To help you better understand the describing functions, some of the ideas in chapter 14 of Glad and Ljung [1] have been implemented in Matlab (2020a or 2020b) and Simulink. Here you can investigate the behavior of a system

$LaTeX: G\left(s\right)\:=\:\frac{K}{s\left(s+1\right)²}$

for various LaTeX: K>0 , with a static nonlinearity LaTeX: f acting on the input (see Figure 14.1 in [1]). In the software, you will find a set of examples that implement the various nonlinearities discussed in [1]. These are

Example 2: is an ideal relay
Example 3: is a cubic function
Example 4: is a relay with a deadzone
Example 5: is a saturation
Example 6: is a pure deadzone
Example 7: is a deadzone with hysteresis

The code for these examples can be found here, and example 7 generates the figure below.

[1] Reglerteori: flervariabla och olinjära metoder, Glad, Torkel and Ljung, Lennart, 2003, Studentlitteratur.