Comparabilty of chaotic behaviour

[AHaumer]

During regression testing we had a short discussion about comparabilty of chaotic behaviour - if I remember correctly, Modelica.Electrical.Analog.Examples.ChuaCircuit. Here we decided to shorten the interval for comparison to exclude chaotic behaviour. In my opinion it is possible in many cases to name parameterization for periodic behaviour which can be compared very well and different parameterization to demonstrate chaotic behaviour. I have implemented a small library ChaoticCircuits to demonstrate Chuas ciruit and three other examples from literature. They run fine in OM and Dymola. I provided comparisonSignals as well as referenceResults in subdirectory Resources/Reference.
@maltelenz could you compare using WSM?
@casella what's your opinion after having a look at the library?
I would suggest to include these models into Modelica.Electrical.Analog.Examples.

[maltelenz]

@AHaumer WSM gets different results for the models in ChaoticCircuits.ColpittsCircuit. Here is a plot of iL at the beginning of AnalyticEquations:

Or at the end:

Similar differences are present in the other variables, and in the IdealCircuit and ImprovedCircuit.

For ChaoticDiode, ChuasCircuit and ChaoticOscillator, the results from WSM match the reference results.

[AHaumer]

Thanks a lot @maltelenz perfect! this shows that {ChaoticDiode, ChuasCircuit, ChaoticOscillator} can be parameterized such way to achieve periodic behaviour, even if you might have to continue the simulation as shown for ChaoticOscillator.AnalyticEquations:

I plotted y w.r.t. x.
Changing the parameterization, we reach chaos. This means that small differences in initial conditions lead to dramatically different results. An of course, only slightly different implementation of the solver also has the same consequences. Therefore regression testing and comparison between tools is only possible for periodic behaviour - or very short duration of simulation.

I believe I have to investigate Colpitt's circuit a bit more to find really periodic behaviour - with the current parameterization chaos is not very visible but present (note the thickened line):

BTW for these diagrams I used different initial conditions; v1.start=5.65 V, v2.start=0.72V, iL.start=0 A.

It seems that the "periodic" results of all examples tumble within a narrow band, but ColpittsCircuit seems to be more sensitive.

[AHaumer]

See also discussion in #4577

[HansOlsson]

I would prefer to keep the models chaotic (if they are intended for that), and adapt the comparisons to only run them a short while.

Obviously one can imagine more advanced solutions - but to me:

• The examples are intended as examples, thus an example of a chaotic system should show chaotic behavior.
• The comparisons are intended to ensure that models don't misbehave too much; we cannot expect bit-equality in simulation results; so only comparing the first part will be good enough to detect that.

Obviously we can do more, but trying to find more stable solution will be a fine balancing act (and it's not clear that periodic solutions are inherently more stable than chaotic ones for these models (*) ) and I think the effort should be spent elsewhere. Comparing the stable attractors themselves is also possible, but that would require a completely different comparison tool - and I don't see the benefit for MSL-development in setting up that.

*: The issue is the sensitivity of the trajectory (for one period in a periodic system). Generally the sensitivity varies during the trajectory, so it may be that the periodic solutions are slightly less sensitive than chaotic ones (as the chaotic ones often pass through some very unstable parts), but I doubt that the periodic solutions are actually stable..

[AHaumer]

@HansOlsson there are oscillators designed for stable periodic behaviour.
However, some scientists investigate them how to obtain chaotic behaviour to exploit chaos in various sciences, e.g. cryptograhpy.

[HansOlsson]

@HansOlsson there are oscillators designed for stable periodic behaviour. However, some scientists investigate them how to obtain chaotic behaviour to exploit chaos in various sciences, e.g. cryptograhpy.

I can understand that it varies, e.g., the Chua circuit is seen as an example of chaotic behavior, whereas the Colpitt oscillator was designed to generate a certain frequency. To me the settings in MSL should match that: so that if you run the Chua circuit you should get chaotic behavior whereas the Colpitt oscillator should generate an oscillation with a specific frequency.

