Chua's oscillator circuit -- Intuitive picture

[Filip Larsen]

For a chaotic model, a single 'blip', at the appropriate place can completely change the behaviour - hence 'the butterfly effect'.

"Completely change the behavior" sounds wrong to me.

The truncation and rounding errors in numerical method are considered a small perturbation as well, that is, the resulting long-term (chaotic) trajectory for a given initial condition will eventually diverge from such trajectory calculated by any other method. So it makes more sense to define dynamical "behavior" of a chaotic system not from its precise trajectory, but from measures of the strange attractor the trajectory produces, like the Luapunov exponent or the geometry of the stable and unstable manifolds [1]. In that way "behavior" can be defined in terms of how the phase space is mixed and folded, which is a necessary dynamics for chaos.

[1] https://www.bohrium.com/en/sciencep...stable_and_unstable_manifolds_of_chaotic_sets - This site is new to me, but at first glance the description here sounds valid. The topic is treated in many text books on chaos.

 

[sophiecentaur]

So it makes more sense to define dynamical "behavior" of a chaotic system not from its precise trajectory, but from measures of the strange attractor the trajectory produces,

That's a good point. Chaotic systems need a different approach from standard deterministic (right word?) systems. I've heard this sort of comment about weather forecasting models which identify which way they're going and switch from one mode to another when it ;looks like a chaotic pattern is approaching.

So you want to build a real electronic Chua oscillator, to generate a real random sequence? Then how will you implement Chua's diode with real components?

Why are you getting combative about this? There a millions of circuits which will never display chaotic behaviour and your average simulator is well suited to them. There will always be a good mapping between reality and simulation there. But is it true that any randomly chosen chaotic system will behave the same as the simulation unless the relevant simulation parameters are all included and not 'assumed' when the simulator enters its stock values?
If you remember, You took exception to my comment about simulations 'not being real' but (unless we are living in 'The Matrix') it is valid. It wasn't a criticism; I was just introducing a caveat.

 

[Baluncore]

Why are you getting combative about this?

It is you who is combative.

 

[Filip Larsen]

Chaotic systems need a different approach from standard deterministic (right word?) systems.

Indeed.

Regarding the name you could call them non-chaotic, classical or perhaps quasi-linear deterministic systems. For dissipative systems these are systems with only the simple phase space attractors you also see for linear systems, such as isolated fix-points (e.g. hard non-linear die tossing) or limit-cycles (e.g. a near-linear driven damped oscillator).

 

[sophiecentaur]

It is you who is combative.

Well, I get cross when someone refuses to accept a perfectly valid comment about the limitations of Simulators. Your comment about simulating a random signal generator with LTSpice is clearly nonsense and I never suggested it. I will calm down when you acknowledge that a simulation is not 'real' in the sense that it is only valid under certain conditions. You want it to be a panacea?

 

[Baluncore]

I will calm down when you acknowledge that a simulation is not 'real' in the sense that it is only valid under certain conditions. You want it to be a panacea?

This thread is about Chua's Oscillator, what do you not understand?

 

[Swamp Thing]

I have installed ngspice (RPi-5), but haven't got around to experimenting with it. However, it's possible to do thought experiments and refine one's ideas while doing other life stuff.

This has led me to discard the idea that there should be a momentary increase in disspiation exactly while the system is passing from one lobe of the attractor to the other (crossing over the separatrix ??).

The motivation for such an idea was the fact that the oscillation needs to regrow itself after those lobe-to-lobe transitions. However, it now seems to me that the dissipation could be occurring during the overshoot after crossing the separatrix (or whatever the barrier between lobes is called). This overshoot takes us into the region where the negative resistance of Chua's diode drops ( --> less gain) while the positive resistance of the conventional resistor of course remains fixed (--> same loss).

I hope to test this using suitable probes and computed plots in a day or two.

Edit: All this thinking about an oscillation quenching and regrowing in sync with a lower frequency square wave has just reminded me of the venerable superregenerative receiver. Could it be that electronic chaos and attractors have been lurking in our midst since the vacuum tube era? Wait, didn't someone mention 1911 in this thread?

 

[Baluncore]

However, it now seems to me that the dissipation could be occurring during the overshoot after crossing the separatrix (or whatever the barrier between lobes is called). This overshoot takes us into the region where the negative resistance of Chua's diode drops ( --> less gain) while the positive resistance of the conventional resistor of course remains fixed (--> same loss).

That is correct.
Energy increases until it climbs out of one attractor, then from the crest of the saddle, it overshoots the destination attractor, encountering the far soft wall, that attenuates or dumps some energy. It is then shepherded between that wall and the saddle, until it can again escape from the local attractor.

 

[Baluncore]

I hope to test this using suitable probes and computed plots in a day or two.

It should be possible with SPICE to plot the sum of all energy circulating or stored in L or C.
Ea = ½⋅L₁⋅I(L₁)² ; Eb = ½⋅C₂⋅V(C₂)² ; Ec = ½⋅C₁⋅V(C₁)²

For some reason that does not seem to work. I get a minimum energy sum on the saddle. Maybe zero potential is relative to the local attractor, rather than the high-ground.

 

[sophiecentaur]

This thread is about Chua's Oscillator, what do you not understand?

And you introduced the idea of a simulation. I simply introduced the necessary caveat when dealing with chaotic systems . (The 'reality' word; If you feel you can argue with that then you should read more about Chaos Theory. Have you actually appreciated my point or have you rejected it out of hand?

