Hello Mohsen:

I believe the short answer is "No". As reflected in many of the other posted responses, positive Lyapunov exponents, by themselves, do not always indicate "chaos". Additional information about the system, and / or texts need to be performed to conclusively diagnose "chaos" in most systems.

The Lyapunov exponent itself merely quantifies the degree of "sensitivity to initial conditions" (i.e., local instability in a state space). Such local instability can arise for a variety of reasons in different types of systems. For example, in nearly all experimental systems, where Lyapunov exponents are estimated from recorded time series data, these data include some level of stochastic noise (which itself might include measurement noise, as well as system noise, or both, etc.). Noise, by itself, will create "sensitivity to initial conditions" that can trigger findings of positive Lyapunov exponents even for systems that are not remotely "chaotic".

For a more detailed discussion, I wrote a book chapter on Lyapunov exponents a few years ago:

http://www.wiley.com/WileyCDA/WileyTitle/productCd-047124967X.html

You can download a PDF of the chapter from the "Publications" section of my web site:

http://www.edb.utexas.edu/faculty/dingwell/pubs_main.html

(Scroll to the bottom to see the link)

This is a complex issue, as indicated by the many responses already posted. So the book chapter is in no way intended to be "exhaustive", but I hope it is informative.

I think that the problem with the second case is that there is not an attractor. You will have chaos when there is an attractor and a positive lyapunov coefficient.

But this point of view can be a little subtle. For instance, you can map your unbounded system that lives in (0,infinity) to the region (0,1) by an appropriate transformation. For instance it can be y(n) = 1/x(n) so x -> 0 is mapped to y -> infinity and x -> infinity is mapped to y -> 0. You can prove that after this mapping your system will have no more a positive lyapunov coefficient.

In other words, you need to map your system into a bounded space and only then test the for sign of the lyapunov coefficient. I think that If your system is chaotic, no one to one mapping will destroy chaoticity.

Best Regards.

Juan

Lyapunov exponent is important in chaotic system. For example in lyapunov circle is expand to ellipse. for more details refere Chaos- Introduction to Dynamical system- Alligood, Chaotic Dynamics- Baker and Gollub, Chaos in Dynamical system- E. Ott. In this book detailed application and explantion also avilable.

In most cases the answer is yes. However this is not always so. Please consult the paper G. A. LEONOV and N. V. KUZNETSOV. TIME-VARYING LINEARIZATION AND THE PERRON EFFECTS. Int. J. Bifurcation Chaos 17, 1079 (2007) for further information about this question.

http://www.math.spbu.ru/user/nk/PDF/2007_IJBC_Lyapunov_exponent_Linearization_Chaos_Perron_effects.pdf

Chaos means unpredictability. Positive Lyapunov exponent is not enough, you also need a bounded phase space. For the logistic system, any finite error after a few iterations grows to the size of the whole system [0,1]. In the other case, the system is infinite. Unless you actually bound it to be finite by wrapping everything around on a torus. Then you get chaos as well.

There's a wealth of good books on chaos. One available online is this:

http://chaosbook.org/

I also like Bob Dorfman's book:

http://www.amazon.com/Introduction-Nonequilibrium-Statistical-Mechanics-Cambridge/dp/0521655897

but it may depend on your background whether it fits.

Hope that helps.

It depends of course on your definition of chaos. A rough definition states that a system is chaotic (in a subset S of state space) if it shows (i) sensitive dependence on the choice of initial conditions and (ii) S is bounded such as to exclude the trivial case of an unstable linear system where trajectories diverge exponentially for all times.

By construction of Lyapunov exponents, property (i) is satisfied if a LE is larger than zero. Property (ii) guarantees that trajectories "fold back" close to where they where before, hence the notion "stretching and folding" for many chaotic systems.

The answer is simply that state space is infinite in the second example, and x_n -> oo. When saying that positive Lyapunov exponents imply chaos, one means compact state space. One could compactify space in the second example by the map y=1/(1+x^2), say, but then the Lyapunov exponent would be

Logistic map for r =1.9 leads to NEGATIVE Lyapunov exponent, equal to, approximatelly, -2.3 .

As to second example You have gave - this system is, in fact, in permannent transient and LE is not well-defined. Indeed, the LE is computed as a derivative averaged by the invariant measure over the bassin of attraction (see, e.g. E. Ott "Chaos in dynamical systems"). The system You gave has no basin of attraction thus LE cannot be computed.

The logistic map for r=1.9 leads to negative Lyapunov exponent

Random and chaotic both time series will give positive Lyapunov exponent but from determinism test you can differentiate. Paper containing detail is : "D. T. Kaplan and L. Glass, "Direct test for determinism in a time series," Physical Review Letters, vol. 68, pp. 427-430, 1992.". Scale dependent lyapunov exponent is better choice for nonstationary and multiscaled time series. Correlation dimension is also good measurement tool.

Hello Mohsen:

I believe the short answer is "No". As reflected in many of the other posted responses, positive Lyapunov exponents, by themselves, do not always indicate "chaos". Additional information about the system, and / or texts need to be performed to conclusively diagnose "chaos" in most systems.

The Lyapunov exponent itself merely quantifies the degree of "sensitivity to initial conditions" (i.e., local instability in a state space). Such local instability can arise for a variety of reasons in different types of systems. For example, in nearly all experimental systems, where Lyapunov exponents are estimated from recorded time series data, these data include some level of stochastic noise (which itself might include measurement noise, as well as system noise, or both, etc.). Noise, by itself, will create "sensitivity to initial conditions" that can trigger findings of positive Lyapunov exponents even for systems that are not remotely "chaotic".

For a more detailed discussion, I wrote a book chapter on Lyapunov exponents a few years ago:

http://www.wiley.com/WileyCDA/WileyTitle/productCd-047124967X.html

You can download a PDF of the chapter from the "Publications" section of my web site:

http://www.edb.utexas.edu/faculty/dingwell/pubs_main.html

(Scroll to the bottom to see the link)

This is a complex issue, as indicated by the many responses already posted. So the book chapter is in no way intended to be "exhaustive", but I hope it is informative.

Dear Mohsen,

Definitely, the answer is not enough positive Lyapunov exponents to gurantee a chaotic behavior. For r=1.9 the Lyapunov exponent is negative. The difference between both examples is that for the logistic map the phase space is bounded and for the second one is unbounded. Chaotic dynamics on non--compact spaces is far to be understood; for instance linear systems can be chaotic and any chaotic systems (on a compact metric space) can be emmbeded in a linear one (with non--compact phase space). Anyway, there are several non--equivalent definitions of chaos and you should precise in which definition you are interested in.

Best regards.

Dear Mohsen,

the best world recognized specialists gave you enough good informations. I would only highlight on one important point for beginner in chaos: do not trust any computation you perform. Read my paper: Can we trust in numerical computations of chaotic solutions of dynamical systems ?

