What limits the oscillations’ amplitude in the single-branch pulsating heat pipe

Abstract

In this paper, we show that the oscillations’ amplitude increases and then saturates during the startup in the single-branch pulsating heat pipe (SBPHP) due to the interplay between an instability mechanism and a limiting mechanism produced by nonlinearities. The SBPHP is a small tube closed at one end (which is heated) and open at the other, in which sits a single vapor bubble followed by a liquid plug. Increasing the heater temperature above a critical value leads to oscillations of the liquid plug. The oscillations’ amplitude initially increases over time and eventually saturates (the system reaches a steady state). To describe this dynamics, we start from a known theoretical SBPHP model. The model is derived from the momentum balance on the liquid plug which includes both the pressure difference between the vapor and the environment and the viscous friction between the liquid and the wall. Pressure in the vapor varies due to compression and expansion of the vapor bubble, but also due to evaporation and condensation as the liquid plug oscillates between the heat source and the heat sink. Both the pressure (described by perfect gas law) and the evaporation and condensation (limited in both the heat source and the heat sink) make the differential equations in the model nonlinear. By applying the center manifold reduction technique followed by the normal form reduction technique, we obtain an approximate analytical solution (describing the position and velocity of the liquid plug, and the mass of vapor, as a function of time). The analytical solution describes well the startup and the subsequent steady state. Using this solution, we show that the initial increase in oscillations’ amplitude is due to the instability mechanism (produced by the interplay of phase change and viscous friction), while the following amplitude saturation is due to the pressure and phase-change nonlinearities, as the system reaches a steady state corresponding to a limit cycle. The limit cycle is created through a Poincaré–Andronov–Hopf bifurcation. We find that one can greatly increase the oscillations’ amplitude in the steady state by increasing the equilibrium instability and by decreasing these nonlinearities. This can be done by increasing the phase change and by decreasing the friction. We conclude that the SBPHP dynamics should be viewed as a combination of an instability mechanism which tends to increase the oscillations’ amplitude and a limiting mechanism produced by nonlinearities, which limits the amplitude.

Appendices

Detailed procedure to derive the normal form

In this appendix, we show how to derive the normal form vector field for r and $\theta$ (Eq. (3.3)) from the vector field for $q_1$, $q_2$ and $q_3$ (Eq. (2.14)). We first use the center manifold reduction to reduce the three-dimensional system to a two-dimensional one (Sect. A.1). We then reduce the two-dimensional system to its normal form, which we can solve analytically (Sect. A.2). The solution is provided for new phase-space variables $z_1$ and $z_2$, related to the original one ($q_1$, $q_2$ and $q_3$) through a series of transformations. In Sect. A.3, we make the inverse transformation and provide the analytical solution for $q_1$, $q_2$ and $q_3$. Our procedure is mainly based on Wiggins [55]. See Rand [32] for an introduction and Moon and Rand [26] for an example of the technique. Because the procedure involves lengthy manipulations, it is best handled using analytical software. The MATLAB code used for this purpose is made available in supplementary materials. In the following, we will not include the full explicit expressions of some of the quantities in the intermediary steps, please refer to the code for those.

1.1 Center manifold reduction

We start from Eq. (2.14), but replace ${\sigma }$ by the bifurcation parameter ${\delta }$, using ${\sigma }={\delta }+{\zeta _{f}}$ and resulting in

$$\begin{aligned} \dot{q}_1&= q_2 , \end{aligned}$$

(A.1a)

$$\begin{aligned} \dot{q}_2&= \left[ 1 - {c_P}\left( \frac{q_1}{1 + q_1}\right) \right] \left( -q_1+q_3\right) -2 \, \zeta _f \ q_2 , \end{aligned}$$

(A.1b)

$$\begin{aligned} \dot{q}_3&= -2 \left( {\delta }+{\zeta _{f}}\right) \left( 1-{c_T}\right) \, q_1 \nonumber \\ {}&\qquad + {c_T}\, {T_{HL}}\,\left( \arctan \left[ \, - \, \left( \frac{2 \left( {\delta }+{\zeta _{f}}\right) }{{T_{HL}}\cos ^2[\psi /2]}\right) q_1 \right. \right. \nonumber \\&\left. \left. \quad -\tan \left[ \frac{\psi }{2}\right] \right] + \frac{\psi }{2}\right) . \end{aligned}$$

(A.1c)

To apply the center manifold reduction technique, we need the real part of the complex conjugate eigenvalues to be 0. Here, however, the real part of the complex conjugate eigenvalues can be positive or negative, if ${\delta }\ne 0$ (see Sect. 2.2). Fortunately, we can transform the system to satisfy the center manifold requirement by a clever trick [55, p.251]: We define ${\delta }$ as a new phase-space variable and add the following differential equation to the system:

$$\begin{aligned} \dot{{\delta }}&= 0 , \end{aligned}$$

(A.1d)

which defines ${\delta }$ as a constant. Because ${\delta }$ is a phase-space variable, the quantities ${\delta }q_1$ are now nonlinear. Thus, ${\delta }$ will not appear in the linear part and will not contribute to the eigenvalues anymore. The real part of the complex-conjugate eigenvalues will be 0. (We will show that in the following.) The new phase space given by Eq. (A.1) is four-dimensional. As a compact notation, we define the vector of phase-space variables $\varvec{x}$ as $\varvec{x}\equiv [q_1,q_2,q_3,{\delta }]^T$ and Eq. (A.1) becomes $\dot{\varvec{x}}=\varvec{f(\varvec{x})}$, where $\varvec{f}$ is the vector field.

1.1.1 Standard form

Before making the reduction, we must transform the system into a standard form. To do so, we first split the vector field as

$$\begin{aligned} \varvec{\dot{x}}&= \varvec{f}\left( \varvec{x}\right) = A \varvec{x} + \varvec{F}(\varvec{x}) , \end{aligned}$$

(A.2)

where $A \varvec{x}$ is the linear part and $\varvec{F}(\varvec{x})$ is the nonlinear part. The matrix A is given by

$$\begin{aligned} A&= \mathrm {D}\varvec{f}(\varvec{0}) = \begin{bmatrix} 0 &{} 1 &{} 0 &{} 0\\ - 1 &{} -2 {\zeta _{f}}&{} 1 &{} 0\\ -2 {\zeta _{f}}&{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{}0\\ \end{bmatrix} , \end{aligned}$$

(A.3)

where $\mathrm {D}\varvec{f}(\varvec{0})$ is the Jacobian of the vector field evaluated at the equilibrium $\varvec{x}=\varvec{0}$. (This is the linearization of the vector field.) The nonlinear part is given by $\varvec{F}(\varvec{x}) = \varvec{f}(\varvec{x}) - A\varvec{x}$. We have that $\varvec{F}(\varvec{0})=\varvec{0}$ and $\mathrm {D}\varvec{F}(\varvec{0})=\varvec{0}$. For small $\varvec{x}$, $\varvec{F}(\varvec{x})$ is small and the system is weakly nonlinear.

