Can we measure the effect of noise on a chaotic signal using the lyapunov exponent?

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Theophile Fozin

Lyapunov exponent is a quantitative measure which express how chaotic is a signal. From the best of my knowledge, it will not be the good candidate for measuring noise in both channels. However, i think the méthod proposed by Li-Jen Shang and Kuo-Kai Shyn in the journal Chaos Soliton and fractals in 2009 will help you to solve your problem. Indeeed, they proposed a method for extracting chaotic signal from noisy environment.

Good luck!

http://www.sciencedirect.com/science/article/pii/S0960077909001222

Je Sen Teh

As mentioned earlier by Theophile, you cannot use Lyapunov exponent to measure or extract noise from a channel. However, the Lyapunov exponent can help you determine if noise has effects on the chaotic signal.

For example, it may be possible (however quite improbable) that noise will cause the chaotic signal to have Lyapunov exponents that are lower than zero, indicating that the signal non-chaotic. On the other hand, the presence of noise may also cause a chaotic map that has windows of periodicity (such as the logistic map) to have Lyapunov exponents that are always positive. If the Lyapunov exponent is used in such a manner, it can help you depict the effect of noise but not measure it.

Edwin A. Umoh

Lyapunov exponent is used for testing for chaos in deterministic systems. As already explained above, LEs are not used for measuring noise or it effects in chaotic systems. Even when a chaotic system has been degraded by the presence of noise, such effect could only lead to either increase or decrease in the value of the LEs.

Sergey Vladimirovich Astakhov

Dear Majid,

I can recommend you to use the measure, proposed in the papers attached. This measure is connected with positive Lyapunov exponents, but enables one to determine the degree of noise influence on both regular and chaotic systems.

Best regards,

Sergey

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