Chua's oscillator circuit -- Intuitive picture

[Swamp Thing]

I have been trying to understand intuitively why Chua's circuit should want to toggle between modes as it does.

The aim is to internalize this concept the way a Steve Mould or a 3Blue1Brown might help me do

Which is closer to the truth:
(A) Rectification of the signal around one bias point builds up a DC shift, which increasingly pushes towards another bias point, and vice versa
(B) Like an actively "de-damped" pendulum on top of a mini rocking chair, with the "rocking arcs" shaped so as to be metastable when vertical

 

[Baluncore]

Which is closer to the truth:

How you conceptualise Chua's oscillator, will depend heavily upon your background, or the field from which your approach.

 

[Filip Larsen]

Since chaos cannot be present without instabilities (e.g. a positive max Lyapunov exponent along the trajectories in the strange attractor) view (B) sounds more directly closer to the mathematical definition of chaos. On the other hand, (A) sort of describes a positive feedback mechanism that "spools up", so assuming that is how the electronics in Chua's circuit actually "work", then this may be just as valid as a practical description of the flipping between the two regions (I have ever only worked on the equations, never with the actual circuit, so I am biased towards the analysis of the equations).

 

[tech99]

As a radio engineer, it seems to me to resemble some behaviour which was well known by 1910. If we have two resonant circuits of the same frequency that are coupled together closely, then if we apply an initial impulse, damped oscillation will occur at one of two frequencies at random.

 

[Swamp Thing]

How you conceptualise Chua's oscillator, will depend heavily upon your background, or the field from which your approach.

Yes, I guess I was framing my question like, "Is a needle more like a burger or more like a soprano?" One could make a case for both by stretching each analogy enough.

 

[sophiecentaur]

I am trying to relate charges on those two capacitors to the populations of Foxes and Rabbits in the first situation which was presented to me as chaotic.

 

[Baluncore]

Here is a simulation in LTspice using real electronic components.

 

[sophiecentaur]

real

Could you drop one of those on your foot?

 

[Baluncore]

Could you drop one of those on your foot?

Yes. Every component part is available commercially, so you can build it, and it will work. There are no tricks being performed by LTspice with controlled sources to synthesise Chua's diodes.

There are two attractors, at V2 = ±1.8 volts. As the signal is amplified by the local gain, the oscillator signal orbits that attractor, until the phasor reaches an amplitude where it can escape across the saddle near zero volts, to the other attractor. While crossing the saddle, the phasor is attenuated, whereupon the process is repeated, each time with a different escape velocity and attenuation. Notice how the first cycle of the phasor has a different amplitude each time, and that there are occasions when it fails to complete the saddle crossing.

 

[Swamp Thing]

Dr Chua reminsces about the invention of his circuit:
https://www.chaotic-circuits.com/wp-content/uploads/2015/01/THE-GENESIS-OF-CHUAS-CIRCUIT.pdf

Page devoted to it (includes an interactive Javascript simulation page and how-to-build info):
https://chuacircuits.com/

 

[Swamp Thing]

While crossing the saddle, the phasor is attenuated

Is the energy dissipated, or is the energy transferred to another element than the one whose voltage is plotted?

 

[Baluncore]

Is the energy dissipated, or is the energy transferred to another element than the one whose voltage is plotted?

I am not sure. It may be dissipated in the linear resistor between the two capacitors, or it could be invested, regenerated in Chua's non-linear diode, which appears as a negative resistance amplifier, obviously with a power supply capable of absorbing energy.

To keep it conceptually simple, I did only plot the C2 component of the phasor. If you run the simulation, you can plot the power dissipated in the linear resistor.

 

[sophiecentaur]

Yes. Every component part is available commercially, so you can build it, and it will work.

That's more of a philosophical idea - like treating any mathematical process as being a 'real' physical process. However accurate, a mathematical model is still not the real thing. I think that dodgy conclusions can be drawn too easily by having too much faith what a simulation actually does if yo are not totally confident that the simulator has not supplied hidden values to parameters that the user didn't supply. PF often receives posts with this problem.
Just being picky, I know but there should always be a caveat about this, particularly in fringe situations (chaos plus noise, for instance)

 

[Baluncore]

However accurate, a mathematical model is still not the real thing.

Someone might want to build a real circuit. That schematic and simulation is the best help I can provide.

I believe it is the schematic and simulation of a physical circuit that worked, and that was then being shared. It will be somewhere in the LTspice archives.

 

[DaveE]

Is the energy dissipated, or is the energy transferred to another element than the one whose voltage is plotted?

In the ideal circuit model:

The only element that can dissipate energy is the resistor. The negative resistor (Chua diode) can only produce energy. These must be in balance on average for the oscillation be bounded and sustain itself. But clearly they both vary chaotically throughout the cycles. Dissipation will be highest when the cap voltages are different; the upper left or lower right areas of @Baluncore's phase plot above. Likewise, energy is input in proportion to C1's voltage, which is only zero momentarily.

Stored energy is always moving around between the reactive elements, of course.

 

[sophiecentaur]

That schematic and simulation is the best help I can provide.