The eigenvalues of A are: $\lambda _1=0$, $\lambda _2=-i$, $\lambda _3=i$ and $\lambda _4=-2{\zeta _{f}}$. We see here that we indeed have the real part of complex-conjugate eigenvalues $\lambda _{2,3}$ being 0. The corresponding eigenvectors are the columns of the matrix

$$\begin{aligned} V = \begin{bmatrix} 0 &{} i &{} -i &{} 1 \\ 0 &{} 1 &{} 1 &{} -2{\zeta _{f}}\\ 0 &{} 2{\zeta _{f}}&{} 2{\zeta _{f}}&{} 1 \\ 1 &{} 0 &{} 0 &{} 0 \\ \end{bmatrix} . \end{aligned}$$

(A.4)

The standard form we seek requires the linear part to be in a block diagonal form [55, p.37], in the Jordan canonical form more specifically. To reach that form, we use a linear transformation given by a matrix ${T_a}$. For A having distinct real eigenvalues $\lambda _j$ with corresponding eigenvectors $\varvec{v_j}$ and distinct complex eigenvalues $\lambda _j = a_j + i \, b_j$ and $\bar{\lambda }_j = a_j - i \, b_j$ with corresponding complex eigenvectors $\varvec{w_j} = \varvec{u_j} +i\, \varvec{v}_j$ and $\varvec{\bar{w}_j} = \varvec{u_j} -i\, \varvec{v}_j$, with $j = k+1,...,n$, the matrix ${T_a}$ is given by ${T_a}=\left[ \varvec{v_1},\varvec{u_1}, \dots , \varvec{v_n}, \varvec{u_n}\right]$ (see Perko [30, p.39,28,154,108] for details). Given the eigenvectors Eq. (A.4), we obtain, for ${T_a}$ and its inverse ${T_a}^{-1}$ (with $\kappa \equiv 1/(1+ 4{\zeta _{f}}^2)$),

$$\begin{aligned} {T_a}= & {} \begin{bmatrix} 0 &{} -1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} -2{\zeta _{f}}\\ 0 &{} 0 &{} 2{\zeta _{f}}&{} 1 \\ 1 &{} 0 &{} 0 &{} 0 \\ \end{bmatrix} \qquad \text {and} \qquad \nonumber \\ {T_a}^{-1}= & {} \begin{bmatrix} 0 &{} 0 &{} 0 &{} 1\\ -1 &{} -2 \kappa \, {\zeta _{f}}&{} \kappa &{} 0\\ 0 &{} \kappa &{} 2 \kappa {\zeta _{f}}&{} 0\\ 0 &{} -2 \kappa \, {\zeta _{f}}&{} \kappa &{} 0\\ \end{bmatrix} . \end{aligned}$$

(A.5)

We apply this transformation to the vector of phase-space variable $\varvec{x}$ and obtain a transformed vector of phase-space variables, we have

$$\begin{aligned} \varvec{y}= & {} {T_a}^{-1}\varvec{x} = \begin{bmatrix} y_1 \\ y_2\\ y_3\\ y_4 \\ \end{bmatrix} \nonumber \\= & {} \begin{bmatrix} {\delta }\\ \kappa \ \left( -q_1 + q_3 - 2 {\zeta _{f}}q_2 - 4 {\zeta _{f}}^2 q_1\right) \\ \kappa \ \left( q_2 + 2 {\zeta _{f}}q_3\right) \\ \kappa \ \left( q_3 - 2 {\zeta _{f}}q_2\right) \\ \end{bmatrix} \end{aligned}$$

(A.6)

and

$$\begin{aligned} \varvec{x} = {T_a}\varvec{y} = \begin{bmatrix} q_1 \\ q_2 \\ q_3 \\ {\delta }\\ \end{bmatrix} = \begin{bmatrix} - y_2 + y_4\\ y_3 - 2 {\zeta _{f}}\, y_4 \\ 2 {\zeta _{f}}\, y_3 + y_4\\ y_1 \\ \end{bmatrix} . \end{aligned}$$

(A.7)

Substitution of $\varvec{x} = {T_a}\varvec{y}$ into the system $\varvec{\dot{x}} = A \varvec{x} + \varvec{F}(\varvec{x})$ (Eq. (A.2) ) leads to

$$\begin{aligned} \dot{\varvec{y}} = \varvec{g}\left( \varvec{y}\right) = J \ \varvec{y} \ + \varvec{G}(\varvec{y}) , \end{aligned}$$

(A.8)

where J is the matrix ${T_a}^{-1} A \, {T_a}$ and $\varvec{G}(\varvec{y}) = {T_a}^{-1} \varvec{F}({T_a}\varvec{y})$. The matrix J is given by

$$\begin{aligned} J = {T_a}^{-1} A \, {T_a}= \begin{bmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} -1 &{} 0\\ 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} -2{\zeta _{f}}\\ \end{bmatrix} . \end{aligned}$$

(A.9)

We can split Eq. (A.8) into three rows as

$$\begin{aligned} \dot{\varvec{y}} = \begin{bmatrix} \dot{\varvec{y}}_p \\ \dot{\varvec{y}}_c \\ \dot{\varvec{y}}_s \\ \end{bmatrix} = \begin{bmatrix} 0 &{} 0 &{} 0\\ 0 &{} J_c &{} 0 \\ 0 &{} 0 &{} J_s\\ \end{bmatrix} \begin{bmatrix} \varvec{y}_p \\ \varvec{y}_c \\ \varvec{y}_s \\ \end{bmatrix} + \begin{bmatrix} \varvec{0} \\ \varvec{G}_c (\varvec{y}_c, \varvec{y}_p, \varvec{y}_s)\\ \varvec{G}_s (\varvec{y}_c, \varvec{y}_p, \varvec{y}_s)\\ \end{bmatrix} , \end{aligned}$$

(A.10)

where

$$\begin{aligned} J_c = \begin{bmatrix} 0 &{} -1 \\ 1 &{} 0\\ \end{bmatrix} \quad \text {and} \quad J_s = -2{\zeta _{f}}. \end{aligned}$$

(A.11)

In Eq. (A.10), the first row having variable $\varvec{y}_p={\delta }$ corresponds to a parameter subsystem, the second row having variables $\varvec{y}_c=[y_2,y_3]^T$ corresponds to a center subsystem since its eigenvalues are complex conjugate eigenvalues with 0 real parts and the third row having variable $\varvec{y}_s=y_4$ corresponds to a stable subsystem since its eigenvalue is real and negative. The parameter, center and stable directions are linearly uncoupled from one another. We have that $\varvec{G}_c=\varvec{0}$, $\varvec{G}_s=\varvec{0}$, $\mathrm {D}\varvec{G}_c=\varvec{0}$ and $\mathrm {D}\varvec{G}_s=\varvec{0}$. Equation (A.10) is the standard form required to apply the center manifold theorem.

1.1.2 Center manifold reduction

There exists a center manifold for the system Eq. (A.10) defined locally by

$$\begin{aligned}&W^c_{loc}(\varvec{0}) = \lbrace \left( \varvec{y}_c, \varvec{y}_p, \varvec{y}_s\right) \in \mathbb {R}^c \times \mathbb {R}^p \times \mathbb {R}^s \mid \nonumber \\&\quad \varvec{y}_s=\varvec{h}\left( \varvec{y}_c, \varvec{y}_p\right) , |\varvec{y}_c |<{\gamma }_c, \ |\varvec{y}_p |<{\gamma }_p, \ \nonumber \\&\quad \varvec{h}(\varvec{0},\varvec{0})=\varvec{0}, \ \mathrm {D}\varvec{h}(\varvec{0},\varvec{0})=\varvec{0} , \rbrace \end{aligned}$$

(A.12)

for ${\gamma }_c$ and ${\gamma }_p$ sufficiently small [55, Sec. 18.2, p.251]. The center manifold is a three-dimensional surface in the four-dimensional phase space. Sufficiently close to the origin $\varvec{y}=\varvec{0}$, “small amplitude periodic orbits are contained in the center manifold” [55, p.247]. Consequently, we will be able to consider only the dynamics in the center manifold and thus reduce the dimensionality of the system. In Eq. (A.12), we have that $\varvec{y}_s=y_4=h\left( \varvec{y}_c, \varvec{y}_p\right)$, the stable direction is reduced to a function h of $\varvec{y}_c=[y_2,y_3]^T$ and $\varvec{y}_p=y_1={\delta }$.

