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RE: Instability In Chaotic Systems: Predicting the Future From Present in a Deterministic Chaotic System
A positive Lyapunov exponent does not always yield a choatic system. For example, consider the map which maps x to 2x where x is in R. The lyaponov exponen is positive but the map is clearly not chaotic.
Thank you for your usual perusal @mathowl.
Before the positive Lyapunov exponent will be tested, the exponential divergence must have been checked, which is a first requirement before computing the Lyapunov exponent. Also, I am using other methods such as the correlation dimension.
The TISEAN software I am using is that developed by Holger Kanz so it is just necessary I discuss what he has to say on the subject matter as referenced in[[1]]
You will agree with me that there are lots of arguments still going on about the best way to characterized chaos. That was why I clearly stated that I am writing in the confines of the methods and tools I am using. (See my last statement before the thank you message)
Any suggestions on the best way to characterize will be appreciated.
So let's take an ODE x'=2x with x in R^2. Could you show that the maximal Lypunov exponent is positive and explain why this system is not chaotic?
Note that if the maximal Lyapunov exponent is positive (and exists in the sense that the limit converges) then there is exponential divergence
I tried to avoid mathematical expressions (I leave that for the mathematicians). Let try to give an equation here.
Consider Xn1 and Xn2 as two points in phase space with distance ||Xn1 – Xn2|| = δ0 << 1
Let δΔn be the distance between the trajectories of these two points some later time Δn
Then:
δΔn = || Xn1 +Δn - Xn2+ Δn||.
The Lyapunov exponent 𝛌 is determined by:
δΔn ≈ δ0e𝛌Δn,
With δΔn << 1 and Δn>> 1
This equation remains valid provided that the limit converges like you have rightly said and then there will be exponential divergence if 𝛌 is positive.
Other methods such has phase space method, false nearest neighbors; correlation dimension etc. will also be used on the data set I am working on.
It is however not all close trajectories that will eventually have this (that is the essence of false nearest neighbor) and so the consideration is only on those that fulfill this requirement.
I think the challenge here is that I am working with real data, i.e. (actual scalar measurement as most physics research using nonlinear dynamical tool), so in a way, we have to use a proper embedding dimension that fits the data. Then we also consider modeling the physical system.
Perhaps when I finish collecting my data and start my analysis, it would be much more clearer the angle from which the TISEAN software will be computing and searching for chaos in the data set.
Thank you and I look forward to our discussion on the chaos this week. I really want to have a robust research and I can use any help I can get.
Edit: to add, I agree that the estimated finite dimension and Lyapunov exponents on their own cannot be a suitable prove for strict nonlinear determinism if a clear scaling region cannot be established. One suggested way to avoid this is to test first if the data could be explained by a linear model. Some literature suggested the use of surrogate data
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You didn't answer my question.
Somehow I missed this. You want me to prove the math model you wrote in your comment right?
Well, perhaps I can do that in a post when I am much stable.