(you can find it freely on HAL

http://hal.archives-ouvertes.fr/index.php?action_todo=search&view_this_doc=hal-00682818&version=1&halsid=s0adclb7c8k3c8bdsc0l63c3o0

best wishes

René

There are at least four questions:

1. Theoretical physical closed systems (see answer from Peter Grassberger)

2. Experimental signal analysis in well defined systems these are near stationary

3. Experimental signal analysis in open nonstationary systems

4. The role of noise in the analyses of experimental data.

I willl never believe in chaotical dynamics after analysis of the single experimental time series, even if LE will be positive.

I can believe in chaotic dynamics, if I repeat the same experiment many times and obtain allways statistical significant similar LE-values.

Because of the real and computational noise you need to perform the test for determinism (for instance surrogate data) for you analyzed signal. You should also take a look on the FFT-analysis. Take care on peaks showing the periodic compounds (for instance 50 or 60 Hz from current supply or 37 Hz from AHU). Artefacts are unfortunatelly very frequent.

Best Regards,

Zbigniew

Hello dear all.

Thanks a lot for the constructive answers. As René Lozi mentioned, the best world recognized specialists have given their professional ideas. As I got from the comments, I summarize the results:

1- Chaos can happen in systems with orders greater than 2 (I think I read somewhere that in discrete time systems or maps, chaos can be observed even in the first order systems). So, we should first of all inspect the order of the system.

2- The logistic map with r=1.9 is not chaotic, as some specialists commented , because its lyapunov exponent is Negative (-2.9 as Jakub Gac reported). I checked the reference (http://www.dt.fee.unicamp.br/~tiago/courses/dinamica_caotica/Lyapunov.pdf) that I read and found it is right, the system is chaotic for 3.5699456. Sorry for this mistake.

3- The system should be bounded or compact in phase space. Otherwise even positive lyapunov exponent doesn't mean chaos in the system. In agreement with Daniel Wojcik , in my simulation I saw that the chaotic system trajectory always belongs to [0,1]. Sometimes by a mathematical map (such as y=1/(1+x^2) ) we can change the state space of unstable system which gives negative LE. But what about the chaotic system?

4- Unstable systems don't have any attractors, although they give positive LEs. But a chaotic one has a strange attractor. One question can be how can we recognize the strange attractor from the response? while as some people mentioned calculations with digital computers, which have a limited number of digits to show numbers, and can lead to a false alarm of chaos in the system. Isn't it better to simulate the system in an analog set-up outside of digital computer and only see the results obtained from an IO card on the computer?

5- Some people believe noise and chaotic signals have to somewhat have similar time domain response and both can give +LEs. So it is so difficult to distinguish the noise and chaos and we need to refer to another space such as spectrum analysis. It seems another tool.

6- Finally as Jonathan Dingwell mentioned in his answer, LE is solely a measurement for sensitivity to initial conditions and is not 100% an indication of chaos. We need additional information about system. Can't we gather all these additional information plus LE into one formula or algorithm and define a new criteria that can definitely determines if the system is chaotic? Since it may be possible that somebody claims that he/she has found a new chaotic system other than the famous ones such as Lorenz, Henon..., we can judge his claim by this new criteria and accept or refuse it certainly.

Sorry for the long letter and thanks again for all the comments and specially for those people who sent useful links. I am studying them and will return soon.

I am looking forward to your constructive answers again.

Best regards.

"Unstable systems don't have any attractors, although they give positive LEs. But a chaotic one has a strange attractor"

What do you mean under "unstable systems"? LE is a measure for a stability. Nonlinear systems are unstable when LE is positive and have unpredictable state after disturbance. And they possess a strange attractor. And what about "snapshot attractors"?

Regards,

Zbigniew

Dear Zbigniew,

for example consider: this system: x_dot=2x which gives x(t)=x(0)*exp(2t). This is obviously an unstable system. Now we put a disturbance on initial conditions, for example consider two nearby trajectories: x1(t)=x1(0)*exp(2t) and x2(t)=x2(0)*exp(2t). Thus by subtracting we'll have: delta_x(t)=delta_x(0)*exp(2t). Now compute LE:

LE=lim(1/t*ln(delta_x(t)/delta_x(0))) while t goes to infinity. In our case this formula gives LE=+2. So this system is quite sensitive to initial conditions and two trajectories will diverge from each other by time. Although the system has positive LE, it is not chaotic. So I conclude lyapunov exponent alone is not enough for chaos. We should check the stability and the equilibrium points and also initial conditions before using LE.

Regards.

Mohsen

A chaotic system has three 'axioms': (1) Sensitive dependence on initial conditions, (2) Topological transitivity (phase space mixing), (3) Periodic points are dense in phase space. Liapunov exponents being positive pretty much guarantees (1) in a deterministic dynamical system. I would like anyone to demonstrate an interesting situation in which we have (1) but not (2) and (3). I don't mean an unstable expanding system like x(n+1)=r*x(n) with r>1, or a stochastic system, which lies outside the frame of discussion proper.

Dear Mohsen,

i'm experimentall physicists and neuroscientists. In the theory you could be right. My comments shows that I am never sure to find a chaotical behavior in all analyses with a positive LLE. So it is nothing against your comment.

Are you got your estimates with a computer? Please do not forget that computers are integers machines. Nothing with "true float real numbers".

BTW - I did not yet found a definition for an "unstable system". I''v learned only Lyapunov:

Regards,

Zbigniew

.

.

(1)LE is positive;

(2)The motion is bounderied.

The motion is chaos.

Hi again.

Dear Zbigniew, I am sorry if I went fast. An unstable system, in a general sense, means a system which has a response which is not bounded in any limit and grows exponentially usually. Since you are a neuroscientists this example may clarify the matter: consider a bacteria (or maybe a neuron I think) which doubles each hour. The population of this bacteria grows fast and is ideally described by a dynamic equation like: x(n+1)=2*x(n). The response of this equation is x(n)=exp(2*n) at hour n, which does not remain in any limits. This is called an unstable system and the response is called unstable response (sometimes this characteristic is called instability also). But because of limitation of food, each hour a fraction of them die. This is usually modeled as a (1-x(n)) factor which means reduction of the population. So the real world equation of growth is something like x(n+1)=r*x(n)*(1-x(n)) which is called logistic map (r is population growth rate). Note that in this equation x(n) is normalized and fractional. This equation can exhibit different manners including chaos depending on r and x(0) (initial conditions).

I hope this has clarified the subject to somewhat. for more information search in google. for example see the links below:

http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node37.html

http://math.la.asu.edu/~chaos/logistic.html

http://en.wikipedia.org/wiki/Logistic_map

Regards

Mohsen

Dear Reza,

I have little information about fractals and fractional dimension. Can you please introduce a good reference on this subject that also describes its relation to chaos?

