In “Analyze This!“, we examined system
Now, let us seek its equilibrium quantitatively.
In theory, one may first solve differential equation
for , using a popular symbolic differential equation solver such as ‘ode2’. Then compute as , followed by .
However,in practice, such attempt meets a deadend rather quickly (see Fig. 1).
Bring in a more sophisticated solver is to no avail (see Fig. 2)
An alternative is getting directly from polynormial equation
We can solve (1) for if . For example, when , has three roots (see Fig. 3).
First two roots are complex numbers. By Descartes’ rule of signs, the third root
is the of equilibrium (see Exercise-1).
Exercise-1 Show the third root is real and positive.
Exercise-2 Obtain from