2}$ . The transition from a local drop number of 1 to a drop number 2 occurs when

(3.27) \begin{equation} a_2^{1^2}=\delta,\quad \text{and }\quad a_1=\delta. \end{equation}

Solving these two equations for $\mu _{\mathrm{in}}$ and $\mu_\mathrm{out}$ then identifies the transition parameter values $\mu _{\mathrm{in},1\to 2}$ and $\mu _{\mathrm{out},1\to 2}$ . In order to solve (3.27), we use the expression for $a_1$ to find $\int _{\mu _{\mathrm{out},1\to 2}}^{\mu _{\mathrm{in}}}\lambda _1(\nu )\mathrm{d}\nu =0$ . Substituting this result into the expression for $\Lambda$ in (3.25), we obtain

(3.28) \begin{equation} \Lambda (\mu _{1^2:2},\mu _{\mathrm{out},1\to 2})=\int _{\mu _{\mathrm{out},1\to 2}}^{\mu _{\mathrm{1^2:2}}}(\lambda _2(\rho )-2\lambda _1(\rho ))\mathrm{d}\rho. \end{equation}

We next assume that $\Delta \mu =\mu _{1^2:2}-\mu _{\mathrm{out},1\to 2}\gt 0$ is small so that we can Taylor expand

\begin{equation*} \Lambda (\mu _{1^2:2},\mu _{\mathrm {out},1\to 2})=-\frac {1}{2}\Lambda _{\nu \nu }(\mu _{1^2:2},\mu _{1^2:2})(\Delta \mu )^2+\mathrm {O}((\Delta \mu )^3).\end{equation*}

Inserting this into (3.26) and partially solving for $\Delta \mu$ , we find

(3.29) \begin{equation} \Delta \mu =\sqrt{\frac{2\varepsilon }{\Lambda _{\nu \nu }}\log \left ( c_{1^2}\delta \sqrt{-2\varepsilon/(\Lambda _{\nu \nu })} \mathrm{err}\,(\Delta \mu/\sqrt{\varepsilon })\right )}. \end{equation}

Assuming that at leading order $\Delta \mu \sim \sqrt{-\varepsilon \log \varepsilon }$ , we find $\mathrm{err}\,(\Delta \mu/\sqrt{\varepsilon })\to \sqrt{\pi }$ , so that, consistently, at leading order

(3.30) \begin{equation} \Delta \mu =\sqrt{\frac{2\varepsilon }{\Lambda _{\nu \nu }}\log \left ( c_{1^2}\delta \sqrt{-2 \pi \varepsilon/(\Lambda _{\nu \nu })}\right )}. \end{equation}

Away from the transition, the amplitude of $a_2(\mu )$ is given at leading order through

(3.31) \begin{equation} a_2(\mu )=\left \{\begin{array}{ll} \mathrm{O}(\varepsilon a_1^2(\mu )), & \mu \gt \mu _{1^2:2},\\[5pt] \exp \left ({\frac{1}{\varepsilon } \int _{\mu }^{\mu _{1^2:2}}\lambda _2(\rho )\mathrm{d}\rho +\frac{1}{\varepsilon } \int _{\mu _{1^2:2}}^{\mu _{\mathrm{in}}}2\lambda _1(\rho )\mathrm{d}\rho }\right ),& \mu \lt \mu _{1^2:2}, \end{array} \right. \end{equation}

or, neglecting $\log \varepsilon$ -terms,

(3.32) \begin{equation} b_2(\mu )=\varepsilon \log (a_2(\mu ))=\int _\mu ^{\mu _{\mathrm{in}}} \Lambda _{\mathrm{max}}(\rho )\mathrm{d}\rho, \qquad \Lambda _{\mathrm{max}}(\rho )=\mathrm{max}\,\{\lambda _2(\rho ),2\lambda _1(\rho )\}=\left \{\begin{array}{ll}2\lambda _1(\rho ),& \rho \gt \mu _{1^2:2},\\[5pt] \lambda _2(\rho ),& \rho \lt \mu _{1^2:2}. \end{array}\right. \end{equation}

For $\mu _{\mathrm{out}}\lt \mu _{\mathrm{out},1\to 2}$ , at leading order the local drop parameter value is given by setting $a_2^{1^2}(\mu )=\delta$ in (3.25), which gives

(3.33) \begin{equation} \int _{\mu _{\mathrm{out}}}^{\mu _{1^2:2}}\lambda _2(\rho )\mathrm{d}\rho + \int _{\mu _{1^2:2}}^{\mu _{\mathrm{in}}}2\lambda _1(\rho )\mathrm{d}\rho \stackrel{!}{=}0, \end{equation}

in agreement with our predictions in Section 3.1 and with direct simulations illustrated in Figure 8.