Dear Chris,

I have similar idea with you. Some references mention another axiom instead of (ii) or (iii). they state that also the system should not be PREDICTABLE. Like Maen, I believe that some systems may have sensitive dependence on initial conditions but may not be chaotic. I am thinking about that and will introduce an example soon.

By the way, can you give an example that satisfies (i), but not (ii) or (iii)?

Dear Mohsen

I think we do not need a discussion about scholar nonlinear systems liike Volterra-Lotka or bouncing ball systems.

I'v answered only tp respone your question supposing you have an experimental problem.

Shortly. the positive local LE from a single time-series is not enough to confirm the determinism of rhe time-series you studied. But it is very helpfull.

Best Greetings,

Zbigniew

Dear Zbigniew,

Thank you for your help. In fact, yes, I had a numerical problem with the logistic system when running simulations in Matlab. The results werenot sometimes as I expected. Generally we can not trust computer simulations for chaos 100%.

in case of LE I agree with you. It is an aid but is not enough and we should take care of other things too.

Thanks again.

Kind regards,

Mohsen

I do agree. LE is not enough. The problem with determining chaos is that there is no precise mathematical definition of chaos thus making a single parametric identification very difficult. As already explained by other researchers, the conditions fulfilling the criteria for chaos may be checked and then conclusions may be drawn. Sometimes using tools to investigate the phase space evolution ( Like RP and RQA) help us better understand the dynamics. Alternatively one can check for topological dimensions as well.

Dear Mohsen,

You could also try alternative algebraic techniques (as mentioned previously, LE is not enough). An example could be:

[1] M. Ragulskis, Z. Navickas. The rank of a sequence as an indicator of chaos in discrete nonlinear dynamical systems. Communications in Nonlinear Science and Numerical Simulations 16 (2011) 2894–2906.

We have enough strong theoretical definiton of chaos: dependence on initial conditions, fractal dimension and the positive Lyapunov exponent, The problem appears only in time series generated by digital or experimental systems. Therefore one should be very carefully about statement on chaos in experimental or digitally generated data. I prefer now to read more about nonlinear and nonstationary systems, noise and less about chaos which is like linearity an idealistic term. In the pratice is LE alone not enough to define chaos. But is very helpfull to investigate the stabilty of some systems.

If you consider a given time series, positive Lyapunov exponent is not by itself a necessary and sufficient condition to conclude that it pertains to chaotic system. I invite you to viisit our site www.sistmp.com . Recurrence Quantification Analysis may be a good method for estimation.

Dear Elio,

your proper link is: http://www.saistmp.com/

The above discussion here is devoted to a single value of LE of a single time series. I myself want to underline the role of noise, possible nonstationarity and statistics. I investigate always the spectrum of local LE, so I can estimate the probability (!) of deterministic behaviour., proving previously a standart frequency spectrum und making at least a surrogate data test. While we are using mathematics and computer tools I am always very carefully to say "I saw a chaos". In the real world there are no "ideals" for linearity, chaos and noise. We are just looking always for a "traces" of determinism, because the determinism is most significant to be applicable. Nothing more.

.

Many commentators have observed the importance of differentiation between random and deterministic nature of the chaotic sequence. A nice reference here could be:

[1] J.M. Amigo, S. Zambrano, M.A.F. Sanjuan. True and false forbidden patterns in deterministic and random dynamics. Europhys. Lett., 79:50001, 2007.

The Lyapunov exponent can be obtained directly from fractal expansions and borders of chaos for each dimension of the studied system rather than being estimated from long raw data series. 1-dimensional unimodal maps are useful. There is a strict relationship between Feigenbaum value F=4.669... and the LE for natural numbers

(the trick is to ignore the basins of stability which are plenty in the logistic map for example). In a way we want to get the big picture of the system, not its self-similar microscales). The key issue is to check for the existence of internal memory (UPOs) and what the system's behavior is like once a UPO is detected (attractors, repellers). The fractal expansions need to be ralated to their triggers because only then we eliminate the problem of multidimensionality. So I consider the primary question posted here as scale-related. The data accuracy or the problem of noise is less important because we base on limits when n goes to infinity (forget what infinity is in practice).

It was shown that fractality may disappear in some bigger scales, and we may easily become irritated that we cannot find or prove what we intended to.

The final thing (but not last) is what would you do, what would it give to know that a system is chaotic in the first place? Can you use the determinants of a chaotic system for some purpose or is it just an academic issue?

Chaos is all about experiments and implementation after all. It is very difficult as plain maths. Fortunately, the heuristics is so fascinating and inspirational for the theory itself. There are plenty of new findings, some still unpublished.

Some more thoughts.

The Lyapunov exponent measures the rate of system's sensitivity to initial conditions, roughly speaking. There is no mention of causality.

For an open system some LEs (again the problem of scales) the problem of causality must be explained somehow and followed, otherwise we can easily come to a blind avenue of exploring the features of overlapping multidimensional projections into a 2-D space, and really see nothing (I mean order, of course).

Then, let us not be misguided by sheer words: those 'initial' conditions really mean

local conditions, we do not wish to go back to the beginning of the calculations all the time.

Then, there is the problem of why systems can evolve LE-wise in one dimension, and why it can evaporate in another dimension after few initial expansions.

No. You must ensure that orbits with positive Liapunov exponents are bounded. If the orbits are bounded and you have at least one positive Liapunov exponent then the system is chaotic.

In our laboratory we are able to establish the following parameters.

given the time series , we operate its reconstruction in phase space. On this basis we may calculate the following variables : the %Rec that is the percent of recurrences that the given time series is able to evidence, the %Determinism that is the %Det that determines the dynamics process in study, the correlation dimension thatt should saturate about a low dimension of the attractor if the given time series is not affected by noise so largely, the Lyapunov spectrum, bifurcations and still more. ... also LE is appropriate. Consequently the determinism may be valuated correctly.

Hi again.

Thanks for all the new comments. It seems the next thing I have to read on chaos is fractals and their dimensions. Dear Elio , I referred to the link (http://www.saistmp.com as Zbigniew corrected). The special issue on Chaos was useful.

As I conclude from the recent comments , we should be very careful when declaring chaos on a system because of computer storing system limitation and noise and other random parameters. At this point I think all of us have a common point on chaotic system i.e. sensitive dependence on initial conditions. This can be both qualitatively and quantitatively (LEs). But another characteristic of the chaotic systems, as some references indicate, is that they are not predictable in long-term (as weather predictions). Here I am wondering if we have a measurement tool for predictability or probability , as Zbigniew mentioned, for chaotic systems? Let me shed some light on this by giving an example in my mind:

Suppose the system of catching a train or an airplane. for example your ticket is for 8 O'clock. Ignoring the Airport formalities, if you reach the station by 7:59, you will be able to travel, otherwise you will be left behind. We all agree that this is a very sensitive system to initial conditions. I model it by mathematical function: y=int(x)+1 namely integer part. For example if you reach on 7.99 that will yield 8, but if you reach on 8.01 it yields 9 and you miss the 8 O'clock train and have to wait till 9. This, intuitively, has the sign of chaos (I do not know how much is its LE) but it is quite predictable. If you give me the time, I will DEFINITELY tell you if you catch the train. Do you agree this misjudgment caused by the lack of a measure for prediction or is it related to attractors and fractal dimensions?

