Dear Sajad,

analog computers are real physical systems.

It is a semantic issue whether you call, for instance, pendulum as an analog computer for Josephson junction or for the charge densitiy wave and vice versa. Analog Computers = Real Systems.

Maybe invest 5$ for NE565 and NE555, and a few resistors and capacitors. Then you have a real system described by driven pendulum equation or Josephson junction. Or, give a few dollars for bass-loadspeaker, piezo-microphone and 3-6mm diameter steel balls. It is also an analog computer imitating the Fermi-Ulam question. But it is really real system!

We have to distinguish between experimental, digital and theoretical approaches. For the theoretical approach we need only our brain (knowledge and logic). For the digital one - our computers. For the real systems - either our subjective rating or ... computers.

Here is the wolf interred ...

Your question is challenging.

since the beginning of study of chaotic systems, physicists and mathematicians are looking for a good real physical or chemical system which can be modeled with a low dilension ODE model.

Lorenz equation is an ODE model of atmosphere which can be only modeled using PDE. Then there is no chance to fit anything.

Rossler model is issued from chemical equations (see

http://nonlineaire.univ-lille1.fr/SNL/media/2010/CR/Letellier.pdf)

with a lot of simplifications.

the Belousov-Zhabotinsky chemical equation very thoroughly studied is very mysterious (nobody knows even the number of ODE needed to model it).

Hence the only one system corresponding to your quest is the Chua Circuit for which we know both the exact model and the electronic value of the signal.

However even in this very simple model, the value of coefficients is not stable in electronic circuit, depending on temperature or the perturbations.

This is, from my point of view, the state of art in models of chaotic systems.

Thank you Professor Lozi (I worried that I haven't said my question right). You are completely right. I hope someone can propose something for me.

A long time ago I posted some chaotic time series on the web:

http://www.physics.emory.edu/~weeks/research/tseries1.html

one of which I believe is low-dimensional chaotic data, from an experiment. (In fact several of the time series are experimental, but I think only one of them is chaotic.) I never did demonstrate completely that it was chaotic, it was only a side project when I was a graduate student, and now I don't do time series analysis. But the data might be of use to you, you can download it from the link.

Sajad, i was in the same situation as you are presently some months ago. Tired of analysing models with no real life application. Currently am working on developing dynamical models for atmospheric variables (which have been reported to be chaotic) using the methods proposed by Bezruchko and Smirnov (Phys. Rev. E 63, 016207). Though, not in the format i wanted (Lorentz, rossler style), it is a start. I would have love to help you out but understanding the equations and methods is a real task for me. There are similar methods for generating difference equations from real time series.

Thank you Samuel. Good luck with that.

The Lorenz model also applies to lasers, hence we call it Lorenz-Haken model. (in fact it also applies to some other systems as the Markus-Lorenz water wheel, etc.).

In the case of lasers, there is a particular laser system that has been demonstrated to exhibit Lorenz chaos quite convincingly: the ammonia far-infrared laser, see Weiss et al., Applied Physics B 61 (1995) 223. However it is not obvious that the Lorenz-Haken equations can be applied to this particular laser. A more refined theory of the laser shows good aggreement with both experiment and the Lorenz model, see Roldan et al. Quantum and Semiclas. Opt. 9 (1997) R1. Take a look also at Gilmore et al. Phys. Rev. E 55 (1997) 2479 for a quantitative comparison of chaotic trajectories (theoretical and experimental) based on a topological analysis.

As for the application of the Lorenz model to other systems, you can find more in Tongen, Am. J. Phys., 81, (2013) 127.

Some time ago we have proposed the 3D-model which is simpler compared to the Lorenz model. Our system contains only one nonlinear therm but demonstrates chaos and open the way for the simle analysis in order to predict areas with chaos in parametric space. The model is described here: http://link.aip.org/link/doi/10.1063/1.1544517

I'm out of my element here, but I immediately thought of building a Lorenz waterwheel. That would seem to be an easy an inexpensive thing to build that should fit in the Lorenz family of systems. I know people build those things all the time. I found this project by some Georgia Bulldogs that seems apropos: http://www.ace.gatech.edu/experiments2/2413/lorenz/fall02/

Hi Sajjad

As a beginner in chaos, can we regard recorded data of electricity price, gold price, oil price and ... as chaotic time series?

If yes I can provide some data of such type for you.

Chaotic pendulum, magnetic pendulum, etc. are simple apparatus that you may build. Indeed you can find many websites provide raw data, like this:

http://www.math.canterbury.ac.nz/~r.sainudiin/lmse/double-pendulum/

Dear Mr. Heydari

Some claims such data may be chaotic and some claim not. However I don't want some real chaotic data. I want some real data with an acceptable model (which should be a white model. Not a black-box model tuned by those data themselves) which can produce them (at least topologically). I think there should be more chance in finding a map rather than flow (based on my experience and Professor Lozi's answer). I hadn't time to investigate the systems proposed by Seng-Kin, Adam, Vyacheslav, Eric and especially Eugenio), but I will do that soon and post the results here.

