Few remarks

First never forget that a positive Lyapunov exponent is not sufficent to conclude. In your case, it is possible to conclude because the data investigated are produced by a deterministic system (Sprott A system). But for experimental data, you need first to a a clue about the underlying determinism...

Second, No transient in conservative system. Each initial conditions lead to a given behavior. For the same parameter values and depending on initial conditions, you may have quasi-periodic regime (torus), periodic trajectory, or chaotic ones.

What counts is the integral of the trace as you mentioned.

Now, frankly, if you compute a Poincaré section and that you have the chaotic sea between the closed loop (signature of the section of a torus), do you really need the Lyapounov to conclude that you have a chaotic regime ? I don't believe so.

At the end, 3D conservative systems are rare, because by definition conservative systems are Hamiltonian (know that a Hamiltonian system can also be dissipative) meaning that the phase space is twice the number of degrees of freedom. Starting with a system with two degree of freedoms, you have thus a 4D phase space. But since you have one constant of motion (conservation of energy), it means that the phase space can be reduced by one, thus leading to a 3D system as the Sprott A system. so, chaotic conservative systems are most of the time written as 4D system (like the Hénon-Heiles for instance).

Christophe

I am not sure what do you mean by 3D conservative system. I have two possible interpretations:

Option A: A Hamiltonian system with three degrees of freedom (a six-dimensional dynamical system)

Option B: A three dimensional volume preserving flow.

Neither have "transient dynamics" in the sense that trajectories cannot be attracted to a torus. Any transient that you might observe must come from your numerical method introducing spurious dissipation. To minimize this risk, the best advice is to employ symplectic integrators or their generalization to volume preserving flows.

Volume preservation or simplecticity provide also a check on the accuracy of the Lyapunov exponent computation: you may compute each exponent independently and then check for the sum against cero.

I meant Option B. You are right about not having transient dynamic. I had forgotten that. I will investigate the way you proposed. Thank you very much.

However, is there any code someone can introduce me for that? I want a code (MATLAB code) that I give it a 3D system with an initial condition and it tell me whether it is chaotic for sure or not. E.g. consider the famous Sprott A system:

x' = y

y' = -x+yz

z' = 1-y^2

It is well known that for (x0,y0,z0) = (0,5,0) the trajectory is chaotic. However for (x0,y0,z0) = (1,0,0) there will be a torus.

Compute the Lyapunov exponents.

You can try this method---0-1 test for chaos. It is directly used to the time series data.

If I'm not wrong, the Sprott A system you display above is not volume preserving, since the trace of the Jacobian is z. So the system you show above can be area preserving, dissipative or expansive depending on the values of z. In second place, I think you should be carefull telling the differentce between convervative or autonomous and area preserving (I say area, because Hamiltonian systems conserve area proyections, so if your system is three dimensional and area preserving, two Lyapunov exponents should have same value with opposite sign, and the other one should be zero), since conservative means that the force field comes from a potencial, while area preserving systems might not come from a potential. E.g. The forced undamped pendulum is area preserving but not conservative. If the trace of the Jacobian of the 3D flow you are dealing with is zero, then you should use a symplectic (fouth order would be nice) integrator and compute the Largest Lyapunov Exponent. But be carefull, because the coexistence of Kam islands (torus) with hyperbolic regions (destroyed tours) make the dynamics dependent on the region of the phase space you are in, according to Kam's theorem. On symplectic integrators for matlab see:

http://www.mathworks.com/matlabcentral/fileexchange/7686

For Hamiltonian chaos I recommend, for instance:

Chaos and integrability in nonlinear dynamics (Michael Tabor)

If your system is not area preserving, a fourth order Runge-Kutta should work, as far as I am experienced and concerned.

I would plot Poincare section and get from it a return map. Arrangement of points on return map would give hints about periodic or chaotic properties of trajectories as well as about completion of transient mode.

I agree. If you don't need to give a quantitative proof of chaos, plotting the Poincaré map is the easiest way to see at glance Kam islands and hyperbolic regions. Although more difficult, you can compute invariant manifolds of a saddle fixed point and see homoclinic intersections, which are the responsible for chaos. Melnikov's analysis is based on such fact, and can be developed for some systems. Power spectra analysis is another quantitative method to detect chaos. By the way, if you don't feel like programming Lyapunov exponents or something else, J.A. Yorke's "Dynamics: Numerical explorations" software is available and free on the internet. Although I recommend programming everything. Anyway, the program is very useful to learn about chaos. You can arrange your own 3D processes with it, and compute many interesting typical features of Dynamical systems, such us periodic fixed points, Lyapunov exponents, invariant manifolds, basins of attraction, bifurcation diagrams, chaotic parameter sets, chaotic saddles, etc...

Dear Alvaro Garcia

I am so happy knowing you. I will be in contact with you. Thank you for those beautiful answers.