It is common to approximate h by a power expansion, we have

$$\begin{aligned} y_4&\approx h, \qquad \text {with : } \nonumber \\ h&= c_1 \, y_1^2 + c_2 \,y_1 y_2 + c_3 \, y_1 y_3 + c_4 \, y_2^2 \nonumber \\&\quad + c_5 \, y_2 y_3 + c_6 \, y_3^2 . \end{aligned}$$

(A.13)

We now must find the coefficients $c_j$. For $W^c_{loc}(0)$ to be a center manifold, h must satisfy the differential equations of the system. By substitution of $y_4=h({\delta },y_2,y_3)$ in Eq. (A.8), we find that h must satisfy the following quasilinear partial differential equation [55, p.252]

$$\begin{aligned} \mathcal {N}\left( h(\varvec{y}_c,\varvec{y}_p)\right)&\equiv \mathrm {D}_{\varvec{y}_c} h(\varvec{y}_c,\varvec{y}_p) \cdot \varvec{g}_c \left( \varvec{y}_c,h(\varvec{y}_c,\varvec{y}_p),\varvec{y}_p\right) \nonumber \\&\quad - \varvec{g}_s \left( \varvec{y}_c,h(\varvec{y}_c,\varvec{y}_p),\varvec{y}_p\right) = 0 . \end{aligned}$$

(A.14)

Collecting terms of like powers, we obtain 6 algebraic equations which we solve for the 6 c’s, leading to

$$\begin{aligned} c_1&= 0 , \end{aligned}$$

(A.15a)

$$\begin{aligned} c_2&= \frac{4}{\left( 1 + 4 {\zeta _{f}}^2\right) ^2} , \end{aligned}$$

(A.15b)

$$\begin{aligned} c_3&= \frac{2}{\left( 1 + 4 {\zeta _{f}}^2\right) ^2} , \end{aligned}$$

(A.15c)

$$\begin{aligned} c_4&= \frac{-{T_{HL}}\, {c_P}+ 2{\zeta _{f}}\, \tan \left[ \psi /2\right] \left( 1+2{\zeta _{f}}^2\right) {c_T}}{2 {T_{HL}}\left( 1 + 5{\zeta _{f}}^2 + 4{\zeta _{f}}^4\right) } , \end{aligned}$$

(A.15d)

$$\begin{aligned} c_5&= \frac{-{\zeta _{f}}{T_{HL}}\left( 1 +2{\zeta _{f}}^2\right) {c_P}+ 2{\zeta _{f}}^2 \, \tan \left[ \psi /2\right] {c_T}}{{T_{HL}}\left( 1 + 5{\zeta _{f}}^2 + 4{\zeta _{f}}^4\right) } , \end{aligned}$$

(A.15e)

$$\begin{aligned} c_6&= \frac{-{T_{HL}}\left( 1 +2{\zeta _{f}}^2\right) {c_P}+ 2{\zeta _{f}}\, \tan \left[ \psi /2\right] {c_T}}{2{T_{HL}}\left( 1 + 5{\zeta _{f}}^2 + 4{\zeta _{f}}^4\right) } . \end{aligned}$$

(A.15f)

By substitution of $\varvec{y}_s=y_4=h({\delta },y_2,y_3)$ in system Eq. (A.10), the center subsystem becomes

$$\begin{aligned} \dot{\varvec{y}}_c&= \varvec{g}_c = J_c \varvec{y}_c + \varvec{G}_c (\varvec{y}_c, \varvec{y}_p, \varvec{h}(\varvec{y}_c, \varvec{y}_p)) \nonumber \\&\qquad \text {with:} \qquad J_c = \begin{bmatrix} 0 &{} -1 \\ 1 &{} 0\\ \end{bmatrix} , \end{aligned}$$

(A.16a)

where $\varvec{y}_c = \left[ y_2 , y_3\right] ^T$ and $\varvec{G}_c$ is a complicated function of $y_2$ $y_3$ and ${\delta }$. This system is now completely decoupled from the stable subsystem (does not depend on $\varvec{y_s}$ at all). The parameter subsystem equation is again

$$\begin{aligned} \dot{{\delta }}&= 0 . \end{aligned}$$

(A.16b)

Equation (A.16) is a three-dimensional phase space (with variables $y_2$, $y_3$ and ${\delta }$). However, since $\dot{{\delta }} = 0$, the motion on the plane $y_2-y_3$ is invariant (remains on the plane), so knowing the value ${\delta }$ is enough to predict the motion on the plane $y_2-y_3$. The dynamics is thus effectively two-dimensional and is given by Eq. (A.16a). Properties of the dynamics in the plane $y_2-y_3$ can change as the height ${\delta }$ changes. In fact, we will find that a Hopf bifurcation occurs at ${\delta }=0$.

1.2 Normal form

Now that we have a two-dimensional system Eq. (A.16a), we can use the normal form reduction technique to obtain an approximative solution for it. In normal form, we want the instability to be captured in the linear part, we want ${\delta }$ to appear in the linear part [55, Eq.(19.2.1,19.2.3), p.278,279]. Unfortunately, we purposefully removed ${\delta }$ from the linear part to apply the center manifold reduction. We can bring ${\delta }$ back by splitting the vector field $\varvec{g}_c$ in Eq. (A.16) again, but without taking ${\delta }$ as a phase-space variable. Doing so, we get

$$\begin{aligned} \varvec{\dot{y}}_c = \varvec{g}_c (\varvec{y}_c) = B \varvec{y}_c + \varvec{H}(\varvec{y}_c) , \end{aligned}$$

(A.17)

where $B=\mathrm {D}_{\varvec{y}_c} \, \varvec{g}_c(\varvec{0})$ is the Jacobian of $\varvec{g}_c$ evaluated at the fixed point $\varvec{y}_c=\varvec{0}$. Also, $\varvec{H} = \varvec{g}_c - B\varvec{y}_c$, $\varvec{H}\in C^1(E)$, $\varvec{H}(\varvec{0})=\varvec{0}$ and $\mathrm {D}\varvec{H}(\varvec{0})=\varvec{0}$. As desired, the eigenvalues of B now depend on ${\delta }$. However, B is not in the Jordan canonical form, so we must perform a linear transformation again, similarly to what was done in Sect. A.1.1.

1.2.1 Standard form for complex conjugate eigenvalues

Here, we transform Eq. (A.17) in its standard form, such that the linear part is in a Jordan canonical form. To do so, we must perform a linear transformation through the multiplication of a matrix ${T_b}$. To find such matrix, we first Taylor expand B for small ${\delta }$. We find

$$\begin{aligned} B= & {} \begin{bmatrix} 0 &{} -1 \\ 1 &{} 0\\ \end{bmatrix}+ & {} \frac{2{\delta }}{1+4{\zeta _{f}}^2} \begin{bmatrix} 1 &{} 0 \\ 2{\zeta _{f}}&{} 0\\ \end{bmatrix} + \mathcal {O}({\delta }^2) . \end{aligned}$$

(A.18)

Considering the truncated expression of B above, we compute the trace ${{\,\mathrm{tr}\,}}(B)$ and the determinant $\det (B)$. Assuming the discriminant ${{\,\mathrm{tr}\,}}(B)^2-4\det (B)<0$ (true for either small $|{\delta } |$ or ${\delta }>0$), the eigenvalues are complex conjugates given by $\lambda =\alpha + i\omega$ and $\bar{\lambda }=\alpha - i\omega$ with the real part ${\alpha =\tfrac{1}{2} {{\,\mathrm{tr}\,}}(B)}$ and the imaginary part ${\omega =\tfrac{1}{2} \sqrt{{{\,\mathrm{tr}\,}}(B)^2-4\det (B)}}$. Taylor expansion of $\alpha$ and $\omega$ in terms of ${\delta }$ leads to

$$\begin{aligned} \alpha&= \frac{{\delta }}{1+4 {\zeta _{f}}^2} + \mathcal {O}({\delta }^2), \qquad \text {and} \qquad \nonumber \\ \omega&= 1 + \left( \frac{2 {\zeta _{f}}}{1+4{\zeta _{f}}^2}\right) {\delta }+ \mathcal {O}({\delta }^2). \end{aligned}$$

(A.19)

The eigenvectors corresponding to the eigenvalues $\lambda$ and $\bar{\lambda }$ are $\varvec{w} = \varvec{u} +i\, \varvec{v}$, and $\varvec{\bar{w}} = \varvec{u} -i\, \varvec{v}$, respectively, with

$$\begin{aligned} \varvec{u} = \begin{bmatrix} \dfrac{\alpha -b_{22}}{b_{21}}\\ 1 \end{bmatrix} \qquad \text {and} \qquad \varvec{v} = \begin{bmatrix} \dfrac{\omega }{b_{21}}\\ 0 \end{bmatrix} , \end{aligned}$$

(A.20)

where the $b_{ij}$ are the elements of the truncated matrix B (Eq. (A.18)).