Figure 8.

Numerical comparisons for the drop-by-1 to drop-by-2 transition. Top: local and global drop numbers depending on $\mu _{\mathrm{in}}$ (left) and $\mu _{\mathrm{out}}$ (right) for several values of $\varepsilon$ ; predicted drop as solid black curve. Bottom: $\mu _{\mathrm{out}}$ versus $\mu _{\mathrm{in}}$ for several values of $\varepsilon$ together with prediction for $\varepsilon =0$ (solid black) (left); drop values $\mu _{\mathrm{in}}$ both local and global as observed in top row shown here as a function of $\varepsilon$ , with linear fit and predicted values of drops as green dot. Extrapolated values at $\varepsilon =0$ from linear fits fall within 10% of the predicted value for local drops and within 2% for global drops (right). Throughout, $\mu =\mu _{\mathrm{c}}+\hat{\mu }$ indicates values relative to criticality.

Figure 9.

Top: dynamics of centre manifold equations (3.20) with instability thresholds for $a_1$ and $a_2$ and $\lambda _{1/2}=0$ , respectively, and subsequent 1:1 and 1:2 resonances; $\mu$ decreases to the left, the vertical plane $a_1=0$ is invariant. Sample dark and bright red trajectories illustrating trajectories exiting in the $a_1$ and $a_2$ directions, respectively. Bottom: dynamics in projective coordinates (3.35) with invariant planes $a_2=0$ and $\xi =a_1^2/a_2=0$ ; trajectories follow the stable branch in a transcritical bifurcation in the horizontal plane with unstable manifold in purple and blue for nontrivial and trivial branch, respectively; trajectories exit in the $a_2$ direction along the unstable manifold, albeit at locations where $a_1^2/a_2\neq 0$ , so $|a_1|\sim \sqrt{|a_2|}\gg |a_2|$ (bright red), or when $a_1^2/a_2\sim 0$ and $|a_2|\gg |a_1|$ (dark red).

Rigorous asymptotics and a geometric view on resonances. We next present a geometric view that also explains why higher order terms are irrelevant. Consider the new variable $\xi =a_1^2/a_2$ , together with $a_2$ , and find the new equations

(3.34) \begin{align} \xi '=&(2\lambda _{1}(\mu )-\lambda _{2}(\mu ))\xi -c_{1^2}(\mu )\xi ^2 +\mathrm{O}(|a_2|)\nonumber,\\[5pt] a_2'=&\lambda _{2}(\mu )a_2+c_{1^2}(\mu )\xi a_2+\mathrm{O}(|a_2|^2), \end{align}

(3.35) \begin{align} \mu '=&\varepsilon. \end{align}

In the invariant plane $a_2=0$ , we find a slow passage through a transcritical bifurcation at the resonance, as $2\lambda _{1}(\mu )-\lambda _{2}(\mu )$ passes through zero. Before the resonance, $2\lambda _{1}(\mu )-\lambda _{2}(\mu )\gt 0$ , and the nontrivial equilibrium $\xi _*=(2\lambda _{1}(\mu )-\lambda _{2}(\mu ))/c_{1^2}$ is exponentially attracting; past the resonance, trajectories follow the now stable trivial equilibrium $\xi =0$ . Normal to this stable family of equilibria, trajectories first decay, then grow in the $a_2$ -direction with the eigenvalue $\lambda _{2}+c_{1^2}\xi _*=2\lambda _{1}$ ; after the passage through the transcritical bifurcation growth is governed by the normal eigenvalue $\lambda _{2}$ , thus reproducing the integral criterion (3.33). A refined desingularisation, analysing for instance the slow passage through the transcritical bifurcation using geometric blowup as in [Reference Krupa and Szmolyan39] should then lead to a leading-order expansion of the form obtained through the direct calculations outlined above; see Figure 9 for an illustration.

