2

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.

Conifold
  • 11,756
dtn
  • 731
  • 1
  • 6
  • 19
  • I will supplement the question. Period $T$ can also be a parameter of the limit cycle. – dtn May 25 '20 at 07:56
  • 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. https://math.stackexchange.com/q/1564464/115115 for an exhaustive answer (the accepted one) on the periods of the Van der Pol oscillator. – Lutz Lehmann May 25 '20 at 10:18
  • 1
    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 '20 at 10:25
  • I answered your question, or is there something you need to clarify? – dtn May 25 '20 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 $\mu$. – Lutz Lehmann May 25 '20 at 10:33
  • I agree that for the existence of the limit cycle, a certain structure of the differential equation is needed. Let's simplify the issue by restricting ourselves to the class of just such equations. The method that you propose is just right for a particular class of equations. If we slightly modify its structure: add degrees to state variables, introduce complex periodic signals, then we need to check whether this method will work. – dtn May 25 '20 at 10:41
  • Did you heard about Lyapunov Quantities and Limit Cycles? Have a look at https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwjxr4-t787pAhUSyIUKHRjGAl4QFjAAegQIARAB&url=https%3A%2F%2Fjyx.jyu.fi%2Fbitstream%2Fhandle%2F123456789%2F36972%2F9789513944957.pdf&usg=AOvVaw2mnckuUJ4JpeGHp6s2iUSh – Cesareo May 25 '20 at 11:07
  • The only "general method" that comes to mind is Lyapunov functions. It would work straightforwardly for something like $\ddot{x}+\mu (x^2+\dot{x}^2-1)\dot{x}+x=0$ because $V(x,\dot{x})=x^2+\dot{x}^2$ decreases along the trajectories for $V>1$ and increases for $V<1$. For van der Pol you'd have to craft a coercive $V$ that decreases when $V>C$ to estimate the outradius, but that's an art. Such Lyapunov functions for limit cycles are discussed in Haddad's Nonlinear Dynamical Systems, for example. – Conifold May 25 '20 at 20:12
  • But what about a bigger number of variables, for example 3 or more? How to study the limit cycles of such systems? – dtn May 25 '20 at 20:29
  • The number of variables doesn't really matter, you just need a function that decreases along trajectories outside of some annular region. Of course, the difficulty of finding a suitable function increases with the number of variables. But if we interpret "analytical method" more strictly, as a clear heuristic that prescribes some more or less definite steps that lead to an estimate in most cases, the answer is no, there is no such thing. Localization problem for limit cycles is unsolved even for polynomial equations in 2D. – Conifold May 26 '20 at 05:32
  • Then, maybe you should pay attention to analytical approximation? Have you ever met the HAM (homotopy analysis method) developed by Liao?

    [http://prof.khuisf.ac.ir/images/Uploaded_files/Liao-Homotopy4245293-03.PDF

     – dtn May 26 '20 at 06:28

0 Answers0