From those, we build the transformation matrix as

$$\begin{aligned} {T_b}&= \left[ \varvec{v},\varvec{u}\right] . \end{aligned}$$

(A.21)

Introducing a new phase-space vector of variables $\varvec{z}$ with $\varvec{y}_c={T_b}\varvec{z}$, Eq. (A.17) is transformed into

$$\begin{aligned} \dot{\varvec{z}}&= \varvec{k}\left( \varvec{z}\right) = J_B \ \varvec{z} \ + \varvec{K}(\varvec{z}) , \end{aligned}$$

(A.22)

where $\varvec{k}(\varvec{z})$ is a new vector field, $J_B={T_b}^{-1} B \, {T_b}$ and $\varvec{K}(\varvec{z}) = {T_b}^{-1} \varvec{H}({T_b}\varvec{z})$. Equation (A.22) is the required form to apply normal form reduction (requirement specified in [55, 197/,148/]). Indeed, we have that the linear part $J_B$ satisfies

$$\begin{aligned} J_B&= \begin{bmatrix} \alpha &{} -\omega \\ \omega &{} \alpha \\ \end{bmatrix} +\mathcal {O}({\delta }^2) , \end{aligned}$$

(A.23)

so $J_B$ is in the Jordan canonical form at first order approximation. Also, the vector field $\varvec{k}$ is $C^r$ with $r \geqslant 5$, centered at the fixed point of interest such that $\varvec{k}\left( \varvec{z}=\varvec{0},{\delta }=0\right) =\varvec{0}$ and finally, $\alpha ({\delta }=0)=0$ and $\omega ({\delta }=0)\ne 0$.

1.2.2 Normal form reduction

One can reduce Eq. (A.22) to its normal form by third order Taylor expansion of the vector field and successive transformations (see Wiggins [55, Eq.(20.2.7) p.379, p.279] for details), leading to

$$\begin{aligned} \dot{z}_1&= \alpha ({\delta }) z_1 - \omega ({\delta }) z_2 + \left( a({\delta }) z_1 - b({\delta })z_2\right) \, \left( z_1^2 + z_2^2\right) \nonumber \\&\quad +\mathcal {O}(|z_1 |^5,|z_2 |^5) , \end{aligned}$$

(A.24a)

$$\begin{aligned} \dot{z}_2&= \omega ({\delta }) z_1 + \alpha ({\delta }) z_2 + \left( b({\delta }) z_1 + a({\delta })z_2\right) \, \left( z_1^2 + z_2^2\right) \nonumber \\&\quad +\mathcal {O}(|z_1 |^5,|z_2 |^5) , \end{aligned}$$

(A.24b)

where a and b are given by the normal form procedure. Equation (A.24) corresponds to Eq. (3.1) in the main text. We care especially about the amplitude of the oscillations, so it makes sense to introduce polar coordinates with $z_1=r\cos (\theta )$ and $z_2=r\sin (\theta )$, where r is the amplitude and $\theta$ is the phase. Introducing those into Eq. (A.24) leads to

$$\begin{aligned} \dot{r}&= \alpha ({\delta }) r + a({\delta }) r^3 +\mathcal {O}(r^5) , \end{aligned}$$

(A.25a)

$$\begin{aligned} \dot{\theta }&= \omega ({\delta }) + b({\delta }) r^2 + \mathcal {O}(r^4) , \end{aligned}$$

(A.25b)

which are differential equations for r and $\theta$. Because we study this system close to the bifurcation (close to ${\delta }=0$), it makes sense to simplify Eq. (A.25) by Taylor expanding its coefficients in terms of ${\delta }$. We get

$$\begin{aligned} \dot{r}&= d {\delta }r + a_0 r^3 +\mathcal {O}({\delta }^2 r, {\delta }r^3, r^5) , \end{aligned}$$

(A.26a)

$$\begin{aligned} \dot{\theta }&= \omega _0 + c {\delta }+ b_0 r^2 +\mathcal {O}({\delta }^2, {\delta }r^2, r^4) , \end{aligned}$$

(A.26b)

which is the desired normal form (corresponds to Eq. (3.3) in the main text).

The parameters d, $\omega _0$ and c are given by (see supplementary material and Wiggins [55])

$$\begin{aligned} d&\equiv \frac{1}{1+4{\zeta _{f}}^2} , \end{aligned}$$

(A.27a)

$$\begin{aligned} \omega _0&\equiv 1 , \end{aligned}$$

(A.27b)

$$\begin{aligned} c&= \frac{2 {\zeta _{f}}}{1+4{\zeta _{f}}^2}. \end{aligned}$$

(A.27c)

The expression of $b_0$ will not be needed but can be found through the normal form procedure [15]. Finally, $a_0$ is given by

$$\begin{aligned} a_0&= a_{0,T} + a_{0,PT} + a_{0,P}, \end{aligned}$$

(A.28a)

with

$$\begin{aligned} a_{0,T}&\equiv \left( \frac{-2 {\zeta _{f}}^3 \left( 1+ {\zeta _{f}}^2 \cos \left[ \psi \right] \right) }{{T_{HL}}^2(1+\cos \left[ \psi \right] ) \left( 1+{\zeta _{f}}^2\right) \left( 1+4 {\zeta _{f}}^2\right) }\right) {c_T}, \end{aligned}$$

(A.28b)

$$\begin{aligned} a_{0,PT}&\equiv \left( \frac{{\zeta _{f}}^2 \sin \left[ \psi \right] \left( 3+4{\zeta _{f}}^2\right) }{4 {T_{HL}}\left( 1+\cos \left[ \psi \right] \right) \left( 1+{\zeta _{f}}^2\right) \left( 1+4 {\zeta _{f}}^2\right) }\right) {c_P}{c_T}, \end{aligned}$$

(A.28c)

$$\begin{aligned} a_{0,P}&\equiv \left( \frac{-{\zeta _{f}}}{8 \left( 1 + {\zeta _{f}}^2\right) }\right) \ {c_P}, \end{aligned}$$

(A.28d)

which corresponds to Eq. (3.5) in the main text.

1.3 Solution in the q variables (back transformation)

We now would like to know what is the analytical solution for the original variables $q_1$, $q_2$ and $q_3$. To obtain it, we must make the inverse transformations, going from $\varvec{z}$ to $\varvec{y_c}$ (normal form) to $\varvec{y}$ (center manifold reduction) to $\varvec{x}$ (standard form) to $\varvec{q}$ (which is easy). Refer to previous sections to reproduce the following results.