3.4 The drop-by-3 transition

Turning to predictions for higher drops, we need to account for more unstable eigenvalues and therefore higher resonances. We first adapt the asymptotic calculation for the 1:2-resonance criteria above to the drop-by-3 transition.

Once $\lambda _{3}$ becomes positive, we track the six-dimensional unstable manifold associated to the first three complex unstable modes in the static problem. Keeping track of what interactions of modes $\mathrm{e}^{\pm 3ix}, \mathrm{e}^{\pm 2 ix}, \mathrm{e}^{\pm ix}$ can produce the original modes $\mathrm{e}^{ix}, \mathrm{e}^{2ix}, \mathrm{e}^{3ix},$ we see that to leading order, the dynamics on this unstable manifold are given by

(3.36) \begin{align} \dot{a}_1 &= \lambda _{1}(\mu ) a_1 + c_{-1, 2} \bar{a}_1 a_2 + c_{3,-2} a_3 \bar{a}_2, \end{align}

(3.37) \begin{align} \dot{a}_2 &= \lambda _{2}(\mu ) a_2 + c_{1^2} a_1^2 + c_{3, -1} a_3 \bar{a}_1, \end{align}

(3.38) \begin{align} \dot{a}_3 &= \lambda _{3} (\mu ) a_3 + c_{1^3} a_1^3 + c_{1,2} a_1 a_2, \end{align}

(3.39) \begin{align} \dot{\mu } &= - \varepsilon, \end{align}

also taking into account the slow evolution of $\mu$ . Error terms can be seen to be small either heuristically or in the analogue of the geometric desingularisation shown in Figure 9; see the discussion below. We again consider this system with initial conditions $a_1 ({-}{T_{\mathrm{in}}}) = a_2 ({-}{T_{\mathrm{in}}}) = a_3 ({-}{T_{\mathrm{in}}}) = \delta$ , $\mu ({-}T_{\mathrm{in}})=\mu _{\mathrm{in}}$ , and look for the first time $T_{\mathrm{out}}$ for which $|a_j({T_{\mathrm{out}}})| = \delta$ for some $j = 1,2$ or $3$ .

The coefficients of the nonlinear terms depend on $\mu$ and hence are also slowly varying, but since the values of these coefficients will not affect our predictions at leading order, we suppress this dependence. Also note that we have included the cubic term $c_{1^3}a_1^3$ in the equation for $a_3$ , since the analysis of Section 3.3 suggests that for some time we have $a_2 \sim a_1^2$ , and so $a_1^3$ terms are in fact on the same order as $a_1 a_2$ for some relevant time. Following a similar line of reasoning, we see as in Section 3.3 that the back-coupling terms involving $a_2$ and $a_3$ in the equation for $a_1$ , and $a_3$ in the equation for $a_2$ , are higher order. We, therefore, neglect these terms and arrive at the system

(3.40) \begin{align} \dot{a}_1 &= \lambda _{1}(\mu ) a_1, \end{align}

(3.41) \begin{align} \dot{a}_2 &= \lambda _{2}(\mu ) a_2 + c_{1^2} a_1^2, \end{align}

(3.42) \begin{align} \dot{a}_3 &= \lambda _{3} (\mu ) a_3 + c_{1^3} a_1^3 + c_{1,2} a_1 a_2, \end{align}

(3.43) \begin{align} \dot{\mu } &= - \varepsilon. \end{align}

We can now solve this system recursively, first solving (3.40) for $a_1$ , then solving for $a_2$ via its variation of constants formula, and then inserting both expressions for $a_1$ and $a_2$ into (3.42) and solving via the variation of constants formula. We shall use the expressions for $a_1$ and $a_2$ derived above and are left with the equation for $a_3$ , which we write as a sum of the solution to the homogeneous equation $a_3^0$ , the solution $a_3^{1^3}$ with inhomogeneity $c_{1^3}a_1^3$ , and the solution $a_3^{1,2}$ with inhomogeneity $c_{1,2}a_1a_2$ . We find

