Dear Sajad,

I agree with Siew and Hans on this interesting question of what is in there and what is not in there. I believe this question has to do with the fundamental nature of the system.

In the case of a parameter-dependent system and given an initial condition, the value we select for the free parameter certainly will dictate the subsequent dynamics (sometimes drastically). Indeed, when we change this value, we may inhibit a certain type of dynamics while promoting another type or even, perhaps, triggering the emergence of yet another type.

However, all these types are integral part of the system. They are already entangled there. The value of the parameter simply plays the role of a catalyst / inhibitor. As such, it may affect drastically the behavior of the system but will not really modify its fundamental nature. In other words, what is already in there will always be in there and what is not in there will not be added.

In some cases, the free parameter may play the role of a temperature. At certain critical temperatures, the system will undergo a phase transition with a profound modification of the underlying symmetry of its dynamics.

About what your question? You can solve this systems numerically and you can see the result at different R. Also you can investigate the regimes of stability of stationary points (saddle, stable focus, unstable focus, etc). For Lorentz system for example, if stationary point is the unstable focus, system shows chaotic behaviour.

You can think of nonlinear dynamical systems like this also as information/entropy processing devices. In such systems, information/entropy is generated and dissipated. For different parameters, these two types of processes compete to different extents. When the system becomes chaotic, my personal understanding is that information dissipation is too rapid for effective information processing. This way of looking at nonlinear systems does not require us to talk about attractors (although it is still useful to talk about them). Note that information dissipation can still occur in systems where energy is conserved.

Chaos can only occur in highly non-linear dynamical systems. Thus, the non-linear behavior causes chaotic responses under certain conditions. Recall, highly non-linear systems are highly dependent on initial and boundary conditions.

Dear Sajad,

I agree with Siew and Hans on this interesting question of what is in there and what is not in there. I believe this question has to do with the fundamental nature of the system.

In the case of a parameter-dependent system and given an initial condition, the value we select for the free parameter certainly will dictate the subsequent dynamics (sometimes drastically). Indeed, when we change this value, we may inhibit a certain type of dynamics while promoting another type or even, perhaps, triggering the emergence of yet another type.

However, all these types are integral part of the system. They are already entangled there. The value of the parameter simply plays the role of a catalyst / inhibitor. As such, it may affect drastically the behavior of the system but will not really modify its fundamental nature. In other words, what is already in there will always be in there and what is not in there will not be added.

In some cases, the free parameter may play the role of a temperature. At certain critical temperatures, the system will undergo a phase transition with a profound modification of the underlying symmetry of its dynamics.

Very interesting question. I would take a more practical approach and solve the problem by noting that there is no such thing as a pure chaotic system. It is a mathematical idealisation. What really would happen in a practical chaotic system is that stochastic noise (that is inherent in all physical systems) will be simultaneously present. One can never ignore this noise, no matter how small, because chaotic systems amplify small perturbations by their very nature.

And remember from information theory, that noise is very information rich. So in any real chaotic system you will see entropy evolve.

Dear Sajad,

I am sorry but I don't find your question sufficiently clear.

Are you speaking of "deterministic chaos systems" specifically (such as Lorenz's system) or more generally of chaos? Actually I would be interested in your definition of the term "chaos".

If you speak of deterministic chaos systems, you surely know that these are totally deterministic. Thus, mathematically speaking their evolution over time should be totally predictible (actually it is not, even when there are mathematical equations such as for Lorenz's system, because the system of equations, which are non linear, is not analytically solvable and requires a computarized treatment of the data with approximations in the calculation, see below).

However deterministic chaos systems are very sentitive to initial conditions: actually it is possible to demonstrate, for a deterministic chaos system like Lorenz's one, that a given uncertainty on initial conditions, as low as possible, exponentially increases over time (this is the reason why a computarized treatment of the data with approximations in the calculation leads to unpredictibility after some time).

Thus, as there are always uncertainties on initial conditions when dealing with real physical systems, physically speaking the states of such systems are not predictible after some time.

I don't know if these assertions answer your question.

Dear Marc,

Thank you for your answer. However:

1. For simplifying, I limit my question to 3D simple determine flows like Lorenz, Rossler, and Sprott systems.

2. I see no relation between sensitivity to initial condition issue and what I want.

3. I believe what I have questioned is an open problem in chaos theory. I sense there can be some improvements in finding the answer by looking it from topological point of view. Some works by Professors Robert Gilmore, Jean-Marc Ginoux, and Christophe Letellier are helpful references.

Dear Sajad,

an equations system like Lorenz's simulates a real chaotic system: and in real systems (be it weather or heart's rythm) chaos comes from the presence of multiple forcing parametres; the combinations of their (possible) values creates a phase-space in which Lorenz's a ( for instance) is embedded. So different values of a simply tell us in which part of the ( very complicated) phase-space we are, either chaotic or not.