Dear Seng-Kin

Is there a white model too?

Dear Sajad,

I suppose you are not rigid with the weather systems. Why then not to try with a simple one like "bouncing ball". It is maybe old but a very "white" one.

You will find some theoretical aspects in the works of Fermi and Ulam, than later Zaslavski and Holmes. In 80's of the 20th Century appeared some publications with experimental data, which fit well to the theory.

Is that what you are looking for?

Dear Dr. Kowalik

You are right and that is what I am looking for. I just want to test my method (see the attached file which is a submitted paper of mine) on a real case. I am not familiar with the cases you mentioned, but I will investigate them. Thank you very much.

Hi Sajad,

I have played with a chaotic pendulum. Have you tried setting up such a pendulum and measuring characteristics like displacement and velocity (angular or linear)? I am not sure how chaotic is this pendulum and how useful would be the data you obtain from it. May be a good start.

I just noticed Seng-Kin has already mentioned about the pendulums.

I agree - the Pendulum is a very good object for studies because its model is applicable also for other physical systems: CDW (Charge Densitiy Waves), Josephson Junction, PLL (Phase-Locked-Loop), so you have many possibilities to find the real experimental data. But one should not forget that the real systems with noise (!) cannot be described by one dynamical measure as a constant value. In nonlinear system we can find many attractors with different dimension or stability measure; independent on space using for their estimation (time, frequency or phase).

One dynamical state of such system can be represented by some dimension, entropy value or Lyapunov exponent. To characterize the whole system with all probable states enforces to speak for instance about a spectrum of lyapunov exponents (no just one singular value!). Noise has there a key meaning.

Follows your consideration on chaotic circuits, my students have used a motor (Permanent magnet synchronous motor) to generate chaos. You may try this.

Dear Dr. Li

Thank you.

Is there any mathematical and computational model for that motor?

If yes, is the result of that experiment match with the computational results?

Dear Dr. Diebner

Currently I am doing 2 projects on parameter estimation of chaotic circuits. However chaotic circuits are like analog computers. I want a different "more real!" case.

Dear Sajad Jafari,

You can find lots of chaotic phenomenon found in practical systems in text books and scientific papers in 1980s-1990s. You know they have difficulties to be identified their mathematical models. Many researchers treats the problems by designing simple system with avoiding unknown elements in the system and keeping the principles of practical system. If you select a practical system, you will take lots of time to identify the whole features of the system, so that it may not follow your expected differential equations. Then, controls are introduced to adjust the system to follow equations.

You had better study that chaos, as strange attractor, was originally found in electrical circuit in 1961. The detail analyses had been completed by observed data obtained in the circuit as a physical system. The important point is that electrical circuits follow physics. It is not the same as simulations, which follow logics and coding.

I hope this reply works.

You can try economic (macro-economic) data (time series), variables such as inflation in incomplete markets, financial indexes, demand-price models, ...

See Shannon, Dynamic Economics.

Dear Sajad,

analog computers are real physical systems.

It is a semantic issue whether you call, for instance, pendulum as an analog computer for Josephson junction or for the charge densitiy wave and vice versa. Analog Computers = Real Systems.

Maybe invest 5$ for NE565 and NE555, and a few resistors and capacitors. Then you have a real system described by driven pendulum equation or Josephson junction. Or, give a few dollars for bass-loadspeaker, piezo-microphone and 3-6mm diameter steel balls. It is also an analog computer imitating the Fermi-Ulam question. But it is really real system!

We have to distinguish between experimental, digital and theoretical approaches. For the theoretical approach we need only our brain (knowledge and logic). For the digital one - our computers. For the real systems - either our subjective rating or ... computers.

Here is the wolf interred ...

Dear Zbigniew.

I am convinced. Thank you very much.

I think that the easiest way to solve your problem is to play with equations and circuits until you get the feel of what you are looking for. Find the simplest mathematical equation that describes chaotic system (say pendulum). Write the equation in differential form (second order nonlinear). Use the link provided here...

http://www.allaboutcircuits.com/vol_3/chpt_8/11.html

and put your equations in terms of electrical components. Now:

1) Use your math tool software (for example, mathcad) to graphically view your equation.

2) Use your circuit simulator (for example, multisim) to graphically view your equation.

3) Build the circuit and use your oscilloscope to graphically view your equation.

Note that real components may show the results not exactly or not close enough to meet your expectations. This is because real components will always introduce some deviations and it depends on level of complexity.

If your equation is simple, your model and real world simulation might have some similar results...

Dear Sajad,

perhaps the dripping faucet [1] could be of some help. Not very difficult to implement experimentally [2]. Instead of using the optical device of [2] I made some trials by letting the drops falling on a microphone while recording with audacity. Signal was not as clean as those obtained with optical detector, but was nevertheless usable for preliminary studies.

