Jay, I'm not entirely sure I have you question correctly fathomed. Are you asking "how can we determine" or "how to express" that A is more chaotic than B? Thanks, Scot
There is also the argument that a system is either chaotic ... or just not. Degrees of chaos are not relevant for some system implementations.
I agree with Jesus in general, entropy would be a good place to start looking, but only if the question is clarified. Given identical starting input conditions. High entropy may easily mistaken for other forms of possibly linear distributions and therefore lead the observer to thinking a system is chaotic, when indeed it is not.
I do not think Jay's question is answerable completely unless he clarifies some details asked for.
A further consideration is that Jay just says "system is more chaotic than another chaotic system"... this is possibly a dangerous way to look at the problem. Chaos may (and often is) be present in a system ONLY within certain defined input conditions. 2 different systems with identical inputs may show non chaotic and chaotic behaviour and yet both be capable of producing chaotic attractors for some set of input conditions.
@ Scot Forshaw
Thank you sir for your response;
Yes, my question is somewhat like: " how can we determine or how to express that system A is more chaotic than system B?"
@ Jesus Gilberto Concepcion,
@ Aleksei Obrosov
Thank you sir for your response.
Please suggest some notes or material on Entropy. I mean how Entropy can be use to express, a system is how much chaotic it is?
Use simple thermodiynamics.to start.
Try to measurre o estimate the unitial state and final state. Obatin and equilibirioum constant and how it changes with temperature. You can obtain and enthalpy. From there you can stimate an entropy value. Then you can refine these values. Happy new year. The differences in entropies can give you an idea of the difference in chaotic conditions.
@ Jesus Gilberto Concepcion
Thank you sir. Happy new year you too sir. I will try.
Thank You all for your responses. I rewrite the question again clearly.
Let the system A and system B are both chaotic dynamical system. Let the first Lyapunov exponents for system A and system B are alpha and beta respectively and say alpha>beta. It appears from the responses that the system A is more chaotic than system B based on Lyapunov exponents characteristics and some available literature (attached here).
Is there any other quantitative measure to compare the chaoticness of the system A & B ?.
Besides positive lyapunov expontents and entropies, you can observe power spectrum: the more chaotic system is, more broad is its spectrum, and show less sharp peaks. Also, correlation dimension and Lyapunov (Kaplan-Yorke) dimension increases.
Some papers that can helps you:
Anishchenko, V. S., Okrokvertskhov, G. A., Vadivasova, T. E., & Strelkova, G. I. (2005). Mixing and spectral-correlation properties of chaotic and stochastic systems: numerical and physical experiments. New Journal of Physics, 7(1), 76.
Anishchenko, V. S., Vadivasova, T. E., Strelkova, G. I., & Okrokvertskhov, G. A. (2004). Statistical properties of dynamical chaos. Math. Biosciences and Engineering, 1(1), 161-184.
Anishchenko, V. S., Vadivasova, T. E., Okrokvertskhov, G. A., & Strelkova, G. I. (2003). Correlation analysis of dynamical chaos. Physica A: Statistical Mechanics and its Applications, 325(1), 199-212.
- Badges
- Science topic
- Similar topics
- Mathematics