(3.44) \begin{align} a_3^0(\mu )&=\mathrm{e}^{\frac{1}{\varepsilon }\int _{\mu }^{\mu _{\mathrm{in}}}\lambda _3(\nu )\mathrm{d}\nu }\delta,\nonumber \\[5pt] a_3^{1^3}(\mu )&=\int _{\mu }^{\mu _{\mathrm{in}}}c_{1^3}(\mu )\mathrm{e}^{\frac{1}{\varepsilon }\Lambda _{{1^3}}(\nu )}\mathrm{d}\nu \delta ^3,\nonumber \\[5pt] \Lambda _{1^3}(\nu )&=\int _\mu ^\nu \lambda _3(\rho )\mathrm{d}\rho + \int _\nu ^{\mu _{\mathrm{in}}}3\lambda _1(\rho )\nonumber \\[5pt] a_3^{1,2}(\mu )&=\int _{\mu }^{\mu _{\mathrm{in}}}c_{1,2}(\mu )\mathrm{e}^{\frac{1}{\varepsilon }\Lambda _{{1,2}}(\nu )}\mathrm{d}\nu \delta ^3,\nonumber \\[5pt] \Lambda _{1,2}(\nu )&= \int _\mu ^\nu \lambda _3(\rho )\mathrm{d}\rho + \int _\nu ^{\mu _{1^2,2}}(\lambda _1(\rho )+\lambda _2(\rho ))\mathrm{d}\rho + \int _{\mu _{1^2,2}}^{\mu _{\mathrm{in}}}3\lambda _1(\rho )\mathrm{d}\rho. \end{align}

Following the same strategy as in the analysis for near the $1^2$ :2-resonance, we evaluate the integrals to leading order near the maximum of the exponential, finding

(3.45) \begin{align} b_3^{1^3}&=\varepsilon \log |a_3^{1^3}|=\Lambda _{1^3}(\mu _{1^3:3})+\mathrm{O}(\varepsilon ^{1^-}), \end{align}

(3.46) \begin{align} b_3^{1,2}&=\varepsilon \log |a_3^{1,2}|=\Lambda _{1,2}(\mu _{1,2:3})+\mathrm{O}(\varepsilon ^{1^-}), \end{align}

where we used the short hand $1^-$ to denote any exponent less than 1, accounting for potential terms of the form $\varepsilon \log \varepsilon$ . Clearly, $b_3^{1,2} \gt b_3^{1^3} \gt b_3^0$ , so that the amplitude of $a_3$ is at leading order given through $\exp (b_3^{1,2}/\varepsilon )$ . The crossover occurs when $a_3\sim a_2$ , that is, at leading order, when

(3.47) \begin{align} &\int _{\mu _{\mathrm{out},2\to 3}}^{\mu _{1,2:3}} \lambda _3(\rho )\mathrm{d}\rho + \int _{\mu _{1,2:3}}^{\mu _{1^2:2}}(\lambda _1(\rho )+\lambda _2(\rho ))\mathrm{d}\rho + \int _{\mu _{1^2:2}}^{\mu _{\mathrm{in},2\to 3}}3\lambda _1(\rho )\mathrm{d}\rho \end{align}

(3.48) \begin{align} \stackrel{!}{=} &\int _{\mu _{\mathrm{out},2\to 3}}^{\mu _{1^2:2}} \lambda _2(\rho )\mathrm{d}\rho + \int _{\mu _{1^2,2}}^{\mu _{\mathrm{in},2\to 3}}2\lambda _1(\rho )\mathrm{d}\rho + \mathrm{O}(\varepsilon ^{1^-}) \end{align}

(3.49) \begin{align} \stackrel{!}{=} &\, \mathrm{O}(\varepsilon ^{1^-}). \end{align}

The two equalities determine $\mu _{\mathrm{out},\,2\to 3}$ and $\mu _{\mathrm{in},\,2\to 3}$ . Linearising at a solution with respect to these two variables, we find a Jacobi matrix with determinant

\begin{equation*} \left |\begin {array}{cc} -\lambda _3(\mu _{\mathrm {out}})+\lambda _2(\mu _{\mathrm {out}})& 3\lambda _1(\mu _{\mathrm {in}})\\[5pt] -\lambda _2(\mu _{\mathrm {out}})& 2\lambda _1(\mu _{\mathrm {in}}) \end {array}\right |=\lambda _1(\mu _{\mathrm {in}})\left ({-}2 \lambda _3(\mu _{\mathrm {out}})+ 5 \lambda _2(\mu _{\mathrm {out}})\right ). \end{equation*}

Assuming the additional non-resonance condition $2\lambda _3(\mu _{\mathrm{out}})\neq 5 \lambda _2(\mu _{\mathrm{out}})$ , we therefore expect to be able to solve for the variables $\mu _{\mathrm{in}}$ and $\mu _{\mathrm{out}}$ with the implicit function theorem with corrections to the transition value as $\mathrm{O}(\varepsilon ^{1^-})$ , rather than $\mathrm{O}(\sqrt{\varepsilon })$ for the 1-2-transition. The results agree with our summary in Section 3.1 and with numerical simulations shown in Figure 10.