I forgot to say that in Sprott A, since the average of "z" in long time is zero, system is area preserving. LEs are :

+0.014

0

-0.014

Just put the '3D conservative system' into matlab, solve it using ODE45 etc and then use this to calculate the Lyapunov exponents; a single positive exponent (you only really need to calculate the maximum one, which is trivial) means chaos, no doubt about it.

Don't make it as complicated as the above.

Few remarks

First never forget that a positive Lyapunov exponent is not sufficent to conclude. In your case, it is possible to conclude because the data investigated are produced by a deterministic system (Sprott A system). But for experimental data, you need first to a a clue about the underlying determinism...

Second, No transient in conservative system. Each initial conditions lead to a given behavior. For the same parameter values and depending on initial conditions, you may have quasi-periodic regime (torus), periodic trajectory, or chaotic ones.

What counts is the integral of the trace as you mentioned.

Now, frankly, if you compute a Poincaré section and that you have the chaotic sea between the closed loop (signature of the section of a torus), do you really need the Lyapounov to conclude that you have a chaotic regime ? I don't believe so.

At the end, 3D conservative systems are rare, because by definition conservative systems are Hamiltonian (know that a Hamiltonian system can also be dissipative) meaning that the phase space is twice the number of degrees of freedom. Starting with a system with two degree of freedoms, you have thus a 4D phase space. But since you have one constant of motion (conservation of energy), it means that the phase space can be reduced by one, thus leading to a 3D system as the Sprott A system. so, chaotic conservative systems are most of the time written as 4D system (like the Hénon-Heiles for instance).

Christophe

I second 0-1test. It is simple enough and will only give you yes or no.

One more remark and some requests

It is not true that conservative systems don't display transient chaos. This is true if they are bounded, but false if escapes are present. For instance, Henon-Heiles systems is a hamiltonian conservative system that, for determined values of the energy, presents three escapes and transient chaos.

I agree that 3D conservative systems are rare. I am not totally sure, but I guess that, in general terms, 3D systems of Sprott's kind, even when area preserving, might not be Hamiltonian (symplectic). I mean, perhaps you can not find a 4D Hamiltonian system associated to the 3D. I have tried to do so for Sprott A case, and have not been able so far. This makes me think that symplectic integrators are of not use in such cases. So, for area preserving not Hamiltoninan cases, some other integrators would be required. I have heard about geometric integration, which is designed to preserve some property of a system. Does anybody know nice literature on this?

And for experimental data, when you say.... you need first to have a clue about the underlying determinism. Does this mean that for noisy systems trayectories can diverge, as for example in a Wiener process, whose deviation increments with time, even if there is no chaos? And if there is some underlying determinism, can you use LE? I guess all this things are main topics in experimental data analysis, embeddings, topological analysis, etc... Could you please recommend me some nice literature where this features are explained in not an excessively mathematical fashion?

Thanks and regards

3D conservative systems naturally arise in analysis of fluid flows. I worked some years ago with so-called Stokes flows and adiabatic chaos in such flows. Indeed, some of them cannot be associated with Hamiltonian flows directly because of the presence of stagnation points. However, in the systems I worked with it was possible to perform some kind of averaging, and the averaged system turned out to be Hamiltonian. If you are interested, you may have a look at works by Neishtadt, Vainchtein and myself. Please do not consider this as self-advertisement :-)

The term conservative is often used with some ambiguity leaving rigor in the hands of context. It might refer to those that preserve energy, in mechanics, but it might also refer to those that preserve a given measure in the space state of more general dynamical systems.

In 3D it is common to speak conservative systems to refer to those that preserve the three-dimensional volume. A common source of 3D volume preserving dynamical systems comes from hydrodynamics: the Lagrangian description of the flow of incompressible fluids. Please note that while in 2D this problem can be easily asociated with a Hamiltonian system (the stream function place the role of H) in 3D this is not the case.

While the dynamics of 3D volume preserving flows with or without periodic time-dependence shares many qualitative features with Hamiltonian systems, they cannot rigorously associated to this class. Therefore, strictly symplectic methods do not apply to this case. Nevertheless, there are generalizations of the symplectic methods that can be applied to ensure volume-preservation in odd-dimensional systems.

With reference to the answer by Seng-Kin Lao, the standard reference for 0-1 test is:

Georg Gottwald and Ian Melbourne (2004) - (Proc. Roy. Soc. A 460, 603–611).

The input is any time series, that may come from a discrete map, a differential equation or an experiment.

The output is a single number - either 0 (for non-chaotic data) or 1 (for chaotic data).

The 0-1 test is said to have some advantages over other methods such as calculating Lyapunov exponents.

Recent paper (by the same authors) on 0-1 test:

On the Implementation of the 0-1 Test for Chaos

[Georg A. Gottwald and Ian Melbourne, 2009]

http://arxiv.org/abs/0906.1418

Best wishes,

Sundar