[1] Yépez, H. N., Brito, A. S., Vargas, C. A., & Vicente, L. A. (1989). Chaos in a dripping faucet. European Journal of Physics, 10(2), 99.

[2] Cahalan, R. F., Leidecker, H., & Cahalan, G. D. (1990). Chaotic rhythms of a dripping faucet. Computers in Physics, 4, 368-382.

Thank you very much Dr. Gilbert. I will see those references.

``Is there any mathematical and computational model for that motor? If yes, is the result of that experiment match with the computational results?"

Dear Sajad,

It is conceivable that truly chaotic natural phenomena are only representable in a mathematical language by functions which are algorithmically verifiable, but not algorithmically computable, in the sense of this paper on the EPR paradox that I recently presented at Unilog'2013 in Rio.

Regards.

Oops! Trying again.

``Is there any mathematical and computational model for that motor? If yes, is the result of that experiment match with the computational results?"

Dear Sajad,

It is conceivable that truly chaotic natural phenomena are only representable in a mathematical language by functions which are algorithmically verifiable, but not algorithmically computable, in the sense of this on the EPR paradox that I recently at Unilog'2013 in Rio.

Paper: http://alixcomsi.com/42_Resolving_EPR_Update.pdf

Presentation: http://alixcomsi.com/42_Resolving_EPR_UNILOG_2013_Presentation.pdf

Regards.

You can try this

John C. Quinn, Paul H. Bryant, Daniel R. Creveling, Sallee R. Klein, and Henry D. I. Abarbanel

Phys. Rev. E 80, 016201 – Published 6 July 2009

Sajad,

There are many non-circuit, non-simulation sources of chaotic dynamics. Physics sometimes has a mentality of "only model the well-behaved" and thus often overlooks it, but here are some physics examples.

1. In celestial mechanics, "Orbital Resonance" (http://en.wikipedia.org/wiki/Orbital_resonance ) often leads to chaotic motions. See also http://www.lpl.arizona.edu/~renu/malhotra_preprints/rio97.pdf . Many talks once used orbit of Io as an example of chaotic motion.

2. Hurricane Sandy is an update on an old theme: Turns out the forecast models were all in agreement that Sandy would stay out at sea, and chaos theory was foundational in the 1970's to creating reliable forecast models. However, a European team realized that the forecast models needed to be updated, and their "math on the fly" led to predictions of landfall in New Jersey and allowed millions to evacuate (otherwise, they would have been there when Sandy hit).

http://www.scientificamerican.com/article.cfm?id=how-math-helped-forecast-superstorm-sandy

http://www.chicagonow.com/chicago-weather-watch/2013/04/fermilab-severe-weather-seminar-dr-louis-uccellini-on-superstorm-sandy/

Nonlinear systems can often be mapped to infinite dimensional linear systems, thus leading to the concept of chaotic dynamics in infinite dimensional linear systems.

3. Quantum Chromodynamics features Quantum Chaos. See http://link.springer.com/chapter/10.1007%2F1-4020-3949-2_19

Biology and biophysics are full of great examples.

1. Most modern defibrillators/pacemakers are based on the assumption that the heartbeat is in a chaotic regime that can be "bifurcated" into rhythm. See this link for an intro used with clinicians: http://www.physionet.org/tutorials/ndc/

2. Fractal Geometry is found throughout physiology -- "fracticality" is a standard measure of cancer cells (search "fracticality of a cancer cell" ). Dynamics related to these fractal biological systems is typically chaotic.

3. The Predator-prey model has only two variables, but increase to Predator-prey-prey and chaotic switching between the two prey occurs. This phenomena can be recreated in a high school laboratory.

I can post more if you like.

Yes we have several REAL time series taken from a chaotic system: I mean the biomedical signals we record in our laboratories and that we have widely published in the literature (mainly EMG a, EEG and ECG).

Dear Sajad,

if you're happy to use circuit data, you may download some (Chua's circuit) from http://www.cpdee.ufmg.br/~MACSIN/services/data/data.htm

There you will find a reference where you can get identified and validated models from these data. Contact me if you need the paper.

On Research Gate you will find the paper

Article: Global models from the Canadian lynx cycles as a direct evidence for chaos in real ecosystems.

I that paper there is a continuous-time model for data (probably chaotic) you can find on the Web.

Regards.

Luis Aguirre

As others have suggested some electrical circuit is probably your best bet, but you might also consider trying to do parameter estimation on the Belousov Zhabotinsky reaction. There isn't "one correct model" for the BZ reaction but there are many of varying degrees of complexity.

I haven't thought about this stuff for a long time. You might be able to obtain chaotic time series from someone. Harry Swinney in Texas, Patrick De Kepper in Bordeaux, Zoltán Noszticzius in Budapest come to mind. They all did very beautiful and careful experiments in the 80's obtaining very clean time series data. If it were me I wouldn't be able to locate the data but those guys are all really well-organized so ...

I think that Patrick has an account on research gate so you can probably contact him easily.

Thank you very much Professor Pearson