Appendix: Derivation of the slow flow equations
In this appendix, we describe the important mathematical steps involved in the derivation of Eqs. (8)–(11). As usual, we expand the solution u, v in a perturbation series,
$$\begin{aligned} u\left( \epsilon ,t\right)= & {} u_0\left( \tau ,\eta \right) + \epsilon u_1\left( \tau ,\eta \right) +\mathcal O\left( \epsilon ^2\right) \,,\nonumber \\ v\left( \epsilon ,t\right)= & {} v_0\left( \tau ,\eta \right) + \epsilon v_1\left( \tau ,\eta \right) +\mathcal O\left( \epsilon ^2\right) \,, \end{aligned}$$
(23)
with the fast time scale \(\tau \) and the slow time scale \(\eta \) given by,
$$\begin{aligned} \tau = t \,,\quad \eta = \epsilon t\,. \end{aligned}$$
(24)
The time derivative operator becomes
$$\begin{aligned} \frac{{\mathrm {d}}}{{\mathrm {d}}t} = \frac{\partial }{\partial \tau } + \epsilon \frac{\partial }{\partial \eta }\,. \end{aligned}$$
(25)
Equations (23)–(25) are then substituted into the rescaled version of Eqs. (3)–(4). Balancing the \(\mathcal O\left( 1\right) \) terms yields,
$$\begin{aligned}&\frac{\partial ^2 u_0}{\partial \tau ^2}+ {\omega _1^2}u_0= 0\,, \end{aligned}$$
(26)
$$\begin{aligned}&\frac{\partial ^2 v_0}{\partial \tau ^2} + 9{\omega _1^2}v_0 = 0\,. \end{aligned}$$
(27)
The corresponding \(\mathcal O\left( \epsilon \right) \) equations read as follows:
$$\begin{aligned} \frac{\partial ^2 u_1}{\partial \tau ^2}+ {\omega _1^2}u_1= & {} -2\frac{\partial ^2 u_0}{\partial \tau \partial \eta } - \frac{\gamma }{2}\left( u_0+v_0\right) ^3 + f_u \,,\end{aligned}$$
(28)
$$\begin{aligned} \frac{\partial ^2 v_1}{\partial \tau ^2}+ 9{\omega _1^2}v_1= & {} -2\frac{\partial ^2 v_0}{\partial \tau \partial \eta } - \frac{\gamma }{2}\left( u_0+v_0\right) ^3 + f_v - 2\delta \frac{\partial v_0}{\partial \tau } - 6{\omega _1}\sigma _2v_0\,. \end{aligned}$$
(29)
The general solution of Eqs. (26)–(27) is \(u_0 = A(\eta ){\mathrm {e}}^{{\mathrm {i}}{\omega _1}\tau }+\mathrm {c.c.}\) and \(v_0 = B(\eta ){\mathrm {e}}^{3{\mathrm {i}}{\omega _1}\tau }+\mathrm {c.c.}\). Substituting this into Eqs. (28)–(29), and requiring the secular terms to vanish, yields
$$\begin{aligned} 0= & {} -2{\mathrm {i}}{\omega _1}A^\prime - \frac{3\gamma }{2}\left[ A^2\overline{A}+\overline{A}^2B+2B\overline{B}A\right] +F_u{\mathrm {e}}^{{\mathrm {i}}\sigma _1\eta }\,,\end{aligned}$$
(30)
$$\begin{aligned} 0= & {} -6{\mathrm {i}}{\omega _1}B^\prime - \frac{\gamma }{2}\left[ A^3+3B^2\overline{B}+6A\overline{A}B\right] -6{\omega _1}\left( \sigma _2+{\mathrm {i}}\delta \right) B \,. \end{aligned}$$
(31)
Herein, \({}^\prime \) denotes derivative with respect to \(\eta \). It should be noted that the term \(F_v\) does not occur in these equations. Therefore a weak fundamental harmonic forcing of the out-of-phase mode has only second-order effects. Without loss of generality of the subsequent investigations, we assume \(F_u=F>0\) as a positive real-valued quantity in the following.
We introduce polar coordinates \(A=a_1{\mathrm {e}}^{{\mathrm {i}}\left( \sigma _1\eta +\beta _1\right) }\) and \(B=a_2{\mathrm {e}}^{3{\mathrm {i}}\left( \sigma _1\eta +\beta _2\right) }\) with the real-valued quantities \(a_1\), \(a_2\), \(\beta _1\) and \(\beta _2\). Substitution into Eqs. (30)–(31) and separation of real and imaginary part finally gives rise to the slow flow equations (8)–(11).