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I was recently going via a book (Strogatz), that mentions Lorenz's attractor, and that it was found out that for values such as $a=10$, $b=\tfrac{8}{3}$, $c=21$, the system behavior is chaotic.

How does one calculate these values for which the system is chaotic?

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  • $\begingroup$ Please ask new questions separately and not by editing old questions. Also, when asking that new question, please provide more details: Why do you want to make a new chaotic system? What are your goals? $\endgroup$ – Wrzlprmft Jun 24 at 14:21
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For the vast majority of systems (that exhibit chaos for some parameter choices), there is no known way to analytically determine when it will be chaotic. The usual approach would be trial and error, i.e., to simulate the dynamics for different choices of parameters and use numerical estimates to see whether it exhibits a chaotic dynamics.

In fact, the classic choice of parameters for the Lorenz system are just what Edward Lorenz happened to use for his weather simulation (when he discovered the chaotic properties of the system). Now, due to its prominence, the Lorenz system is subject to heavy analytic investigations, which yielded proofs of some of its properties [1], [2], but as far as I know there is not yet a way to determine for which parameter choices it is chaotic (but then I fully admit that I only superficially understand these works). Even if they did, they are far from being a practicable tool to find chaotic parameter regimes of arbitrary systems or being the first thing everybody should know about chaos theory.

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  • $\begingroup$ What about if someone comes up with a new system that exhibit chaos? does that indicates he/she has done simulation for various random values of parameters (hit&trial)? $\endgroup$ – akhil krishnan May 4 at 6:54
  • $\begingroup$ @akhilkrishnan: What about if someone comes up with a new system that exhibit chaos? does that indicates he/she has done simulation for various random values of parameters (hit&trial)? – More or less; the parameter choices are usually not completely random. Note though that some system exhibiting chaos is usually not big news anymore. Finally, please see my edit. $\endgroup$ – Wrzlprmft May 4 at 8:41
  • $\begingroup$ Ok.. I got it!... $\endgroup$ – akhil krishnan May 4 at 9:51
  • $\begingroup$ Again, are their any intuitive ways to point out these values? $\endgroup$ – akhil krishnan May 4 at 11:17
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After doing some studies on this subject for months now I did received the answer to the question that I've been looking out for(I'm really Shameful for myself now).Ok then, here it goes as the phenomena of Chaos is Unpredictable so one can NEVER determine the exact values of Chaotic System Parameters for which the system exhibit Sensitive dependence on Initial Conditions. So, what Wrzlprmft said was exactly the correct answer to this question, any how I do wanted to add some more points to this discussion, while you run the simulations then the system (that one is Interested in studying i.e in this case it is Lorenz System) begins to show sensitive dependence on Initial conditions by exhibiting strange attractors these strange attractors can be due to the sensitive dependence on initial conditions (i.e the system can't trace its own trajectory back as we reverse time) or these strange attractors may NOT BE Necessarily CHAOTIC in that case one has to take separate values of System parameters, this can be confirmed via Lyapunov Exponent, Logistic scale Bifurcation Map etc.

All dynamical systems are characterized by a set of three parameters (X,T,flow), here X stands for the topological space that one is Interested in, T means time set which possess 2 distinct types of symmetry one is continuous time symmetry & the other is discreet -time symmetry(the type of symmetry depends on the interest of study for scholars which he/she may likes).

NOTE: Here I've considered continuous-time symmetry.

In case, if the reader is Interested then the system can produce Strange attractors due to time delay this phenomenon is called as Chaotic Branching.In here, for various different types of Topological structures can give rise to some other system attractors (say Lorentz attractor in this case) BUT as their Fixed points are NOT the same they will possess different topological structural properties.

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  • $\begingroup$ so, the Lyapunov exponent measures an estimate of separation of nearby solutions as they are integrated forward in time, but also the system might not be able to trace its own past trayectories as there might be also a Lyapunov exponent for backward time integration, which I assume would coincide with the forward-time exponent only for time-symmetric chaotic dynamical systems $\endgroup$ – lurscher Oct 28 at 0:05
  • $\begingroup$ @ lurscher Yes!. I've also edited my post, I think it might be helpful. $\endgroup$ – akhil krishnan Oct 28 at 8:39

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