The geometric picture for the drop-by-3 transition. We can follow the strategy for the analysis of the drop-by-2 transition and introduce projective variables that encode the quotients of amplitudes and resonances. For instance, set $ \xi _2=a_1^2/a_2, \xi _3=a_1a_2/a_3,$ writing for short $\lambda _j=\lambda _{j}(\mu )$ , suppressing $\mu$ -dependence in the nonlinear terms, and suppressing higher-order terms in $a_1$ , to find

\begin{align*} a_1'&=\lambda _1 a_1,\\[5pt] \xi _2'&=(2\lambda _1-\lambda _2)\xi _2 -c_{1^2:2} \xi _2^2,\\[5pt] \xi _3'&=(\lambda _1+\lambda _2-\lambda _3)\xi _3 - c_{1,2:3}\xi _3^2 - c_{1^3:3}\xi _3^2\xi _2. \end{align*}

We previously analysed the dynamics in the $(a_2,\xi _2)$ -subsystem. In the regime $\mu \gt \mu _{1^2:2}$ , $\xi _3$ follows the nontrivial stable branch with $\xi _3=(\lambda _1+\lambda _2-\lambda _3)/(c_{1,2:3} - c_{1^3:3}\xi _2)$ . For $\mu _{1,2:3}\lt \mu \lt \mu _{1^2:2}$ , $\xi _2\sim 0$ and $\xi _3=(\lambda _1+\lambda _2-\lambda _3)/c_{1,2:3}$ become the nontrivial stable branch, which at $\mu _{1,2:3}$ exchanges stability with the trivial branch $\xi _3=0$ in a transcritical bifurcation. In particular, $\xi _3\sim 0$ for $\mu \lt \mu _{1,2:3}$ . To reproduce the explicit computations above, we recover growth by analysing exponential growth in the normal direction of $a_1$ , and of $a_{2/3}$ through the values of $\xi _{2,3}$ .

Figure 10.

Numerical comparisons for the drop-by-2 to drop-by-3 transition. Top: local and global drop numbers depending on $\mu _{\mathrm{in}}$ (left) and $\mu _{\mathrm{out}}$ (right) for several values of $\varepsilon$ ; predicted drop as solid black curve. Bottom: $\mu _{\mathrm{out}}$ versus $\mu _{\mathrm{in}}$ for several values of $\varepsilon$ together with prediction for $\varepsilon =0$ (solid black) (left); drop values $\mu _{\mathrm{in}}$ both local and global as observed in top row shown here as a function of $\varepsilon$ , with linear fit and predicted values of drops as green dot. Extrapolated values at $\varepsilon =0$ from linear fits fall within 1% of the predicted value (right). Throughout, $\mu =\mu _{\mathrm{c}}+\hat{\mu }$ indicates values relative to criticality.

3.5 Drop-by-4 and beyond

Predicting higher drops appears to be cumbersome in general. We outline here the calculations for the transition from a drop-by-3 to a drop-by-4. The relevant amplitude equations are

(3.50) \begin{align} \dot{a}_1 &= \lambda _{1}(\mu ) a_1, \end{align}

(3.51) \begin{align} \dot{a}_2 &= \lambda _{2}(\mu ) a_2 + c_{1^2} a_1^2, \end{align}

(3.52) \begin{align} \dot{a}_3 &= \lambda _{3} (\mu ) a_3 + c_{1,2} a_1 a_2+ c_{1^3} a_1^3, \end{align}

(3.53) \begin{align} \dot{a}_4 &= \lambda _{4} (\mu ) a_4 + c_{1,3} a_1 a_3+ c_{2,2} a_2^2 + +c_{1^2,2}a_1^2a_2+ c_{1^4} a_1^4, \end{align}

(3.54) \begin{align} \dot{\mu } &= - \varepsilon. \end{align}

Higher modes will be irrelevant until growth dominates the source terms from resonant interactions. In order to determine the onset of growth in the mode $a_4$ , one would inspect the lowest order source terms, $a_2^2$ and $a_1a_3$ , and find the associated resonances, $\mu _{2^2:4}$ and $\mu _{1,3:4}$ , which give a lower bound for the crossover to a drop-by-4.