As always, It is my best pleasure to here about your constructive comments.

Mohsen

Please go on www.saistmp.com and download Non Linear Methodologies ...... . I repeat .. th ebest way is to uyse QRA ,CZF or CKZF methods.

Dear Maen,

I disagree with

"...noise reduction techniques are Indispensable to be able to determine whether the the system is chaotic or non chaotic".

You may not know just what the noise is and what is not if you do not know the system's equations, or you know it but the number of variables is beyond any practical calculable sense.

Would high frequency oscillations near a stable point be noise to you or perhaps not and instead you would eliminate sudden bursts of local volatility?

You risk losing the entire determinism in the system by filtering data or applying some other data manipulation. Then it would be naive to seek chaotic behavior at all.

This problem is well discussed in the literature.

Data in time series never must be manipulated and particularly by filtering operations. A reconstruction of the attractor features in phase space must be attempted and thus by application of the Recurrence Quantificatiuon analysis one has a good probability to correctly estimate. the %Determinism. Still estimation of Max Line enables to evaluate LE. Chosing a large dimension embedding noise obviously is largely included. Also analysis of Correlation Dimension with the saturation behaviour for increasing embedding dimension largely informs if data are seriously affected or not by noise. . The use of surrogate data techniques gives another important indication

See my previous comment. The fist one the orbits are bounded, in the second the orbits are not bounded.

Dear Paul,

many thanks for your comment - manipulating experimental data means always manipultating of the results. But unfortunatelly we are limited by an experimental measurement (data collecting) due to devices which never work in a full frequency range (0 to oo Hz) and not within an infinte time range.and never in the real-time.

They inject also their own noise and are than digitized into pseudo-reai- time series for the computational analyses. It fortunately does not means that we can not indicate the "traces" of nonlinear, determinitic beahvior.

But we can only suppose - there is a some probability of the deterministic nonlinear (sometimes chaotical) behaviuor- but only on the base of the statistics and correlation wit h a some investigated event,.

Dear Zbigniew,

You wrote

"... we are limited by an experimental measurement (data collecting) due to devices which never work in a full frequency range (0 to oo Hz) and not within an infinte time range.and never in the real-time."

Well, I have always pondered how much experimental work is limited to devices and the laboratory, how little to other data sources.

I accept the problem of data quality in general, and data reliability. But I would also like to say that you may switch to investigate financial markets series (charts) for the presence of deterministic chaos, which are free and reliable and real time not pseudo-real time and absolutely nonlag information sources.

Then, having found some evidence of deterministic chaos there, you may ask yourself questions nobody has asked before. Like why Lyapunov exponents 1.76 and 2.21 set a border for chaotic behavior (vide attachment).

The notion of Lyapunov exponent is related to statistical properties of trajectories (via Oseledec theorem). Lyapunov exponents (there are as many exponents as the dimension of the system, although one usually only consider the largest one) are computed via time averages along trajectories. In this way, they depend on the trajectory (the initial conditions). However, in compact system, one can find nice probability measures called ergodic. If mu is such a measure the Lyapunov exponents are constant on a set of mu-probability 1. mu is somewhat selected by dynamics. In the example of the logistic map, mu is a smooth distribution which weights all the phase space. In this case, having a positive Lyapunov exponent means that any point selected at random will exhibit initial condition sensitivity.

There is a nice although paper by Eckmann and Ruelle (ergodic theory of chaos and strange attractors, 1985) which nicely explains these notions as well as the connections with fractal properties.

I inssit. If one uses time series experimental data the best method is to use Recurrence Quantification Analysis and Variogram Techniques.

Dear Paul,

I do not think that we really disagree together.

Look maybe here: http://www.ane.pl/pdf/6019.pdf

Even the pseudo-time-series can and should be investigated in the sense of their stability. And I'm doing this myself (for instance with analysis of psychotherapy timeseries). In all my comments here I'd like to emphasize the difference between a mathematical theory, numerical modelling and an analysis of the real experimental data also about the influence of noise and nonstationarity on such analyses. Nothing else.

For all young chaos scientifists I recommend to begin with a book of Heinz Georg Schuster: Deterministic Chaos - an introduction.

I have experience of examination of time series possibly chaotic from many years. Also I started with usual examinations. Phase space reconstruction , correlation dimension, lyapunov spectrum, entropy and still more. Possibly you may take vision of a lot of papers that I have puiblished on this matter by only and in collaboration with prof. Joseph P. Zbilut that unfortunately no more is with us and that I consider my great friend as well as a founding father in chaos theory and investigation of time series by appropriate techniques. No doubt that some inappropriate techniques continue to be published on journal of valuable interest and often it is concluded that a given time series is deterministic-chaotic as well as using Lyapunov or correlation dimension as standard methods. Unfortunately this is the same counterpart of what it continues to be in R-R time series. These are time series of interest in clinical electrophysiology. obtained by ECG recording. This is one of the most complex signals existing in nature and , as the cardiologists, physiologists and psychophysiologists well know, it is of great importance in research as well as in diagnosis and therapy in many conditions of organic as well as mental, psychologic and psychiatrix disorders. The usual performed analysis is the so called HRV that in case the scholar that are reading this contribution, will remember soon as the most important in this field of studies. Well. I have said that R-R is the most complex signal existing in nature . It is clearly non linear , non periodic , non stationary, possibly chaotics. Cardiologists, physiologists , psychological and psychiatric scholars continue to perform analysis by using the Fast Fourier Transform. All we know that FFT may be applied only if the given signal is periodic, linear and stationary. No room for FFT in analysis of the R-R. Ok. If you open every day journals , also of the highest scientific value, you continue to see that authors examine R-R signals using FFT. They apply this technique also if the limitations of such procedure are evident and they are increased when you have to examine pathological conditions, as well as when you have to select a proper window, when you are forced to resample at a given frequency and still more. FFT studies continue to be dominant in a lot of applications of the HRV studies. I repeat : scholars that follow HRV studies well know its importance and the constant use of the FFT. Zbilut and I , in order to avoid such inappropriate tendency, published years ago a new method elaborated by me. It was published on Chaos , Solitons and Fractals and prepared as software from the Nevrokard company where one may download the software also free of charge. However, the tendency remains: FFT is dominant.

What is the reason of my such so long discussion. Science aften goes by paradigmatic tendencies. The same thing happens when we have to study non linear possibly chaotic time series. The tendnecy , still again , is dominant , to study the correlation dimension , the lyapunov spectrum and some other parameter. Also in this case the limitations are evident. You have correct estimations only when you start with a model based on differential equantions but when you have only experimental data , the risk to give inappropriate results is very high. There is no matter to avoid such basic situation. Certainly we have many methodlogical indications .. but no more.

