What are the necessary qualitative criteria for the existence of limit cycles

Question

The literature on the existence of limit cycles is replete with complex and abstract mathematical analysis. Or for specific systems of specific structure, specific mathematical criteria that determine the existence or non-existence of limit cycles. But (for many years) I've not been able to find any references that address, in more general, qualitative (non-mathematical) language, what minimum physical elements are necessary in a system for a limit cycle to exist.

My thoughts are a system must be structured (1) to contain at least two different states in which energy can be stored and which is able to flow periodically between them, and (2) to contain a non-linearity that facilitates regular orbits without either growth or decay and (3) sufficient initial amount of energy or else constant net energy such that input balances loss.

Nonlinearity is a key element since you cannot practically realize an oscillator without decay or growth.

Are these the essential key elements or can the necessary elements be described more in a more fundamental way?

Anyone know of references that treat this question?

Answer 1

Your (1) and (3) are along the right lines, but I'm not sure what (2) really means and some limit cycles don't look much like "regular orbits" anyway.

There is always a danger in trying to produce a very general explanation of something - hence the (probably apocryphal) quote "This theory is so perfectly general that no particular application of it is possible" - but let's try anyway ;)

You can state a very general and useful way to think about limit cycles like this. It is not completely "qualitative" since it needs some way to "measure" what is going on in the system - but in some sense, if you can't measure anything about the system, you can't describe its behaviour at all!

First, identify some single parameter that measures the "amplitude" of the system's behaviour.

The "amplitude" could measure things like displacement, velocity, gas pressure, etc, in a mechanical system, a voltage or current in an electrical system, the concentration of some component of a chemical reaction, etc, etc. It could also measure something that is not what an engineer or physicist might call an "amplitude" - for example the frequency of vibration of a system. (Consider questions about limit cycles like "why does the sound of brakes squealing have (almost) a constant pitch, independent of the speed of the vehicle"?)

Then, describe how the amount of "stuff" going into and out of the system depend on the "amplitude". I use the word "stuff" rather than "energy" because this might not be exactly what a physicist would describe as "energy" - though often it will be exactly that. In some systems, it might be literally "physical stuff," like the rate of flow of a material.

Now, plot two lines on a graph, showing the amount of "stuff going in" and "stuff coming out" of the system, as functions of the "amplitude".

The two lines might be any shape of curve, depending on the particular system, but to get a limit cycle they must have these three properties:

• the graphs need to intersect at some point

• for amplitudes smaller than the intersection point, more "stuff" is going in than is coming out

• for amplitudes above the intersection point, more "stuff" is coming out than is going in.

In fact the graphs might have more than one intersection point, which can describe complex behaviour and several different limit points with different amplitudes.

At any particular amplitude, if "stuff out" is bigger than "stuff in", the amplitude will tend to decrease over time, but if "stuff out" is less than "stuff in", the amplitude will tend to increase. Limit cycles can occur at the points where the two flows of "stuff" are equal.