It is convenient to track approximations to the logarithms of the amplitudes, $b_k=\varepsilon \log a_k$ , so that for instance

(3.55) \begin{align} b_1(\mu )&=\int _\mu ^{\mu _{\mathrm{in}}} \lambda _1(\mu )\mathrm{d}\mu,\nonumber \\[5pt] b_2(\mu )&=\int _\mu ^{\mu _{1^2:2}} \lambda _2(\mu )\mathrm{d}\mu +2b_1(\mu _{1^2:2}),\nonumber \\[5pt] b_3(\mu )&=\int _\mu ^{\mu _{1,2:3}} \lambda _3(\mu )\mathrm{d}\mu +b_1(\mu _{1,2:3}) +b_2(\mu _{1,2:3}). \end{align}

For $b_4$ , we need to consider all resonant terms in the equation for $a_4$ . At leading order, we find that $b_4$ is given as a maximum of the expressions obtained by treating all terms separately,

\begin{align*} b_4^{1,3}(\mu )&=\int _\mu ^{\mu _{1,3:4}} \lambda _4(\mu )\mathrm{d}\mu +b_1(\mu _{1,3:4})+b_3(\mu _{1,3:4}),\\[5pt] b_4^{2,2}(\mu )&=\int _\mu ^{\mu _{1,3:4}} \lambda _4(\mu )\mathrm{d}\mu +2b_2(\mu _{1,3:4}),\\[5pt] \ldots & \end{align*}

and similar expressions for cubic and quartic resonant terms. At any given $\mu \lt \mu _{1,3:4}$ , the amplitude is given by the maximum $b_4=\max _m b_j^m$ where $m$ runs through all resonant source terms.

It turns out that $b_4^{2,2}$ maximises the amplitudes, and we can find the value of the transition from a drop-by-3 to a drop-by-4 by solving $b^4_{2,2}(\mu _{\mathrm{out}})=0$ together with $b^3(\mu _{\mathrm{out}})=0$ for $\mu _{\mathrm{out}}$ and $\mu _{\mathrm{in}}$ . For finite $j$ , this equation can easily be solved numerically, and the predicted drop transitions compare well with numerical simulations as shown in Figure 11.

Generalising to higher transitions appears straightforward in principle but cumbersome. One obtains successively $\log$ -amplitudes $b_1,b_2,b_3,b_4,\ldots,b_\ell,\ldots$ for a given $\mu _{\mathrm{in}}$ as functions of $\mu$ , maximising at each $\ell$ and each $\mu$ over all potential resonant source terms. Solving for $b_\ell (\mu )=0$ as a function of $\mu$ given $\mu _{\mathrm{in}}$ then yields the local drop parameter value $\mu _{\mathrm{out}}$ . The argmax of all the exit values $\mu _{\mathrm{out}}$ then determines the local drop number for a given $\mu _{\mathrm{in}}$ . We caution, however, that at some point the skew product structure may not be preserved since, for instance, terms of the form $a_3\bar{a_2}$ in the equation for $a_1'$ may dominate the linear term $\lambda _1a_1$ .

Figure 11.

Drop-to numbers for initial patterns with $j=3,\ldots,8$ varying $\mu _{\mathrm{in}}$ , plotted against $\mu _{\mathrm{in}}-\mu _{\mathrm{c}}$ (top) and $\mu _{\mathrm{out}}-\mu _{\mathrm{c}}$ (bottom). Shown are both local (blue) and global drops (yellow), which always yield an equal drop or a drop by one less than the local drop. The actual drop-to number is the integer part of the plotted value (which is slightly shifted to improve readability). Red markers indicate the theoretical prediction for drops.