First, we have that $\varvec{y}_c = {T_b}\varvec{z}$. Then, we can obtain $\varvec{y}$ from $\varvec{y}_c$ based on the following, where we used row vectors [1, 0] and [0, 1] to select $y_{c,1}$ and $y_{c,2}$, respectively:

$$\begin{aligned} \varvec{y}&= \begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ y_4 \\ \end{bmatrix} = \begin{bmatrix} {\delta }\\ y_{c,1} \\ y_{c,2} \\ h({\delta },y_{c,1},y_{c,2}) \\ \end{bmatrix} \\&= \begin{bmatrix} {\delta }\\ [1,0] \cdot {T_b}\varvec{z} \\ [0,1] \cdot {T_b}\varvec{z} \\ h({\delta }, [1,0] \cdot {T_b}\varvec{z} , [0,1] \cdot {T_b}\varvec{z}) \\ \end{bmatrix}. \end{aligned}$$

We then obtain $\varvec{x}$ from $\varvec{y}$ (based on Eq. (A.7))

$$\begin{aligned} \varvec{x}&= {T_a}\varvec{y} = \begin{bmatrix} -y_2 + y_4\\ y_3 - 2 {\zeta _{f}}\, y_4 \\ 2 {\zeta _{f}}\, y_3 + y_4\\ y_1 \\ \end{bmatrix} \\&= \begin{bmatrix} -[1,0] \cdot {T_b}\varvec{z} + h({\delta },z_1,z_2)\\ [0,1] \cdot {T_b}\varvec{z} - 2 {\zeta _{f}}\, h({\delta },z_1,z_2) \\ 2 {\zeta _{f}}\, [0,1] \cdot {T_b}\varvec{z} + h({\delta },z_1,z_2)\\ {\delta }\\ \end{bmatrix} . \end{aligned}$$

$\varvec{q}$ simply corresponds to the first three elements of $\varvec{x}$, we have

$$\begin{aligned} \varvec{q}&= \begin{bmatrix} q_1\\ q_2\\ q_3\\ \end{bmatrix} = \ \begin{bmatrix} -[1,0] \cdot {T_b}\varvec{z} + h({\delta },z_1,z_2)\\ [0,1] \cdot {T_b}\varvec{z} - 2 {\zeta _{f}}\, h({\delta },z_1,z_2) \\ 2 {\zeta _{f}}\, [0,1] \cdot {T_b}\varvec{z} + h({\delta },z_1,z_2)\\ \end{bmatrix} , \end{aligned}$$

(A.29)

where h is given by the following (based on Eq. (A.13) with $y_1={\delta }$)

$$\begin{aligned} h({\delta },z_1,z_2)&= c_1 \, {\delta }^2 + c_2 \,{\delta }y_2(\varvec{z}) + c_3 \, {\delta }y_3 (\varvec{z}) \nonumber \\&\quad + c_4 \, {y_2(\varvec{z})}^2 + c_5 \, y_2(\varvec{z}) y_3(\varvec{z}) + c_6 \, {y_3(\varvec{z})}^2 , \end{aligned}$$

(A.30)

with $y_2(\varvec{z})=[1,0] \cdot {T_b}\varvec{z}$ and $y_3(\varvec{z})=[0,1] \cdot {T_b}\varvec{z}$.

Now, because the solutions are $z_1=r \cos (\theta )$ and $z_2=r \sin (\theta )$, the quantities $[n,m] \cdot {T_b}\varvec{z}$ are linear combinations of sines and cosines. Thus, we can write each $q_i$ (with $i=1,2,3$) as Fourier series with terms up to second-order harmonics (these being produced by the terms $c_4 {y_2}^2$, $c_5 y_2 y_3$ and $c_6 {y_3}^2$). We have

$$\begin{aligned} q_i&= \tfrac{1}{2} \, a_{i 0} + a_{i 1} \cos \left( \theta \right) + b_{i 1} \sin \left( \theta \right) \nonumber \\&\quad + a_{i 2} \cos \left( 2\theta \right) + b_{i 2} \sin \left( 2\theta \right) , \end{aligned}$$

(A.31a)

with the coefficients computed as usual when dealing with Fourier series:

$$\begin{aligned} a_{i 0}&= \frac{1}{\pi } \int _{-\pi }^{\pi } q_i ({\bar{r}},\theta ) \ d\theta \end{aligned}$$

(A.31b)

$$\begin{aligned} a_{i k}&= \frac{1}{\pi } \int _{-\pi }^{\pi } q_i ({\bar{r}},\theta ) \cos (k\theta ) \ d\theta \end{aligned}$$

(A.31c)

$$\begin{aligned} b_{i k}&= \frac{1}{\pi } \int _{-\pi }^{\pi } q_i ({\bar{r}},\theta ) \sin (k\theta ) \ d\theta \end{aligned}$$

(A.31d)

Because we care mostly about the amplitude of each terms, it makes sense to rewrite this Fourier series in an amplitude-phase form as

$$\begin{aligned} q_i&= \frac{A_{i 0}}{2} + \sum _{k=1}^{n} A_{i j} \sin \left( k\theta +\varphi _{i k}\right) \end{aligned}$$

(A.32a)

with the following coefficients:

$$\begin{aligned} A_{i 0}&= a_{i0} \end{aligned}$$

(A.32b)

$$\begin{aligned} A_{i k}&= \sqrt{{a_{i k}}^2+{b_{i k}}^2} \end{aligned}$$

(A.32c)

$$\begin{aligned} \varphi _{i k}&= \mathrm {atan2}\left( a_{i k},b_{i k}\right) \end{aligned}$$

(A.32d)

Using Taylor expansions in terms of ${\delta }$ and ${\zeta _{f}}$, we obtain the following:

$$\begin{aligned} A_{1 0}&= \left( \frac{{c_P}T_{HL}- 2 {c_T}{\zeta _{f}}\tan \left[ \psi /2\right] }{T_{HL}}\right) \, r^2 \nonumber \\&\quad +\mathcal {O}({\delta }^2,{\zeta _{f}}^2,{\delta }{\zeta _{f}}), \end{aligned}$$

(A.33a)

$$\begin{aligned} A_{1 1}&= r \ +\mathcal {O}({\delta }^2,{\zeta _{f}}^2,{\delta }{\zeta _{f}}), \end{aligned}$$

(A.33b)

$$\begin{aligned} A_{1 2}&= \frac{\left( {\delta }+{\zeta _{f}}\right) \left( {c_P}T_{HL}-2{c_T}{\zeta _{f}}\tan \left[ \psi /2\right] \right) }{2T_{HL}} \, r^2 \nonumber \\&\quad +\mathcal {O}({\delta }^2,{\zeta _{f}}^2,{\delta }{\zeta _{f}}), \end{aligned}$$

(A.33c)

which led to the approximations given by Eq. (3.6), where we also considered $\psi =0$.

Effect of deviation of the equilibrium from the inflexion point ($\psi$)

In the main text, we discussed the dynamics based on the vector field $\dot{r} = d {\delta }r + a_0 r^3$ (Eq. (3.3a)) for $a_0<0$. In Sect. 3.3.2, we showed that we indeed have $a_0<0$, if we assume $\psi =0$. In this appendix, we investigate the general case, where $\psi \ne 0$. We first described how we obtained the evaporation rate equation and how we introduced $\psi$ (see Sect. B.1). We then explain what $\psi$ means (a measure of the deviation of the equilibrium relative to the inflexion point of the wall temperature profile) and evaluate what values it can take, depending on $T_{g,sat,0}$, $T_H$ and $T_L$ (Sect. B.2). We then evaluate the sign of $a_0$, as a function of $\psi$ and other parameters (Sect. B.3). Although we expect to have $a_0<0$ in typical conditions, we find that we can indeed have $a_0>0$ for large enough $\psi$. Finally, we study what the dynamics looks like when $a_0>0$ (Sect. B.4).