Finally, what is the lesson that is derived to me from years of experience. In order to identify deterministic chaos, basic deterministic features, probabilities and so on you need to have a robust method ( as it is the case of the CZF method for the HRV that I mentioned previously for the R-R). The first rule when we examine time series possibly chaotic is that we musy learn to look at the inner structure of the signal. CZF looks at the inner structure of the CZF ( again look at our site www.saistmp.com) , generally speaking one must use the Recurrence Quantification Analysis , the method discovered from Zbilut and Webber , because it looks at the inner structure of the signal and with a robust method. Using RQA you evaluate Recurrences, Determinism, Laminarity, Entropy, dimension of the attractor, , trapping time, trend , LE , consequently you evaluate probabilities.... in brief you have a large spectrum of appropriate informations obtained by a robust method.

Is there a better measurement tool than lyapunov exponent for chaotic systems?

Answer: Yes , it is the Recurrence Quantification Anallysis.

Still another comment. Note that we have not deterministic chaos only. We have Deterministic chaos, noise, non-deterministic chaos that is based on violation of Lispchitz condition in equation differential model of the system. In this case noise has a very robust and important function. The matter is not so simple as it could appear at the first stage. Research on non determinsitic chaotic systems is based as example searching singularties substantially given from a very large Lyapunov exponent-obviously- (going to infinity). Still another question. When using the RQA one may divide the given time series by epochs and thus evaluating lyapunov by epochs , estimating better its behaviour step by step. I see great advantages in this method.

I would like to point out the problem of "perturbations" and the sensitivity to initial conditions. The Lyapunov exponent is calculated keeping in view those "perturbations", hopefully small, the smallest the better (?).

What about the bahavior of the system when "perturbations" are not infinitisimally small but, on the contrary, they are BIG, or very BIG? Can the system be sensitive to such big perturbations and can it exhibit deterministic chaos behavior?

I know that it can and I can provide enough evidence for that. This implies that the condition for witnessing a chaotic behavior based on the Lyapunov exponent based on sensitivity to small perturbations is a special case confined to very small scales

which may be difficult to monitor in reality whereas we fail to grasp some bigger picture of the dynamics and its determinants.

In reality chaotic systems undergo perturbations of different size/energy. So far, you managed to refer to small perturbations only hoping that those can evolve exponentially and that they solely account for ALL chaotic expansions, which I know is not true. This could explain why a positive Lyapunov exponent you consider is an insufficient condition for determining chaos. It is no wonder at all.

Intuitively one should be able to imagine that an exponential divergence from initial conditions following small perturbations eventually leads to inability to follow its trajectory over time, and then what will you do with its determinants? Eventually you will know less than in the beginning which is contrary to what you had been expecting from a chaotic system.

And let us go back to the problem of causality. If it is ignored that you risk analyzing overlapping dimensions and it is possible that when they overlap and you merely see their projection you will not observe deterministic chaos, whereas 1-dimensional maps will show deterministic chaos.

As such the Lyapunov exponent does NOT refer to a trigger of changes in the system but merely to sensitivity to an UNKNOWN CAUSE and UNKNOWN SIZE/ENERGY. You certainly do not want to study small causes only, do you?

Talking about the causality in the biggest possible scale (everybody saw it). Take NIKKEI 225, study it carefully to understand that it was not the earthquake and the tsunami in Japan which caused the crash in 2011 but it was the earthquake in New Zealand on February 22 which triggered an escape of investors from TOKYO.

One hour before the earthquake in Japan NIKKEI 225 had dipped exactly to 3.5699 of a trigger down from Feb. 22. See the logistic map what that 3.5699 means. Chaos is not about averages or rounded figures. It is exact. Its borders are set exact.

Then refer to Mandelbrot set to see where 3.5699 is located on the graph. Then ponder for some time why the market crashed 14.208 times the size of the trigger down and not more. Then take more time to think why it recovered from the -25% slump.

Study NIKKEI 225 from 2011 for deterministic chaos in its purest sense.

Dear Elio,

You indicated that Recurrence Quantification Analysis is a better tool. It is used in many references in your lab's website too. Can you please introduce a good reference for reading it?

Dear Paul,

I think, in chaotic systems, big perturbs probably will yield negative or zero LEs, because the final position stays limited. What do you think about that?

yes, r=3.5699 is the point that logistic map becomes chaotic. It is related to population growth usually. How is it related to earth quack?

Dear Elio,

do you know better measure for the stability of a nonlinear system than LE? I address this question more to our young followers, less to you. A nonlinear pendulum can still work in a linear stable mode as a clock.

I do not personally believe in an existance of fully linear systems. All are chaotic. All behave determinisitic compunds. All are influnced by noise. But we do not really know what the noise means.

As a senior "chaos researcher" (beginning 1979) I feel to be indepted to get my knowlegde even without my own profit in this discussion forum.

Sure, I hope to learn also something from you and all other disputants.

My definition of chaos (for open systems) is different than that of Strogatz:

It is a d-dimensional non-linear fractal system in which in at least one dimension

the dynamics is that of a unimodal map, the causal trigger occurs, and the unit fractal evolves universally at an exponential rate uniquely linked to Feigenbaum constant, and the Lyapunov exponent is greater than 0.44.

Each trigger is related to its own dimension.

The causal trigger is a more general concept of the "perturbation".

UPOs, triggers and fractal expansions are the determinants of such (chaotic) systems.

Triggers are independent, they do not interfere with each other. The inner built memory never decays over time (it resembles solitons). A fractal expansion can take place even in no time as well as take some time (indeterminable).

Chaotic systems are predictable within the span of one fractal expansion but nothing can be said about the duration of time. Trigger size is random and it is a finite number.

Dear Mohsen,

Fortunately big perturbations do not lead to negative or zero LEs.

On the contrary, they lead to linear growth of LEs during fractal expansions.

Dear Elio,

I read RQA method in "Coupling patterns between spontaneous rhythms and respiration in cardiovascular variability signals" I found in http://www.saistmp.com. But it has very briefly introduced the method and has more emphasis on the case study. I found a link to Zbilut's paper in Wikipedia. which one of his papers do you recommend to read for becoming familiar with the method?

Sure, like many other young researchers, I am happy to benefit from the senior experts' experiences like you and Zbigniew. Your guides will help us improving and speeding up our work and prevent repeating some mistakes that we may encounter in the research.

Hi Moshen,

I quite do not understand what do you mean when you wrote " yes, r=3.5699 is the point that logistic map becomes chaotic. It is related to population growth usually. How is it related to earth quack? "

Concerning the logistic map, things are rather clear. Say x_n+1= r x_n (1-x_n) where x_n is in [0,1] and r in [0,4]. For r values 0

Hi Miguel.