Looking more at the graph of the Colpitt oscillator above my first guess is that it isn't chaos, but a superposition of a periodic solution and a very slow transient (explaining why you get Moiré pattern for large stretches; something that would seem odd for chaos).

[AHaumer]

I adapted a little bit the original Modelica.Electrical.Analog.Examples.ChuaCircuit, comparing the results with Dymola2025x and OpenModelica v1.25.0:
For periodic behaviour the maximum relative deviation is about 0,5%. Is this good enough for comparability?
Changing one parameter (R=1500 Ohm instead of R=1900 Ohm) reveals chaotic behaviour. Yes the results run apart after short time.
We could even reach a steady-state solution with constant states using R=2000 Ohm.
Here are some small differences during initial transients but both reach the same terminal states.

I have attached the model and a reference result obtained for period behaviour with Tolerance=1e.7,
@maltelenz may I kindly ask you to have a look how close WSM's results are to the reference?

ChuaCircuit.zip

[HansOlsson]

I adapted a little bit the original Modelica.Electrical.Analog.Examples.ChuaCircuit, comparing the results with Dymola2025x and OpenModelica v1.25.0: For periodic behaviour the maximum relative deviation is about 0,5%. Is this good enough for comparability? Changing one parameter (R=1500 Ohm instead of R=1900 Ohm) reveals chaotic behaviour. Yes the results run apart after short time. We could even reach a steady-state solution with constant states using R=2000 Ohm. Here are some small differences during initial transients but both reach the same terminal states.

Ah, so different parameters - not just different starting point.

As indicated I don't think that makes sense as example, as we would have a big explanation saying "this is normally seen as a chaotic circuit; but it is not chaotic for the given parameters, to get chaos you have to set these parameters...".
That would be a step backwards - an example should demonstrate things as default, not requiring modifications.

Basically: I view examples as examples that as default should demonstrate something when you just simulate it, not primarily as testing models.

[HansOlsson]

BTW: After simulating Colpitt oscillator I feel confident to say that it is not chaotic for those parameters, it's just a transient (and depending on tolerance, numerical noise) that keeps if from a perfect periodic solution.

[AHaumer]

For me a compromise looks fine: Default parameters for periodic behaviour, special parameters for chaos.
Anyhow it's interesting to study the travel into chaos.
Regarding Colpitt:
M.P. Kennedy reported 1994 Chaos in the Colpitts oscillator
Buscarino describes it in his Concise Guide to Chaotic Electronic Circuits (Chapter 1.3)
Maybe I have to work on parameterization ...

[AHaumer]

I compared the analytic equations of Chua's Circuit and Colpitt's Oscillator, plotting the phase diagram v2 vs. v1:
Chua with R=1900 Ohm (periodic) and R=1500 Ohm (chaotic):

Looking a bit closer: Chuas goes between 1900 and 1800 Ohm with bifurcation from periodic to chaotic.
Colpitt with RE={50, 100, 200, 300, 500, 2000, 5000} Ohm:

Looks like bifurcation with a chaotic phase between 100 and 300.

[maltelenz]

I adapted a little bit the original Modelica.Electrical.Analog.Examples.ChuaCircuit, comparing the results with Dymola2025x and OpenModelica v1.25.0: For periodic behaviour the maximum relative deviation is about 0,5%. Is this good enough for comparability? Changing one parameter (R=1500 Ohm instead of R=1900 Ohm) reveals chaotic behaviour. Yes the results run apart after short time. We could even reach a steady-state solution with constant states using R=2000 Ohm. Here are some small differences during initial transients but both reach the same terminal states.

I have attached the model and a reference result obtained for period behaviour with Tolerance=1e.7, @maltelenz may I kindly ask you to have a look how close WSM's results are to the reference?

ChuaCircuit.zip

WSM results are extremely close to the reference. I didn't check exhaustively, but some manual sampling towards the end were within 0.3%.

[AHaumer]

Thanks a lot @maltelenz I appreciate your support!
This proves: At least this model can be compared with parameterization for periodic results, and bring chaotic results with changing one parameter. On the path from periodic to chaotic behaviour one could even see different modes (bifurcation) - this parameter variation has to be done by the user.