# What is the meaning of “smeared limit cycle”?

I'm reading the paper Phase dynamics of coupled oscillators reconstructed from data by Kralemann et. al. (2008), which is about representing phenomena that exhibit a stable limit cycle (i.e. non-linear systems with a periodic orbit or “limit cycle”, where nearby trajectories decay into the limit cycle) as a generalised phase oscillator. They make reference to a smeared limit-cycle, and I’m not sure what they mean with smeared. Can anyone explain this?

## 1 Answer

From the introduction of said paper (boldface mine):

For the rest of the paper we restrict our consideration to the case of weakly and moderately coupled noisy-chaotic oscillatory systems in the asynchronous regime—i.e., to the case when the amplitudes are enslaved and the phase portrait of the individual system resembles a smeared limit cycle. In this case, the coupled system can be adequately described by the phase variables.

A system whose dynamics can entirely be described by one phase variable can also be described by a limit cycle. Now, the basic premise of this paper is that the respective systems can still be adequately (though not fully) be described by one phase variable. This is what the authors denote as a smeared limit cycle.

With other word, through coupling, noise or chaos, there exist possible states that are slightly off the limit cycle, which makes the set of possible states of the system slightly more than a simple limit cycle, i.e., a smeared limit cycle. What is important for the purposes of the paper is that the coupling mainly affects the system along the limit cycle, i.e., it causes a phase offset. This assumption would not hold true anymore if the deviations from a limit cycle were stronger and the dynamics could not be adequately described by a simple phase anymore (but you would need two or more variables).

• So just to hammer this home, the smear refers to the offsets that are transverse to the limit-cycle, which are assumed to have good stability/convergence properties, and thus we can limit ourselves to analysing the effect that the coupling has on the phase, i.e. roughly parallel to the limit-cycle. Correct? – Steve Heim May 16 '16 at 9:44
• @SteveHeim: Correct. – Wrzlprmft May 16 '16 at 9:50