You are absolutely right about the effects of changing value of r. I just wrote that in response to one of the comments of Paul Koszarny . You can find it in previous comments. As he wrote, according to butterfly effect, " it was not the earthquake and the tsunami in Japan which caused the crash in 2011 but it was the earthquake in New Zealand on February 22 which triggered an escape of investors from TOKYO.

One hour before the earthquake in Japan NIKKEI 225 had dipped exactly to 3.5699 of a trigger down from Feb. 22. See the logistic map what that 3.5699 means."

It seems possible if the the governing rule in the earth quack is the logistic map. It shows how much chaos is important in our life.

This is just a metaphor. And certainly metaphors are useful in order to get insights into understanding certain natural phenomena. But there is not a model of earthquakes based on the logistic map.

Some U.S. seismologists are skeptical that there have been documented cases of strange animal behavior prior to earthquakes. The United States Geological Survey, a government agency that provides scientific information about the Earth, says a reproducible connection between a specific behavior and the occurrence of a quake has never been made.

So far so good about animal behavior.

But what about people?

The only opportunity for the man to express his anxieties in a herd is a traded financial market.

I wish to be understood correctly - I do not study quakes. But I do study chaotic systems of various kinds every day. And I would like to make it simple and precise. Chaos is ALWAYS about precision and order. Chaos provides forensic evidence.

Last year during various heuristic studies of chaos I decided to publish a short forecast of NZDUSD (the kiwi against the greenback), the pair is a part of the global currency market. NZDUSD is popular among traders for some reasons. When the calculations and the analysis were ready it was Saturday, markets were dormant.

I wrote that NZDUSD was going to plummet due to the occurance of a trigger down in the system and I marked the first moderate price target of the predicted fall.

On Monday night New Zealand was hit by the biggest earthquake since 1968. The New Zealand dollar crashed against the USD.

The prediction was correct and the trigger expanded fractally in a universal way. The system's evolution was perfectly orderly. The day of the quake was February 22.

It would hardly make any big story alone. Yet on March 11 Japan was hit by a terrible earthquake and a most devastating tsunami. I rushed to study NIKKEI 225 hourly charts. I was looking for a trigger down while the index was inevitably going downwards. To my amazement I discovered that the causal trigger was giant and had occurred on February 22!

NIKKEI 225 closed on that first disastrous day at exactly 3.5699 times the size of the trigger down (which was -6% in percentages). Eventually the fractal expansion was completed when the index plummeted 14.208 times the size of the trigger down (this is called a catastrophic fractal expansion) which corresponded to a further -19% crash.

The NIKKEI 225 behaved exactly in an orderly manner, like a perfect chaotic system. Why investors had begun fleeing Tokyo 17 days before the disaster? WHAT had scared them or WHO?

Refer to TEPCO and their special report.

Why the index closed exactly at 3.5699 on the day of the tsunami? Not higher, not lower but exactly pip to pip at 3.5699? Recall that 3.5699 is a limes at n going to infinity. Recall that 4.669=F is a limes too. In layman's words for n to go to infinity you need A LOT to be happening almost at the same time. Chaos is not about sporadic, casual events. It is not about a road incident but about all roads and all incidents, all vehicles and all pedestrians taken together at the same time.

The death toll in Japan was 19,300.

On futures contracts one could have profited handsomely (his margin multiplied).

But this is not the key issue here.

How come investors' wealth was apparently better protected than people's lives?

I claim that was the biggest and the nastiest scandal in the history of stock exchanges and nuclear energy.

Chaos is forensic and there are people who hate it for that . You cannot wipe out anything from the system which provides evidence of phase transitions. You know when it rained, when it snowed or when the sky was stark blue. You can tell day from night.

I am glad that Mohsen asked the question about the LE problem in the first place.

Chaotic behavior can be spotted in a variety of systems. The phenomenon of crashes and crises requires conceptual reworking completely. Similarly to the demise of the HEM hypothesis, quite a lot of myths are bound to become extinct.

I wonder how much it would practically help you to accept the evidence that something is deterministic chaos and what would you do with free time availed that way?

In fact theoretical physicists have not doubts about the existence of chaotic-deterministic systems in Nature. Some researchers indicate that we have still also chaotic non deterministic chaotic systems and the theoretical physics as well the experimental evidences supporting such systems are very robust.

I retain to have answered to Paul Koszarny.

I am of the opinion that insted of lyapunov exponent, one can use cellular automata to get all kinds of the behaviour.

In particular chaotic behaviour is also observed thro' cellular automata. Please try.

Yes the positive lyapunov exponent is an indication of chaos

Indication yes but not enough. You need also to prove a determinism. Surogate data test or recurrence plots.

In my opinion it is not sufficient to use recurrence plots. The Recurrence Quantification Analysis (RQA) is necessary. RQA wa introduced by J.P Zbilut and C.L. Webber years ago and in its present formulation it enables to estimate the %Recurrence, the %Determinism, the %Laminarity , the Trapping Time, the entropy and the Max Line that leads to the evaluation of LE Lyapunov. A number of my papers and coauthor Zbilut contains RQA analysis. It is a robust but flexible method and obviously a reconstruction of the given time series data in phase space is required.The other important feature is that the given time series data may be explored by epochs and ( another important feature!) the method does not require transformation or manipulation of the starting data. Two time series may be also simultaneously explored by using the KRQA. The RQA was initailly formulated by the authors to be applied mainly to physiological studies but subsequently a very large spectrum of applications has been performed from medicine , to biology, to chemistry, to physics. An excellent method !

Hello,

Be careful Mehdi, the Lyapunov exponentS are not limited to study attraction to or repulsion from a fix point, but characterize the average contraction/expansion rate in the tangent space of more complex objects than fixed points (typically strange attractors). They are closely related to the geometry of the attractor. For example, an estimate of the fractal dimension of a strange attractor is given by the Kaplan Yorke formula involving positive and negative Lyapunov exponents.

Greetings

Bruno

Hello sir,

for a chaotic system, lyoponent exponent is positive. But if you take any set of data at random, calculate lyaponov exponent. It may be positive. But system need not be chaotic.

So in your statement converse is not true.

Try

n.gururajan.

If using the so called method of Recurrence Quantification Analysis , the so called RQA method that was introduced from Webber and Zbilut years ago, you have to look at tle Max Line . Note that RQA gives also some other important variables to look in into the inner structure of the signal as the %Recurrence, the %Determinism, the % Laminarity , the Trapping Time, The entropy and the Trend. My modest opinion is that , as previously said, this is the most important tool to investigate signals without restrictions and approximations.

RQA offers also a complete view about instabilities by studying as example %Laminarity.

To answer the question in the most specific way possible: chaos has many ``definitions'' but I believe a good combination is

1) Positive Lyapunov exponents

2) bounded orbits

The second condition is the one that excludes things like the repulsive harmonic oscillator or the unstable recursion you mention.