1.1 Derivation of the evaporation rate expression, with the deviation parameter $\psi$

Here, we describe the original expression from the evaporation rate from Tessier-Poirier et al. [49] (Sect. B.1.1) and show how we obtain the new expression used here, which introduce $\psi$ (Sect. B.1.2).

1.1.1 Expression of the evaporation rate

The dimensionless evaporation rate $\dot{q}_3$ was derived in Tessier-Poirier et al. [49, Eq.(10b) p.10] as (we adapted the nomenclature to match the one used in the main text)

$$\begin{aligned} \dot{q}_3&= {T_{HL}}\, \arctan \left[ \ - | \widetilde{T'_{wc}} |\, \left( q_1 - \widetilde{x_{c}}\right) \ \right] + {\widetilde{C_{th}}}. \end{aligned}$$

(B.1)

In Eq. (B.1), the parameters are the phase-change limit ${T_{HL}}$, the dimensionless wall axial temperature gradient at the inflexion point $| \widetilde{T'_{wc}} |$, the dimensionless position of the inflexion point relative to the equilibrium $\widetilde{x_{c}}$ and ${\widetilde{C_{th}}}$, a constant which represents the evaporation rate when the meniscus is at the inflexion point, when $q_1=\widetilde{x_{c}}$. Those parameters are defined as

$$\begin{aligned} {T_{HL}}&\equiv \frac{T_H - T_L}{\pi \ m_{g,0} \, \omega _n \ H_v \, R_{th}} , \end{aligned}$$

(B.2a)

(B.2b)

$$\begin{aligned} \widetilde{x_{c}}&\equiv \left( \frac{1}{L_{g,0}} \right) \ x_{c}, \ \text {where} \ x_{c}= \left( \frac{T_H-T_L}{\pi \, |T'_{wc} |}\right) \ \nonumber \\&\qquad \tan \left[ \frac{\pi \left( T_{g,sat,0} - \tfrac{1}{2} \left( T_H+T_L\right) \right) }{T_H-T_L}\right] , \end{aligned}$$

(B.2c)

$$\begin{aligned} {\widetilde{C_{th}}}&\equiv \frac{\tfrac{1}{2} \left( T_H+T_L\right) - T_{g,sat,0}}{m_{g,0} \, \omega _n \ H_v R_{th}} . \end{aligned}$$

(B.2d)

The dimensionless axial temperature gradient at the equilibrium $|T'_{w,0} |$ is an important parameter, which can be expressed as a function of the gradient at the inflexion point as

(B.3)

Finally, note that linearization of Eq. (B.1) leads to $\dot{q}_3 = -2 \, {\sigma }\, q_1$ (Eq. (2.9c)) where

$$\begin{aligned} {\sigma }&\equiv \frac{1}{2} \ \frac{{T_{HL}}\ | \widetilde{T'_{wc}} |}{1+ \left( \widetilde{x_{c}}\ | \widetilde{T'_{wc}} |\right) ^2} =\ \frac{L_{g,0} \ |T'_{w,0} |}{2 \ m_{g,0} \, \omega _n \, H_v \, R_{th}}, \end{aligned}$$

(B.4)

the last equality involving dimensional quantities, obtained by substitution.

1.1.2 New form for the evaporation rate

Here, we will show how to transform Eq. (B.1) into the form given by Eq. (2.7c). We prefer this latter form because it involves less parameters and those are easier to study. Setting $\dot{q}_3=0$, we find that

$$\begin{aligned} | \widetilde{T'_{wc}} |\, \widetilde{x_{c}}=\tan [-{\widetilde{C_{th}}}/T_{HL}] \end{aligned}$$

(B.5)

Substitution of this into the equation for $\dot{q}_3$ eliminates the parameter $\widetilde{x_{c}}$, we have

$$\begin{aligned} \dot{q}_3&= {T_{HL}}\, \left( \arctan \left[ \ - | \widetilde{T'_{wc}} |\, q_1 - \tan \left[ \frac{{\widetilde{C_{th}}}}{T_{HL}}\right] \right] + \frac{{\widetilde{C_{th}}}}{{T_{HL}}}\right) . \end{aligned}$$

In $a_0$ (Eq. (A.28)), the quantity ${\widetilde{C_{th}}}$ only appears in the ratio $2{\widetilde{C_{th}}}/T_{HL}$, as argument of cosine and sine functions. It therefore makes sense to define the quantity $\psi \equiv 2{\widetilde{C_{th}}}/T_{HL}$, which we use to eliminate ${\widetilde{C_{th}}}$, leading to

$$\begin{aligned} \dot{q}_3&= {T_{HL}}\, \left( \arctan \left[ \ - | \widetilde{T'_{wc}} |\, q_1 - \tan \left[ \frac{\psi }{2}\right] \right] + \frac{\psi }{2}\right) . \end{aligned}$$

Knowing that $| \widetilde{T'_{wc}} |$ is ultimately expressed in phase-change coefficient ${\sigma }$ when the equation is linearized to $\dot{q}_3=-2 \, {\sigma }\, q_1$, it makes sense to replace it right away with ${\sigma }$. Substitution of $| \widetilde{T'_{wc}} |\, \widetilde{x_{c}}=-\tan [\psi /2]$ in Eq. (B.4) gives

$$\begin{aligned} {\sigma }&= \frac{1}{2} \ \frac{{T_{HL}}\ | \widetilde{T'_{wc}} |}{1+ \tan ^2[\psi /2]} = \tfrac{1}{2} \ \cos ^2\left[ \psi /2\right] \, {T_{HL}}\ | \widetilde{T'_{wc}} |, \end{aligned}$$

(B.6)

where we use the trigonometric identity $1+\tan ^2 \theta = \sec ^2 \theta = 1/\cos ^2 \theta$.

Solving Eq. (B.6) for $| \widetilde{T'_{wc}} |$, we get $| \widetilde{T'_{wc}} |= 2{\sigma }/{T_{HL}}\, (\cos ^2\left[ \psi /2\right] )$. Substitution of the results in the $\dot{q}_3$ equation above, we obtain

$$\begin{aligned} \dot{q}_3&= {T_{HL}}\, \arctan \left[ \ - \left( \frac{2{\sigma }}{T_{HL}\cos ^2\left[ \psi /2\right] }\right) \, q_1 \right. \nonumber \\&\left. \quad - \tan \left[ \frac{\psi }{2}\right] \right] + \frac{\psi }{2} \, {T_{HL}}, \end{aligned}$$

(B.7)

which is Eq. (2.7c) in the main text.

1.2 Studying the $\psi$ expression

In Sect. B.2.1, we explain what $\psi$ means physically. We also derive an expression of $\psi$ in terms of $T_{g,sat,0}$, $T_H$ and $T_L$. In Sect. B.2.2, we evaluate what values $\psi$ can take.

1.2.1 $\psi$ as a measure of the deviation of the equilibrium from the inflexion point

Both $\widetilde{x_{c}}$ and ${\widetilde{C_{th}}}$ are dimensionless measures of the deviation of the equilibrium relative to the inflexion point. This is obvious for $\widetilde{x_{c}}$, since $x_{c}$ is directly the distance between the equilibrium and the inflexion point. The quantity ${\widetilde{C_{th}}}$ is also a measure of the deviation since it is proportional to $\tfrac{1}{2} \left( T_H+T_L\right) - T_{g,sat,0}$, where $\tfrac{1}{2} \left( T_H+T_L\right)$ is the temperature at the inflexion point and where $T_{g,sat,0}$ is the saturation temperature, which is the temperature at the equilibrium (where $\dot{q}_3=0$). Thus, ${\widetilde{C_{th}}}\ne 0$ only when the equilibrium is away from the inflexion point, when $T_{g,sat,0}\ne \tfrac{1}{2} \left( T_H+T_L\right)$. In fact, we see from Eq. (B.2c) that $x_{c}$ is also expressed as a function of $\tfrac{1}{2} \left( T_H+T_L\right) - T_{g,sat,0}$. Now, the new quantity $\psi$, which we use to eliminate both $\widetilde{x_{c}}$ and ${\widetilde{C_{th}}}$, also expresses the deviation of the equilibrium relative to the inflexion point. Given $\psi =2{\widetilde{C_{th}}}/T_{HL}$, $\psi$ is also proportional to $\tfrac{1}{2} \left( T_H+T_L\right) - T_{g,sat,0}$ and the conclusions for ${\widetilde{C_{th}}}$ above also hold for $\psi$.