Absolutely. My issue is that all simulations are unreal and, without verification, they may not do what you expect. I'm sure you didn't start your technical life via simulations and you will have your own built in defence against the possible results from incomplete data.
Science and Engineering are often over-simplified and we have a duty to make that clear in our advice

 

[DaveE]

Absolutely. My issue is that all simulations are unreal and, without verification, they may not do what you expect. I'm sure you didn't start your technical life via simulations and you will have your own built in defence against the possible results from incomplete data.
Science and Engineering are often over-simplified and we have a duty to make that clear in our advice

Simulations can be very valuable when used appropriately. That's the problem; people think it's an easy answer to every circuit question. They are not. They are a tool that can help an engineer that knows the subject matter and knows their limitations. I would break down their appropriate use into three basic catagories:

1) Circuits that are incredibly well characterized (usually by others) at a lower cellular level, but complex from a global perspective. IC design is the quintessential example. These simulators aren't perfect, but are astoundingly good and can save tons of money and time in that industry.

2) Circuits where the designer understands the big picture but has some specific questions that are difficult to calculate. They are only as accurate as the designer needs. In my experience most of the effort in this case is in building and verifying the models used and testing the simulations against known scenarios before you ever get to running the simulation you wanted at the beginning. Some things aren't real, but the engineer got the answer they needed.

3) Circuits where you just don't really care about accuracy much, your just checking simple functionality. You run the simulator on your circuit schematic to check for gross errors; like I wanted negative feedback but I built positive feedback. This is like the spell checker version for schematics.

Short version: A good tool that isn't that easy to use really well, kind of like oscilloscopes. It will lie to you, but if you know how much, you might not care.

 

[Baluncore]

My issue is that all simulations are unreal and, without verification, they may not do what you expect.

That simulations are real simulations, is a truism. Simulators do not model reality, they model a conveniently reductive, idealised system.

Nothing is certain without verification, but absolute certainty is impossibly expensive. Perfection is the enemy of progress. A simulation does not need to be as perfect as reality, it only needs to be third best, which is good enough for now.

You may well have lost faith in your ability to use simulators, but I have spent the last 50 years simulating many things, and my simulations usually show what I expect. I visualise and simulate circuits in my head as I draw or capture the schematic. I use things like LTspice, to fine-tune the numbers. When a simulation deviates from reality, sufficiently for there to be a problem, it is quite obvious to me.

 

[Swamp Thing]

Re. the discussion about 'reality' of this simulation. If I understand correctly, the beauty of this class of systems is that it is very robust against what would usually be called simulation errors. Of course the quantitative behavior is in any case highly dependent on initial conditions. But the emergent patterns should be replicated even if there are minor differences. That is why things like Feigenbaum's numbers are so universal. You may get the bifurcation at a different value, but you should get it, unless something is terribly wrong. Once you get bifurcations you should see close conformance to Feigenbaum's const etc.

 

[Swamp Thing]

The only element that can dissipate energy is the resistor. The negative resistor (Chua diode) can only produce energy. These must be in balance on average for the oscillation be bounded and sustain itself. But clearly they both vary chaotically throughout the cycles. Dissipation will be highest when the cap voltages are different; the upper left or lower right areas of @Baluncore's phase plot above. Likewise, energy is input in proportion to C1's voltage, which is only zero momentarily.

Can you test the hypothesis that there should be a current spike in the positive resistance during the jump from one attractor to another?

 

[berkeman]

Thread is closed for Moderation.

 

[berkeman]

A reply using AI as a technical reference has been deleted, and the thread is reopened provisionally. Please remember that AI references are not allowed in the technical forums. Thanks.

 

[DaveE]

Can you test the hypothesis that there should be a current spike in the positive resistance during the jump from one attractor to another?

I'm not sure what you mean by "a spike", the circuit oscillates chaotically, so everything is changing. But, generally, you won't get a current spike out of an inductor. The negative resistance current is proportional to the voltage on C1, which also doesn't change quickly. OTOH, the resistor sits between two capacitors which have little resistance to current spikes. So yes, it must, I think.

I could test your hypothesis, but:
1) I don't feel like it.
2) @Baluncore gave us a simulation you can run with LTSpice (free!). For chaotic circuits, I'm not at all sure how to test it on paper. Use the computer.
3) I think @Baluncore already did obliquely with his plot of C2 voltage. There's some pretty high dv/dt in that plot and the causative current has to come from somewhere; it's probably not just from the inductor.

 

[Baluncore]

Can you test the hypothesis that there should be a current spike in the positive resistance during the jump from one attractor to another?

You can do that using LTspice.

There is no current spike between the capacitors during transition. The LC tank, C2, has only about ±0.6 volt, near zero, in quadrature with C1, so the higher voltage on C1, ±1.4V, results in a low bipolar current through R, that falls gently through zero during transition.

The current needed to reverse the C1 voltage comes from part of Chua's diode circuit during the C1 transition, however, (since the slope dv/dt is similar with a fixed C1), that has a similar magnitude to the current in C1 during oscillation. Part of Chua's diode drives the oscillator gain, while the other part flips polarity like a Schmitt trigger, deciding which attractor is dominant at that time.

You really need to dive into LTspice. When it comes to entertainment, LTspice beats any computer based game, and it gives positive rewards in understanding and education.