There are, however, systems that do not have bounded orbits ,which are often called chaotic. Thus scattering off a three-disk system is called chaotic, because it has a very complex orbit structure. In particular it has periodic orbits and their number increases exponentially with their length. This is often taken to be yet another characterization of chaos. To the best of my knowledge (1) and (2) usually lead to this proliferation of periodic orbits for the bounded case. Whether or not this is an actual theorem is not clear to me.

Of course, here I am only talking about systems which are defined by a map or a system of ordinary differential equations. Thus I took determinism for granted. In experimental systems, the presence of noise cannot be discounted and its influence must be assessed separately. The very concept of Lyapunov exponent is not, I believe, really useful if the noise component is dominant.

Certainly it is true. Rather lyapunov exponent ( of course we have to look not to Lyapunov exponent but to the Lyapunov spectrum) we have to look at the correlation dimension that estimates with SATURATION the cases of chaos dynamics. In addition an extimated dimension greater than four of five gives immediately indication that noise may have a great influence in that case. Of course Lyapunov exponet is given also in the case of noise or noise corrupted data.

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The best thing to do is ti use recurernce quantifiication analysis that is able to look at the inner strucuture of the time series data.

Finally, the first thing to ascertain is the presence of non linear mechanisms. Also surrogate data help! But Average Mutual Information may be considered a good and valuable technique.

No. There is a classical example of a time dependent linear system with a positive Lyapunov exponent. To be chaotic, there must exist a compact positively invariant set in phase space (necessary).

I do not intend the No! The problem is hjust here.... to ascertain the presence of non linearities ....I insist .. with experimental data the AMI is a satisfactory criterium.... it preserves non linearities when using surriogate data. It cannot give certainty as any other method but helps very much in the analysis.

Obviously one has to consider when starting from a well established model with a well defined set of differential equation and when instead one has only experimental data that are , particularly in biomedical signal, a superposition of linear and non linear mechanisms plus added noise of meausrements and noise arising from the inner structure of the signal and having a very important constriuctive role. This si the question.

Dear Elio,

in experiments we have to do with four kinds of noise:

(1) environmental noise,

(2) measurement's (instrument's) noise

(3) the system intrinsic noise.

(4) noise introduced due to the computational methods applied for estimations.

What is your method to distinguish and separate these all kinds of noise?

No. In dim 1 there is the classic positive topological entropy implying chaos. But we can have chaos at zero topological entropy. In continuous maps the existence of an orbit with odd period implies chaos, this is not again, an equivalence.

Read the papers by Smital and his students and also the papers by the Ukrainian school after Sharkovsky, and papers by Kolyada and his colleagues.

While in general a positive Lyapunov exponent need not imply chaos, there is an important (and perhaps very general) class of cases where it does. Namely, if an attractor supports a physical invariant measure (aka Sinai Bowen Ruelle measure), call it mu, then the Pesin entropy formula holds:

sum of the positive Lyapunov exponents with respect to the measure mu equals the measure theoretic entropy with respect to mu

By the variation principle, this implies if there are positive Lyapunov exponents, then the dynamics restricted to the attractor has positive topological entropy which certainly would satisfy most definitions of chaos.

A nice discussion of these results by two leading world experts (Yakov Pesin and Boris Hasselblatt) can be found at

http://www.scholarpedia.org/article/Pesin_entropy_formula

The conventional explanation of the mechanism of irreversibility or motion of the system to the chaos is basically based on the property of the exponential instability of Hamiltonian systems and the hypothesis of the existence of the fluctuations. Essence of the explanation is as follows. Poincare theorem of reversibility of Hamiltonian systems argues that there is a very large but finite time, during which the system will again be held arbitrarily close near the starting point of the phase space. But if averaged over an arbitrarily small neighborhood of the phase space where the system has a place, due to the exponential instability it will not return to the initial state.

According to this explanation of the mechanism of irreversibility the existence of exponential instability in Hamiltonian systems is necessary because it would be impossible to explain the second law of thermodynamics without it.

But the deterministic explanation of irreversibility also was submitted. It is follows from the equations of motion of structured particles (Somsikov V.M. How to construct of mechanics of the structured particles. arXiv:1205.2922v1 [physics.gen-ph] 8 May 2012). This mechanism is explained by the irreversible transformation of the motion energy of structured particles into the internal energy. In the frame of this explanation the existence of the Lyapunov exponent does not means chaos or irreversibility.

1. First of all, it is very important what is your definition of chaos. Possible definition of chaotic bounded set in phase space is instability by Zhukovsky of all trajectories from the set

https://www.jyu.fi/science/muut_yksikot/summerschool/en/history/en/history/Summer-School2011/courses/COM/notes/note3.pdf

2. In general case, positive larges Lyapunov exponent (LLE) does not imply instability by Lyapunov. There are known Perron effect of LLE sign reversal. The problem lies in the fact that while usually LLE is intrudes as the characteristic of trajectories of considered nonlinear system, LLE computation is based on consideration of linearized system. But Perron effects are quite rare, see e.g. ergodic theory and Lyapunov regularity property.

http://www.math.spbu.ru/user/nk/PDF/Lyapunov_exponent.pdf

G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron effects, International Journal of Bifurcation and Chaos, Vol. 17, No. 4, 2007, pp. 1079-1107

There is, indeed, a lot of confusion and this is due to ambiguities regarding the Lyapunov exponent of a map or a flow, that are glossed over. The Lyapunov exponent we're talking about is an average value, the average taken along the flow or trajectory. This is so, until the system reaches an attractor, whereupon the divergence ceases, since the attractor cannot reach infinity. So one issue is, do we detect this plateau, or not. Another issue is, whether the average value is, also, the typical value for the exponent. There do exist theorems (by Osledets and Furstenberg) regarding the infinite time limit-but the finite time fluctuations, which have been studied, are also interesting and harder to study. The bottom line is that it provides a strong hint, but only that, regarding whether the further properties of a chaotic system are, indeed, present.

For instance, it can come about that the average value is positive, but the distribution is ``broad'' and the typical value is, in fact, negative.

Certainly true. We have always problems in estimation of LLE in finite time series given as example by experimental data. Of course examination of Lyapunov spectrum should not be used in this case.

I don't understand why the Lyapunov spectrum shouldn't be examined-on the contrary it should be analyzed more fully, i.e. one shouldn't compute, simply, the (maximal) Lyapunov exponent, but its complete distribution, that is the relevant quantity to analyze in numerical simulations or experiments. There are, by now, known pitfalls in looking just at the greater Lyapunov exponent (cf. the work of Osborne and Provenzale in fluid mechanics, for instance) but it is known how to avoid them.

Dear Zbigniew,

sorry for my delay in answer. Glial cells in the brain. They have a detailed role in brain dynamics. They give Lyaounov positive exponents. Consider un expected and unavoidable noise in another set of data. They give positive Lyapunov exponent.