Let’s now express $\psi$ as a function of the temperature $T_H$, $T_L$ and $T_{g,sat,0}$. Starting from $\psi \equiv 2{\widetilde{C_{th}}}/T_{HL}$, after substitution of ${\widetilde{C_{th}}}$ (Eq. (B.2d)) and $T_{HL}$ (Eq. (B.2a)), we get

$$\begin{aligned} \psi&= \frac{2 \pi }{\left( T_H-T_L\right) } \ \left( \frac{T_H+T_L}{2} - T_{g,sat,0}\right) . \end{aligned}$$

Additional manipulations leads to

$$\begin{aligned} \psi&= \pi \ \left( \frac{T_H+T_L}{T_H-T_L} - \frac{2 \, T_{g,sat,0}}{T_H-T_L}\right) , \end{aligned}$$

where $\psi$ depends on three parameters, $T_H$, $T_L$ and $T_{g,sat,0}$. Dividing numerator and denominator of each terms by $T_H$, we obtain

$$\begin{aligned} \psi&= \pi \ \left( \frac{1+T_L/T_H}{1-T_L/T_H} - \frac{2 \, T_{g,sat,0}/T_H}{1-T_L/T_H}\right) \nonumber \\&\quad \text {with: } 0< \left( \frac{T_L}{T_H}\right)< \left( \frac{T_{g,sat,0}}{T_H}\right) < 1 \end{aligned}$$

(B.8)

which now depends only on two groups of parameters, the ratios $T_L/T_H$ and $T_{g,sat,0}/T_H$. The inequalities above are obtained by considering that $0<T_L<T_{g,sat,0}<T_H$ (and temperature expressed in kelvin so always positive). Equation (B.8) for $\psi$ corresponds to Eq. (2.8c) in the main text.

1.2.2 Values $\psi$ takes, as a function of $T_{g,sat,0}$, $T_H$ and $T_L$

We now want to investigate what values can $\psi$ take. We will find that $\psi$ ranges between $-\pi$ (when $T_{g,sat,0}=T_H$, the equilibrium is as far as possible in the heat source region) and $+\pi$ (when $T_{g,sat,0}=T_L$, the equilibrium is as far as possible in the heat sink region).

Figure 10 shows $\psi /\pi$ values as a surface along the ratios $T_L/T_H$ and $T_{g,sat,0}/T_H$. Given the inequalities $0<T_L<T_{g,sat,0}<T_H$, we have boundaries in Fig. 10. First, the condition $T_{g,sat,0}=T_H$ (the equilibrium is far into the heat source) gives the boundary $T_{g,sat,0}/T_H=1$ which is a straight horizontal line in Fig. 10, and we find that $\psi =-1$ (Eq. (B.8)) there. Second, the condition $T_{g,sat,0}=T_L$ (the equilibrium is far into the heat sink) gives the boundary $T_{g,sat,0}/T_H=T_L/T_H$ which gives the diagonal line in Fig. 10, and we find that $\psi =+1$ (Eq. (B.8)) there. The inequalities $0<T_L/T_H<1$ and $0<T_{g,sat,0}/T_H<1$ (Eq. (B.8)) close the surface in Fig. 10.

It is useful to define an additional line corresponding to the case where the equilibrium is perfectly centered on the inflexion point ($x_{c}=0$). We then have $T_{g,sat,0}=(T_H+T_L)/2$ and one can easily verify that $\psi =0$ as expected, from Eq. (B.8). Also, dividing $T_{g,sat,0}=(T_H+T_L)/2$ by $T_H$, we find that

$$\begin{aligned} \left( T_{g,sat,0}/T_H\right)&= \tfrac{1}{2} \left( T_L/T_H\right) + \tfrac{1}{2} \qquad \text {for } \psi =+\pi /2 , \end{aligned}$$

(B.9)

which corresponds to a function of the form $y=mx+b$ in the graph Fig. 10. Additional isolines (for constant $\psi$ values) can be added to the graph by solving Eq. (2.8c) for $T_{g,sat,0}/T_H$, we obtain

$$\begin{aligned} \frac{T_{g,sat,0}}{T_H}&= \left( \frac{1 + (\psi /\pi )}{2}\right) \frac{T_L}{T_H} + \frac{1 - (\psi /\pi )}{2} ,\nonumber \\ \end{aligned}$$

(B.10)

which clearly produce isolines of the form $y=mx+b$ in Fig. 10.

For $\psi =\pm \pi /2$, we have

$$\begin{aligned} \left( T_{g,sat,0}/T_H\right)&= \tfrac{3}{4} \left( T_L/T_H\right) + \tfrac{1}{4} \qquad \text {for } \psi =+\pi /2 , \end{aligned}$$

(B.11a)

$$\begin{aligned} \left( T_{g,sat,0}/T_H\right)&= \tfrac{1}{4} \left( T_L/T_H\right) + \tfrac{3}{4} \qquad \text {for } \psi =+\pi /2 , \end{aligned}$$

(B.11b)

which are also shown in Fig. 10.

Fig. 10

(Color online). $\psi /\pi$ as a surface function of parameters $T_L/T_H$ and $T_{g,sat,0}/T_H$; the lower right corner is not allowed because we must have $T_{g,sat,0}>T_L$ ; we find that $-1<\left( \psi /\pi \right) <+1$

1.3 Sign of $a_0$

Here, we evaluate the sign of $a_0$ in the general case where $\psi \ne 0$. First, consider the phase-change nonlinearity only (Appendix. B.3.1). For the pressure nonlinearity only, $a_0$ does not depend on $\psi$ and we have $a_0<0$ always. We then study the sign of $a_0$ for both nonlinearities (Appendix B.3.2).

1.3.1 Phase-change nonlinearity

For the phase-change nonlinearity only (${c_P}=0$, ${c_T}=1$), we have $a_0=a_{0,T}$ (Eq. (A.28)). We clearly have from Eq. (A.28) that $a_{0,T}> 0$ only if $1+{\zeta _{f}}^2 \cos \left[ \psi \right] <0$¹⁰. Two necessary conditions for having $a_{0,T}> 0$ are having ${\zeta _{f}}>1$ (however, ${\zeta _{f}}$ is usually much smaller than one) and having either $\psi <-\pi /2$ or $\psi >+\pi /2$, so that the equilibrium is far enough from the inflexion point. Solving $1+{\zeta _{f}}^2 \cos \left[ \psi \right] =0$ gives the curves¹¹${\zeta _{f}}= + \sqrt{-1/\cos [\psi ]}$ where $a_{0,T}=0$ and which delimits regions of different signs for $a_{0,T}$. Those boundaries are the same whatever the value of $T_{HL}$ is.

We plot those curves in Fig. 11, with ${\zeta _{f}}$ and $\psi$ as the axis. The region in the center corresponds to $a_{0,T}<0$. There are two regions where $a_{0,T}>0$ in the top corners, delimited by the curves ${\zeta _{f}}=+\sqrt{-1/\cos [\psi ]}$.

