# Analytic estimates of limit cycle parameters

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.

• I will supplement the question. Period $T$ can also be a parameter of the limit cycle.
– dtn
May 25, 2020 at 7:56
• Are you interested in the perturbation analysis where $\mu$ is small, or in the slow-fast dynamic you get when $\mu$ is large? See my comments to this question and the links in them for some partial answers, and esp. math.stackexchange.com/q/1564464/115115 for an exhaustive answer (the accepted one) on the periods of the Van der Pol oscillator. May 25, 2020 at 10:18
• I will look at these links. I construct various differential equations, not only Van der Pol equations, but also others where a limit cycle exists (for example, the Duffing Oscillator). I am not interested in the answer to the question: "the existence and decay of the limit cycle, depending on the value of the parameters." I'm interested in ways of more or less universal estimation of the amplitude and frequency of the limit cycle, which is suitable for a wide class of differential equations.
– dtn
May 25, 2020 at 10:25
• I answered your question, or is there something you need to clarify?
– dtn
May 25, 2020 at 10:25
• This can be seen as easy or complicated, often it is both. Most "random" ODE systems do not have limit cycles. They may have solutions that look for some while like a limit cycle but in the end this again deteriorates (like in the Lorenz attractor, or even with less structure). To have a guaranteed limit cycle needs a strong structural reason. This reason then often also suggests a nearby system with a computable cycle so that the wanted system can be treated as perturbation of it, as was done is the cited link with separate approaches for small and large $\mu$. May 25, 2020 at 10:33