Obviously alwyas one may estimate Lyapunov spectrum every time he is able to ARRIVE to such estimation but of course it is rather obvious that a Lyapunov spectrum connected to a set of experimental data is one thing and the spectrum of a map with its definite analytic behaviour is another thing.

If your system is Hamiltonian and if the motion is regular

then the largest Lyapunov exponent is zero, otherwise on chaotic orbits it is positive.

This is true except for some peculiar regular

hyperbolic structures, such as whiskered tori.

Take the case of the pendulum

which is an integrable systems and therefore the Lyapunov exponents should be zero for all the orbits, actually you have a positive Lyapunov exponent on one peculiar orbit: the separatrix.

No, it only means the system has instabilities..in order to get a methid for estimating Lyapunov exponents in empirical data in which you do not know the basic equation (or even when no suche equation does exist) you can use the RQA approach described in the attached paper

The greatest Lyapunov exponent is an average value. If this average value is, also, the typical value, with respect to the invariant measure, then it does imply chaotic behavior. If the typical value is very different from the average value, it could happen that the average value is positive, but the typical value negative, or zero, in which case the system isn't chaotic. 

There are two type of chaos. It is ordinary or stochastic chaos, and deterministic chaos. The first type of chaos for Hamilton's systems connected with the external noise. The Hamilton’s systems with noise and with the positive Lyapunov exponent always equilibrates. For Hamilton’s systems with the positive Lyapunov exponent and without noise (the deterministic chaos) the chaos have another nature. This chaos always has a place for the system from the potential interaction material points with big enourh number of material points.

Stam:

The remark you make is quite true as a general point. However, Oseledec's theorem (the multiplicative ergodic theorem) guarantees that Lyapunov exponents exist and are almost surely independent of initial conditions if the system has an invaraint measure. Since this is a common property, in many cases the Lyapunov exponent is constant with probability one, at least when referred to the invariant emasure. The issue you raise thus  does not arise frequently.

No, a positive Lyapunov exponent does not implies chaos. We can have a trajectory with a positive Lyaponov exponent that is not chaotic. The trivial example is the system x_(n+1)= Ax_(n), where A is the 2x2 diagonal matrix. If det A!=0, |a11|1, a21=a12=0, the point (0,0) is a saddle and the Lyapunov exponents of the trajectories  != from the  fixed point are l1=log(|a11|0. This system is not chaotic.

Why not? If an initial perturbation can be exponentially amplified, the system is chaotic. So it's essential to *define* first, what chaotic behavior is and *then* discuss the many, different, ways, this can appear. Else it becomes metaphysics.

Regarding Oseldets' theorem: indeed, it pertains to the infinite time limit-but, once, more, it states that this average value exists; it doesn't ensure that it is the typical value and that is the issue. There are many cases, in particular in disordered systems, where it is possible to distinguish the average from the typical value of the Lyapunov exponent, e.g. opscience.iop.org/0305-4470/23/13/042

This is from Oseledec:

The trivial example is the system x_(n+1)= Ax_(n), where A is the 2x2 diagonal matrix. If det A!=0, |a11|1, a21=a12=0, the point (0,0) is a saddle and the Lyapunov exponents of the trajectories  != from the  fixed point are l1=log(|a11|0. This system is not chaotic. It has a positive exponent and is not chaotic.QED

This is metaphysics, absent a definition of what is or isn't chaotic and appealing to authority, that expresses an opinion, rather than a technical result,  isn't helpful. It isn't a proof, but a statement.  IF chaos is defined as the exponential amplification of initial perturbations, THEN certain conequences follow, that are completely impersonal. IF it's defined in another way, other consequences follow. So please provide a *definition* of what's chaotic behavior, so it's possible to check, whether a positive value for the greatest Lyapunov exponent is consistent with the definition. That's all, nothing more.  

Qualifying a system as ``trivial'' is a metaphysical statement-it doesn't mean, by itself, that such a system can't exhibit a partucular property. Indeed, that is a remarkable property of maps, i.e. iterations:  that they can exhibit behavior that, for flows, requires a much richer structure (e.g. that the number of degrees of freedom be greater than the constraints from conservation laws).

I am showing you a true chaotic system based on the Henon map. It is real. Of course, it required retrieving the necessary information with topological tools. There you have four triggers (causets) which are large perturbations of the system. You must be familiar with both Feigenbaum constants: F=4.669... and F(alfa)=2.5029... . For b=0, the Henon map reduces to the quadratic map. The modified Henon map incorporates both Fs. For a > 1.4272 the attracting periodic orbit disappears. There are three distinct fractal borders pertaining to Henon 1.4272 in the attached chart. They are marked by an ellipse. Three, because for each trigger that border is in a different place.

Trigger 2 (yellow) marked exactly Henon 1.4272, reversed to -0.64 (this constant you cannot know but it is universal) and then expanded to 3.5699 (this constant you must know).

Trigger 1 (red) marked exactly Henon 1.4272 and expanded fractally to 2.4220 (this constant you cannot know, but you can find out what it is).

Trigger 1 and trigger 2 set a cotarget which numerically is 2.4220 and 3.5699 (triggers treated as unit triggers).

Trigger 3 expanded a little more than to 2.4220. Then it reversed to -0.64.

Trigger 4 expanded a little more than to 3.5699.

The second Feigenbaum constant related to squeezing forces is well visible for trigger 1 and 4.

On top of this, there are three UPOs.    

The system was analysed with 144-dimensional embedding.

What must strike at a glance is the multidimensionality, though the Henon map is two-dimensional, we have (restricted to the size of the chart) 4 two-dimensional systems active at the same time.

Moreover, the triggers never get perturbed by each other, there is no interference, no decay, no loss of internal memory.  

I wonder how long you will stick to the Lyapunov exponents and the theory of small perturbations? The Lyapunov exponent is useless for large perturbations. You cannot come to any consistent conclusions without the knowledge of the nature of causets.  

Do you need the definition of a bicycle to ride a bike?   

Does positive Lyapunov exponent always mean chaos? No. 

I showed you an example that it is possible to track the trajectories of different fractal developments and for that part I think it becomes obvious that the Lyapunov exponent cannot distinguish such different fractal developments and thus must lead to entirely wrong conclusions.  

I hope you get a good night sleep now (without Lyapunov).  

Dear Stam Nicolis . I am agree with you. In addition, let me add, that for answer this question, I think, need more deeply understand the nature of the conflict between Poincaré and Boltzmann than it is explained now. And for this you should understand the restrictions of the formalisms of classical mechanics connected with the hypotheses of the holonomic constraints. This hypothesis was used to obtain the Lagrange equation (see H. Goldstein: Classical mechanics). As far as I remember, C. Lanczos, in his book: «The variational principles of mechanics» emphasized the lack of the answer on this question.