However, it now seems to me that the dissipation could be occurring during the overshoot after crossing the separatrix

The graph of the atyractor is deceptively smooth looking b ut the behaviour may be far from that. Imo, the dissipation will occur throughout; why should it not? The graph that's plotted will only tell you the dynamics of the system for one particular starting point as it progresses from a particular condition. There's only a tiny amount of energy that determines which way the system progresses when the path is at the "separaterix" it can 'just' roll over into there other side or f'all back.' (The double pendulum behaviour shows this nicely and it can be fun to predict which way the pendulum will go when right at the top. That will determine which of the wildly different possible behaviours will follow after the pendulum has 'decided'.
I rather think that the path of a CH can be influenced greatly by the presence of noise, particularly at those decision points. It would be easy to look at the oscillator performance and see how it's affected by added noise or a chance impulse in the right place (the butterfly).

 

[Baluncore]

Have you actually appreciated my point or have you rejected it out of hand?

https://en.wikipedia.org/wiki/Simulation_hypothesis

 

[sophiecentaur]

https://en.wikipedia.org/wiki/Simulation_hypothesis

That page doesn't mention chaos so how is it obviously relevant? Our thread is about a chaotic system so choose a suitable reference that explicitly includes it. There are number of places in that Wiki page that hint at the complication of chaotic systems but it doesn't choose to discuss it. - says it's perhaps too hard to include.
Basically, if the states in a simulation model include strange attractors then there will be initial conditions where the results can be thrown way out with small differences in initial conditions. It will be possible that running the simulator would very likely give the wrong prediction for a 'real' system. Do you really disagree?

 

[Filip Larsen]

It will be possible that running the simulator would very likely give the wrong prediction for a 'real' system.

It seems you still are hooked up on the idea that the precise trajectory for a chaotic system is "the way" to predict its dynamical behavior, but it rarely makes sense to classify behavior of a chaotic system from one specific trajectory.

Chaotic systems are also often modeled via a parameterized set of differential or difference equations, and all the usual selection of applicable numerical methods can as such be used produce trajectories from initial conditions as a kind of basic building block in further analysis for each specific system. As mentioned earlier, mapping out the stable and unstable manifolds reveal a lot of the dynamics and on top of that you can then apply continuation techniques to map out the dynamical behavior as a function of parameter changes. That is, many systems capable of exhibiting chaos for some parameter values will exhibit classical stable solutions for other parameter values so it becomes relevant to map out how the manifolds emerge, move around and possibly vanish again in parameter space. The result of such analysis is more or less independent of which otherwise applicable numerical solver is used since it captures the systems behavior rather than one specific trajectory.

For examples on this method relevant to current discussion, search for papers on continuation analysis of Chua's circuit, e.g like [1]. Some even research for physical realizable tuning capabilities for Chua's diode [2] that would allow tuning physical experiments to parameter values as a experimental setup mimicing continuation analysis.

[1] https://arxiv.org/abs/1312.1933 - Bifurcation structures and transient chaos in a four-dimensional Chua model.
[2] https://arxiv.org/pdf/2308.08964 - Physical Implementation of a Tunable Memristor-based Chua’s Circuit.

Edited for spelling.

 

[DaveE]

It will be possible that running the simulator would very likely give the wrong prediction for a 'real' system.

This is a strawman argument. Simulators never represent "real" systems (chaotic or not) with perfect accuracy. Engineers never create models of reality that are perfect, it's just too hard and wasteful. The simulator and it's inputs are valuable as an approximation, like nearly every analysis tool outside of pure math. You won't impress most people by finding minor errors in people's models, they already know they aren't perfect, they likely already know how and why too.

 

[Baluncore]

https://en.wikipedia.org/wiki/Simulation_hypothesis

That page doesn't mention chaos so how is it obviously relevant?

It talks about reality, which includes chaos.
You did not even read the reference to the "simulation hypothesis" that began ... "The simulation hypothesis proposes that what one experiences as the real world is actually a simulated reality, such as a computer simulation in which humans are constructs. There has been much debate over this topic in the philosophical discourse."

It will be possible that running the simulator would very likely give the wrong prediction for a 'real' system.

Today, in my metaphysical philosophy, I believe that our reality is nothing but part of a simulation. As such, the concept of running a simulation, within a simulation, recursively, makes every simulation part of the "greater simulation", or should that be, the "greater reality". Your argument that a simulation is not real becomes preposterous in that light. In my simulation, you are but an organ grinder object. I am not your monkey constructor.

 

[sophiecentaur]

This is a strawman argument. Simulators never represent "real" systems (chaotic or not) with perfect accuracy.

It would be oversimplification to equate the two situations and, of course, any use of a simulator should be used with care. But chaos is a bit different; introducing noise is a great way of showing limits of a test circuit and its effect can be quantified - likewise varying component values - but can this work for a simulated chaotic circuit?

 

[sophiecentaur]

Today, in my metaphysical philosophy, I believe that our reality is nothing but part of a simulation.

That is almost certainly out of bounds here.

the reference to the "simulation hypothesis"

Yes, I read it.

It talks about reality, which includes chaos.

So why do you not discuss that and show how your thinking relates to your use of the description of a 'real' component value in the context of chaos. If you have a point to make then don't rely on the reader to attempt to find that in a reference.

 

[sophiecentaur]

It seems you still are hooked up on the idea that the precise trajectory for a chaotic system is "the way" to predict its dynamical behavior,

Yes, you have a very good point there. Thanks for your response. I take all this post on board and it is very informative. The reason for my getting involved with this is in answer to @Baluncore 's talk of using LTSpice and the assumption that you just plug it in and can know about the behaviour of the oscillator. As you say, much deeper examination of the circuit behaviour is needed than just plotting the voltage variation somewhere for a particular set of parameters.

It seems that, as with all other chaotic systems that can be modelled, one can't just build a physical (real) model and take real-time measurements, expecting therm to be what the simulation gives you.