According to Eguíluz et al, amplification of an incoming audio signal by the human ear can be described by a supercritical Hopf-bifurcation

$$\dot x(t) =(\lambda-i\omega_0-|x(t)|^2)x(t)+S(t),$$

where $\lambda<0$ is the bifurcation parameter located close to the bifurcation point (i.e. $\lambda\approx0$), $\omega_0>0$ is the Hopf frequency of a particular hair cell in the cochlea, and the non-homegeneous term $$S(t)=F\cdot(\cos \omega t+i\sin\omega t)$$ represents the incoming signal with amplitude $F>0$ and frequency $\omega\approx\omega_0$. One can show that for $F$ not too small $$x(t)\approx F^{1/3}\cdot(\cos \omega t+i\sin\omega t).$$ So far, so good.

However, I am a bit puzzled by the fact that both the input $S$ and the response $x$ are complex variables, and hence correspond to two-dimensional real signals. I would intuitively expect a simple audio signal to be just of the real/one-dimensional form $F\cos\omega t$. Since no different frequencies or different amplitudes are involved, I do not think that this additional dimension is due to overtones. But how am I to interpret it?

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    $\begingroup$ @ZaellixA If we view $\Bbb C$ as a real vector space, then it is isomorphic to $\Bbb R^2$. In this sense, a complex function can always be interpreted as a two-dimensional real function. Thanks for your comment, though. I will make the question more precise. $\endgroup$ Feb 29 '20 at 15:47
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    $\begingroup$ Yep, thanks for that. As I said I am no expert and so far I have used complex numbers in audio just as a means of easing out calculations and formulation. In general, you always end up with taking real parts of every complex quantity when you want to find out the actual quantities of interest. By using complex numbers you can quite easily find out the effect a system has on amplitude and phase without considering trigonometric functions. I believe that this is the case here too. Kinda the same as considering a complex force function in vibrating systems. $\endgroup$
    – ZaellixA
    Feb 29 '20 at 15:55
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    $\begingroup$ I am not familiar with this topic, but it is very common in studying waves (pressure, light, etc.) to use complex functions instead of real ones, with the intention of taking the real part at the end of the calculation to get the physical answer. $\endgroup$
    – KF Gauss
    Feb 29 '20 at 15:55
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    $\begingroup$ @KFGauss Thanks, I almost thought something like this. So I should think of the actual input being $F\cos \omega t$ and the output being $\approx F^{1/3}\cos \omega t$? The imaginary part is just there to make the model work? For the sinusoidal input, it is just a phase-shifted version of the real part, anyway, so it does not seem like actual additional information. $\endgroup$ Feb 29 '20 at 16:00
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    $\begingroup$ @ZaellixA Yes, thank you! The phase shift $\phi$ is indeed present here anyway, but it goes to zero as $\omega$ approaches $\omega_0$. $\endgroup$ Feb 29 '20 at 16:17

The equation $$\dot z = (\mu +\mathfrak{j}\omega_0)z-|z|^2z+Fe^{\mathfrak{j}\omega t} \tag {1}\label {1}$$ from Eguiluz, etc., is one of the simplest model of entrainment, that is of a phase lock "loop".

The function $z_1(t)=A \, e^{\mathfrak{j}\omega t}$ for some $A$ is a stationary solution of $\eqref{1}$ since $|z_1|^2=|A|^2$ and $\dot z_1 = A \mathfrak{j}\omega\, e^{\mathfrak{j}\omega t}$, therefore $A\mathfrak{j}\omega e^{\mathfrak{j}\omega t}= (\mu +\mathfrak{j}\omega_0) A e^{\mathfrak{j}\omega t} -|A|^2Ae^{\mathfrak{j}\omega t}+Fe^{\mathfrak{j}\omega t}$ and upon dividing both sides with the exponential $e^{\mathfrak{j}\omega t}$ we get $$A\mathfrak{j}\omega = A(\mu+\mathfrak{j}\omega_0 - |A|^2) +F \tag{2}\label{2}$$ This equation $\eqref{2}$ is a complex cubic in the unknown complex amplitude $A$, and can be solved given $\mu, \omega, \omega_0$ and $F$.

If we set the time reference so that $F$ be real then the phase of $A$ represents the phase delay of the entrained oscillation $z_1(t)=Ae^{\mathfrak{j}\omega t}$ of complex amplitude $A$ relative to that of the driving force.


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