Dear professor Diebner,

I agree to your description of generic chaos. I like the interaction and open system viewpoint. However, I think that mathematical point of view is more important and basic. I think there are rules (mathematical rules) for everything in this world. I even sense that many of those rules which cause very complicated phenomena may be very simple (this is why I like chaos theory). Some may be disagree, but I see no harm in asking about chaos in a pure mathematical way.

Thank you very much for the answers.

Yours sincerely

Maybe I couldn't say what I want well. Of course they are related and compatible (or even sides of one thing). However I dream about finding a simple quantitative answer.

To be concrete, the existence of a chaotic behavior results from a collection of unstable periodic orbits (chaos has nothing to do with noise or any stochastic processes since arising from 100% deterministic system). What changes between R=26 and R=28, is the population of unstable periodic orbits. The bifurcation between this two parameter values (which is not a very common one, in fact, a subcritical Hopf bifurcation) remains to be extensively studied in terms of periodic orbits. In a general way, to investigate what change when a parameter is varied, just track the periodic orbits. Then, you have to investigate if there is a stable orbit which attracts the trajectory or not, how numerous are the unstable periodic orbits. How is the attraction basin around this collection of unstable periodic orbits, and so on. A simple quantitative answer is, I believe, quite utopic (at least today).

Lorenz differential equation (LDE) has only three fixed points when x’,y’ and z’ are equal to zero. However for certain parameters in LDE in which LDE goes to chaos mode, we get very large number of fixed points (much much greater than 3) in the strange attractor that does not correspond to x’=y’=z’=0. Where do these fixed points come from?

To the initiate, the differences between solutions to equations with chaotic behavior and solutions to equations without chaotic behavior can be mystified. Only special values of input parameters yield chaotic output, and thus this question is a good one because the emergence of what appears to be unpredictability in output is not expected, especially when (as far as empiricism tells us) most other parameter values fail to yield what appears to be unpredictable output.

I have a non-perjorative explanation to the question that allows us to see the question to be a red herring (again, without judgement). Consider the following:

(1) 100% deterministic equations almost always have deterministic output, but sometimes they do not. Thus All DE have output that is either deterministic, or non-deterministic. The general class thus being "DEs", not "DEs with D output".

(2) So far we cannot predict which, why, or where non-determinism will reliable emerge from 100% deterministic equations (and thus the question).

(3) We perceive, but do not know, that deterministic output is 'the norm', and that non-determinism is somehow unusual due to its apparant rarity. The apparent rarity exists due to our species' preference for well-behaved equations that give reliable output.

(4) Our perception of scarcity of sets thus being completely subjective, if we ignore relative frequencies of members of the two classes, we can more generally define deterministic output as a set of special cases of output from deterministic equations (no one would disagree that instances of non-deterministic output are special cases).

(5) Perhaps we have explored mostly the types of equations that mostly lead to deterministic output because we seek utility and avoid nonsense. (i.e., we prefer elegant solutions that lie to us).

(6) Perhaps the universe is filled with equations that we do not yet know of that have input parameter sets that lead to non-deterministic output.

(7) Would it not be reasonable then to say that every set of input parameters combined with deterministic or non-deterministic output is a special case of relationships among input parameters, relations expressed via the equation, and output?

(8) A paradox of sort exists because input parameter values that lead to what we think of a non-deterministic output ALWAYS lead to non-deterministic output, making that output both predictable and deterministic.

(9) A thought experiment in which a student is only taught first about equations that have non-deterministic output for certain input parameters could change, for them, the definition of canonical, usual, or normative, and for us, help us realize the subjectivity of empirical experience on those definitions.

(10) Perhaps also many more equations which are 100% deterministic in operation have non-deterministic output for pairs or trios of input parameter values, we simply have not happened upon them in the particular domain of inquiry in which those equations are routinely employed.

(11) If chaos, for example, is a special case, then so are all non-chaotic behaviors. Compare round vs. eliptical orbits; do we know which are the special case? A perfectly round orbit can be modeled by an equation for an elipse, but is the reverse true? And so perhaps deterministic output is a special case and we are blinded by its frequency to confuse the special case with the general, which is "equations".

(12) An AI such a Genetic algorithm could be made to learn equations that tend to apparently non-deterministic output from specific input parameters and millions or billions of such equations could be found to exist. It would of course encounter many that failed to be shown to be apparently non-deterministic, but that exercise should inform us on the existence of non-deterministic output and learn meta-categories of types of such behaving equations.

For more perspective on our species' biases and knowledge, visit UnbreakingScience.com