Suppose we have a two-dimensional system of differential equations, say, the well-known Van der Pol oscillator:
$$ \dot{x}=y, \dot{y}=\mu (1-x^2)y-x $$
Everyone knows that the study of limit cycles is a very complex problem. Each of them is unique in its own way, and there is no universal set of parameters characterizing each of them. As I understand it, in most cases the limit cycles are studied by numerical and graphical methods.
Are there approximate analytical methods that allow at least an average estimation of the amplitude and frequency of the limit cycle (for complex limit cycles, these concepts are very vague)?
Let me explain what I mean by the amplitude and frequency of limit cycles. The limit cycle of the Van der Pol oscillator has a very characteristic shape, therefore, parameters such as amplitude and frequency are not applicable to it. On the other hand, the amplitude can be considered the radius of the circle beyond which the limit cycle does not extend, and the frequency is the number of complete passage along the path of the limit cycle per second.