Fig. 11

(Color online). Boundaries for $a_{0,T}=0$ (Eq. (A.28)); $a_{0,T}<0$ in the blue region and $a_{0,T}>0$ in the orange region

Fig. 12

(Color online). Boundaries for $a_0=0$ (Eq. (A.28)), for both pressure and phase-change nonlinearities (${c_P}=1$, ${c_T}=1$), are computed numerically for increasing $T_{HL}$ for (a) to (f); $a_0<0$ in the blue region and $a_0>0$ in the orange region

1.3.2 Pressure and phase-change nonlinearities

We now compute the boundaries $a_0=0$ for both pressure and phase-change nonlinearities (${c_P}=1$, ${c_T}=1$) numerically. We find that they now depend on the value of $T_{HL}$ (see Fig. 12). From Eq. (A.28), we see that $a_{0,T}\propto 1/{T_{HL}}^2$, $a_{P,T}\propto 1/{T_{HL}}$ and $a_{0,P}$ is not a function of $T_{HL}$. Thus, for very small $T_{HL}$, the term $a_{0,T}$ dominates and the boundaries (Fig. 12a) look very much like the one for the phase change only (Fig. 11). As we increase $T_{HL}$, the boundaries move, the region for $a_0>0$ in the left corner moves to the right, the one on the right corner moves to the left and a new one appears in the bottom right (see Fig. 12a–c). The two regions for $a_0>0$ in the right eventually merge (Fig. 12c and d). As $T_{HL}$ is further increase, the $a_{0,P}$ terms starts to dominate, so the regions where $a_0>0$ must eventually disappear. This is what we observe as both regions for $a_0>0$ move out of the graph (Fig. 12d–f). The region for $a_0<0$ eventually takes all the available space for large $T_{HL}$.

Fig. 13

(Color online). a Vector field $\dot{r}(r)$ from $\dot{r}=d{\delta }r + a_0 r^3$ (Eq. (3.3a)), for $a_0=0$ or $a_0>0$. For $d{\delta }>0$ and $a_0>0$, the amplitude grows faster than in the linear case and do not saturate. For $d{\delta }<0$ and $a_0>0$, an unstable fixed point exists at $r= { r_{LC} }$, which corresponds to an unstable (repulsive) limit cycle in the $z_1$ and $z_2$ variables. b Vector fields $\dot{r}(r)$ including a negative cubic term; a stable limit cycle may exists

1.4 Dynamics for $a_0>0$

Let’s now discuss how the dynamics is affected by having a positive term $a_0$ in the vector field $\dot{r}=d{\delta }r + a_0 r^3$ (Eq. (3.3a)). For $a_0<0$, the function $\dot{r}(r)$ was bending downwards. Now, for $a_0>0$, the function $\dot{r}(r)$ curves upwards (see Fig. 13a), the nonlinearities are not saturating, but instead, tends to increase $\dot{r}$. The stability of the equilibrium point at $r=0$ is still controlled by ${\delta }$. Although the slope at $r=0$ is given by $d{\delta }$, the sign of the slope only depends on ${\delta }$ because $d>0$ always (so $d{\delta }>0$ and ${\delta }>0$ means the same thing as well as $d{\delta }<0$ and ${\delta }<0$). For $d{\delta }>0$, the amplitude grows faster than the linear case ($\dot{r}$ is larger for $a_0>0$) and without bounds, the system never reaches a limit cycle. For $d{\delta }<0$, the equilibrium is stable, but an unstable fixed point (corresponding to a unstable limit cycle in either the $z_i$ or the $q_i$ variables) exists at $r= { r_{LC} }$. For $r< { r_{LC} }$, the amplitude decreases toward $r=0$ and for $r> { r_{LC} }$, increases without bounds. As ${\delta }$ goes from positive to negative, the unstable limit cycle is created, a process called a subcritical Poincaré-Andronov-Hopf bifurcation.

Now, we have to keep in mind that the full vector field also includes higher order terms ($\mathcal {O}(r^5)$, see Eq. (A.26)). The equilibrium stability is still controlled by ${\delta }$. The truncated form $\dot{r}=d{\delta }r + a_0 r^3$ is sufficient to predict the limit cycle bifurcation but higher order terms are, however, required to predict the dynamics further. In Fig. 13b, we show how the vector fields $\dot{r}$ can look like when a negative quintic term is included. At small r, the vector fields looks like Fig. 13a, but differs qualitatively for larger r. At $d{\delta }$ negative and large enough, there is only a stable equilibrium (see $\dot{r}=-3r+r^3-0.2r^5$ curve). At $d{\delta }$ negative but small enough, there is one unstable limit cycle but also one larger stable limit cycle, due to the quintic term (see curve $\dot{r}=-r+r^3-0.2r^5$). At positive $d{\delta }$, there is now only one large stable limit cycle (see curve $\dot{r}=r+r^3-0.2r^5$). Thus, adding a quintic term may prevent the amplitude from growing without bounds, by creating a stable limit cycle surrounding the unstable limit cycle. Let’s note that there is still the creation of an unstable limit cycle as ${\delta }$ becomes negative, through a subcritical Poincaré–Andronov–Hopf bifurcation. However, there is an additional, global, bifurcation that can occur. For $d{\delta }$ negative and sufficiently small (curve $\dot{r}=-3r+r^3-0.2r^5$), increasing $d{\delta }$ leads to the creation of an unstable limit cycle and a larger stable limit cycle (curve $\dot{r}=-r+r^3-0.2r^5$). This process is called a a saddle-node bifurcation of cycles (see Strogatz [44, Subcritical Hopf Bifurcation p.251, Saddle-node Bifurcation of Cycles p.261]).

Fig. 14

(Color online). Oscillation amplitude r (based on the amplitude in $q_1$) from numerical simulations of Eq. (2.14), for various initial conditions and for ${c_P}={c_T}=1$, ${\zeta _{f}}=1.5$, $T_{HL}=0.1$ and $\psi =0.9\pi$. The results show similar behavior as in Fig. 13b. a $\Pi =0.95$ (so $d{\delta }=-0.0231$), for $r(\tau =0)$ small enough r decreases toward 0, but for $r(\tau =0)$ large enough, r increases or decreases toward a constant value. This suggests the existence of a small unstable limit cycle surrounded by a larger stable limit cycle. b $\Pi =1.05$ (so $d{\delta }=+0.0231$), r increases or decreases toward a constant value. This suggests the existence of a stable limit cycle

The behavior we observe numerically for $a_0>0$ qualitatively corresponds to the behavior with a negative quintic term described above. Figure 14 shows numerical simulations of Eq. (2.14) for various initial conditions. In Fig. 14a, $d{\delta }$ is small and negative, the dynamics suggest the existence of a stable equilibrium, an unstable limit cycle and a larger stable limit cycle, just like the curve $\dot{r}=-r+r^3-0.2r^5$ in Fig. 13b. In Fig. 14b, $d{\delta }$ is positive, the dynamics suggest the existence of a unstable equilibrium, and one large and stable limit cycle, just like the curve $\dot{r}=r+r^3-0.2r^5$ in Fig. 13b.

Let’s now provide a few final remarks on the vector field with a negative quintic term as shown in Fig. 13b. If the system behaves as the curve $\dot{r}=-r+r^3-0.2r^5$, the unstable limit cycle acts as a barrier: Self-oscillations are possible even though the equilibrium is stable, but an initial push beyond the unstable limit cycle is needed. Also, the system described by the vector field with a negative quintic term has an hysteresis: As $d{\delta }$ is increased above 0, large oscillations starts (given a small perturbation) but can only be turned off by bringing $d{\delta }$ below some negative value. The described behavior with a negative quintic term could be made possible by other nonlinearities than the one considered in this paper. For example, having a phase change which suddenly increases when the amplitude is large enough could produce such behavior.