I am reading Introduction in Nonlinear Dynamics for Physicists and I am confused by the section on annihilation and creation operators being used in analyzing oscillators.
When analyzing the equations
$\frac{dx}{dt} = y$
$\frac{dy}{dt} = -x + \mu[-q(x)y+f(x)]$
They introduce the complex variable $a(t)$ and make the substution:
$x(t)=a(t)*e^{it}+a^**e^{-it}$
$y(t)=i*a(t)*e^{it}-i*a^**e^{-it}$
They then make the substitution and have terms like $q(a,a^*,e^{\pm it})$
I have two questions:
Why is this substitution made?
What does the notation $q(a,a^*,e^{\pm it})$ mean?