Lyapunov Exponent and degree of chaos - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:35:41Z http://mathoverflow.net/feeds/question/71837 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71837/lyapunov-exponent-and-degree-of-chaos Lyapunov Exponent and degree of chaos Vinaya Shrestha 2011-08-01T22:09:49Z 2011-08-02T03:30:47Z <p>I am aware that having positive Lyapunov exponents in a system signifies that a system is chaotic. However, I would like to know if there is a means to know the degree of chaos in the system from the Lyapunov exponents. For example, does it signify anything if a system has 10 positive Lyapunov exponents out of 25, or all positive Lyapunov exponents. Thanks.</p> http://mathoverflow.net/questions/71837/lyapunov-exponent-and-degree-of-chaos/71840#71840 Answer by Helge for Lyapunov Exponent and degree of chaos Helge 2011-08-01T22:34:43Z 2011-08-01T22:34:43Z <p>Consider the map $T: \mathbb R^4 \to \mathbb R^4$ given by $$T x = \begin{pmatrix} 1 &amp; 1 &amp; 0 &amp; 0 \\ 1 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; \cos(\theta) &amp; \sin(\theta) \\ 0 &amp; 0 &amp; -\sin(\theta) &amp; \cos(\theta) \end{pmatrix}$$ Then two Lyapunov exponents are $\neq 0$, and two are zero. Furthermore, one sees that the dynamic splits into a chaotic part given by $\begin{pmatrix} 1 &amp; 1 \\ 1 &amp; 0 \end{pmatrix}$, the cat map, and by a completely regular one given by a rotation by $\theta$.</p> <p>Hence, you want all Lyapunov exponent non-zero to get fully chaotic dynamics.</p> <p>Of course, one might interpret all these things differently depending on what one means by chaos.</p> http://mathoverflow.net/questions/71837/lyapunov-exponent-and-degree-of-chaos/71854#71854 Answer by Yuri Bakhtin for Lyapunov Exponent and degree of chaos Yuri Bakhtin 2011-08-02T03:30:47Z 2011-08-02T03:30:47Z <p>The (Kolmogorov--Sinai metric) entropy is a measure of chaos in a dynamical system w.r.t. an invariant measure. For a broad class of dynamical systems it is equal to the sum of all positive Lyapunov exponents. I would recommend reading <a href="http://www.scholarpedia.org/article/Pesin_entropy_formula" rel="nofollow">http://www.scholarpedia.org/article/Pesin_entropy_formula</a></p> <p>Quoting this article, </p> <p><i>The content of this formula is that the entropy of a measure is given exactly by the total expansion in the system.</i></p>