3.6 Drop number transitions for large $\text{j}$

In the limit of large $j$ , the criteria for transitions can be evaluated explicitly. Using the expansion for eigenvalues (2.5), we can find the resonances as defined in (3.3) explicitly. Setting $\mu =3j^2-\frac{1}{2}+\hat{\mu }$ , we have in decreasing order,

\begin{equation*} \hat {\mu }_{1^2:2}=-3, \quad \hat {\mu }_{1^3:3}=-6, \quad \hat {\mu }_{1,2:3}=-\frac {15}{2}, \quad \hat {\mu }_{1,3:4}=-14, \quad \hat {\mu }_{1,4:5}=-\frac {45}{2}, \end{equation*}

and

\begin{equation*} \hat {\mu }_{1^4:4}=-10, \quad \hat {\mu }_{1^2,2:4}=-\frac {114}{10}, \quad \hat {\mu }_{2^2:4}=-\frac {27}{2}, \quad \hat {\mu }_{1,3:4}=-14, \ldots \end{equation*}

Evaluating the log-amplitudes in (3.5)–(3.8) explicitly for eigenvalues as in (2.5), we also find explicit crossover points $\hat{\mu }_{\mathrm{in},1\to 2}$ , $\hat{\mu }_{\mathrm{out},1\to 2}$ , for drop-by-1 to drop-by-2 and $\hat{\mu }_{\mathrm{in},2\to 3}$ , $\hat{\mu }_{\mathrm{out},2\to 3}$ for drop-by-2 to drop-by-3 as

(3.56) \begin{equation} \hat{\mu }_{\mathrm{in},1\to 2}=3,\quad \hat{\mu }_{\mathrm{out},1\to 2}=-3, \qquad \qquad \hat{\mu }_{\mathrm{in},2\to 3}=15,\quad \hat{\mu }_{\mathrm{out},2\to 3}=-12. \end{equation}

Note that the change to a drop-by-3 occurs well before the amplitude $a_4$ becomes relevant.

The drop values $ \hat{\mu }_{\mathrm{out},\ell }$ for a drop by $\ell$ as a function of $\hat{\mu }_{\mathrm{in}}$ are given by

(3.57) \begin{align} \hat{\mu }_{\mathrm{out},1}&=-\hat{\mu }_{\mathrm{in}}, &\qquad 0\lt \hat{\mu }_{\mathrm{in}}\lt \hat{\mu }_{\mathrm{in},1\to 2}, \end{align}

(3.58) \begin{align} \hat{\mu }_{\mathrm{out},2}&=\frac{1}{2} \left ({-}3 - \sqrt{-9 + 2 \hat{\mu }_{\mathrm{in}}^2}\right ), &\qquad \hat{\mu }_{\mathrm{in},1\to 2}\lt \hat{\mu }_{\mathrm{in}}\lt \hat{\mu }_{\mathrm{in},2\to 3}, \end{align}

(3.59) \begin{align} \hat{\mu }_{\mathrm{out},3}&=\frac{1}{6} \left ({-}24 - 2\sqrt{3} \sqrt{-33+ \hat{\mu }_{\mathrm{in}}^2}\right ), &\qquad \hat{\mu }_{\mathrm{in},2\to 3 }\lt \hat{\mu }_{\mathrm{in}}\lt \hat{\mu }_{\mathrm{in},3\to 4 }. \end{align}

For the drop-by-3 to drop-by-4 transition, we find, based on the source term $a_2^2$ and the associated 2,2:4-resonance,

(3.60) \begin{align} \hat{\mu }_{\mathrm{in},3\to 4}&=3 \sqrt{\frac{1}{2} \left (159+28 \sqrt{14}\right )}\nonumber \\[5pt] &=34.4521\ldots,\nonumber \\[5pt] \hat{\mu }_{\mathrm{out},3\to 4}&=\frac{1}{4} \left ({-}30-\sqrt{2 \left (9 \left (159+28 \sqrt{14}\right )-297\right )}\right )\nonumber \\[5pt] &=-23.6125\ldots. \end{align}

The transition associated with the resonant term $a_1a_3$ is located at

\begin{align*} \hat{\mu }^{13}_{\mathrm{in},3\to 4}&=\frac{1}{2} \sqrt{21 \left (147+8 \sqrt{201}\right )}\\[5pt] &=36.9757\ldots,\\[5pt] \hat{\mu }^{13}_{\mathrm{out},3\to 4}&=\frac{1}{4} \left ({-}30-\sqrt{21 \left (147+8 \sqrt{201}\right )-519}\right )\\[5pt] &=-25.0887\ldots, \end{align*}

a larger value of $\mu _{\mathrm{in}}$ . The drop values for finite $j$ have significant corrections, in particular for larger drop numbers or when the drop parameter value is close to the existence boundary.