I am working on the following delay differential equation

$$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$
where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$.

I know that there are three equilibrium points to equation (1) by solving
$$0=(1-\alpha)f-f^3\tag{2},$$
namely $f=0 \vee f=\pm\sqrt{1-\alpha}$.

I am however specifically interested in oscillatory solutions of this equation (close to and above the [neutral curve](https://math.stackexchange.com/questions/847091/how-to-obtain-asymptotes-of-this-curve)).

My question is about the maximum value that $f$ can obtain for such a periodic solution, for given $\alpha$ and $\delta$. Basically I want to determine the amplitude of this oscillation. **Is there a method (or approximation), that allows me to determine the maximum value of $f$?**

Some of my thoughts:

I understand that the maximum value is at least $f>\sqrt{1-a}$, and from some numerical solutions I suspect that the maximum value is about twice this equilibrium value. However, I would like to have a more informed answer.

For example, for $\alpha=\frac{3}{4}$ and $\delta=2$, I numerically obtained the solution for different initial conditions $y\_0$, and, as long as the initial condition is not the equilibrium point, it ends up in the same stable oscillation mode.

![Example of delayed differential equation](https://i.stack.imgur.com/YR3xs.png)

After finding the there is some bound present (see my incomplete answer below), I numerically investigated the behavior along the neutral curve:

$$ \delta=\frac{\arccos\left(\frac{3\alpha-2}{\alpha}\right)}{\sqrt{\alpha^2-(2-3\alpha)^2}},$$

and in the below figure I plotted the thus obtained $f\_{max}$ as a function of $\alpha$. (Please be aware that the results are numerically obtained, and I did not carefully analyze the accuracy yet).

![Maximum function value wrt alpha](https://i.stack.imgur.com/38f1V.png)

Apparently, I found for the maximum function value the following relation

$$f\_{max}=\sqrt{2(1-\alpha^2)},$$

which fits perfectly between $\sqrt{1-\alpha}$ and $\sqrt{1+\alpha}$ on the given domain for $\alpha$. This cannot be a coindicent, can it?