Why only odd harmonics after non-linear amplification? Why do I get only odd harmonics appearing? This not only happens at saturation of the transfer curve, but also in the linear portion of the curve.

I make a input signal that has too high power that it will saturate the transfer curve. The transfer curve below is for Amplitude modulation, I also have a transfer curve of the output phase change for each input signal level.

This is basically how I do it

My input signal, x(t), is the sum of two sine waves


For each point on the transfer curve above I apply Amplitude and Phase modulation by using this array

modulating_phasor[] = (power_out / power_in) * exp(i*phase_deviation)


Then I look at my input signal and lookup where its value matches the input power of the transfer curve and use that location in the modulating_phasor array

output_signal[] = x[t] * modulating_phasor[location]


This makes the output signal complex array so I take the absolute value and for parts that should be negative, I multiply that location on by -1 by finding the correct location on input signal. 

I cant upload my code... but you get the idea of what I am doing right? 

Now this is my output and no matter what my input level on my signal is i.e. whether I am in the non-linear part or linear part, I always get odd harmonics. I've also tried a different transfer curve. Only get odd again.
An ideal amplifier would have a transfer characteristic of $f(x)=Ax$: the input signal comes out amplified and otherwise undistorted. A real amplifier will deviate from this and go into saturation. We could model it by a polynomial $f(x) = \sum_{n} a_n x^n$. Now, what we would still expect is that the amplifier treats positive and negative values the same way, i.e., it is symmetric in the sense that $f(-x) = f(x)$. This leads to the requirement $a_n = 0$ for all even $n$. 

Now, one can show that raising a sinusoidal with frequency $f_0$ to the $n$-th power generates harmonics up to order $n$, however, preserving their oddity:


$n = 2$ generates frequencies 0 and $2f_0$
$n = 3$ generates frequencies $f_0$ and $3f_0$
$n = 4$ generates frequencies 0, $2f_0$ and $4f_0$
$n = 5$ generates frequencies $f_0$, $3f_0$ and $5f_0$


and so on. It's a consequence of the binomial theorem and quite easy to show (see, e.g., here).

In short: A transfer function that is odd symmetric will generate only odd harmonics. This is why you are seeing only those.