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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.

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What definition of "degree of chaos" do you want to use? Without defining that it might be difficult to answer your question. –  Ryan Budney Aug 1 '11 at 22:13
    
sorry for using ambiguous language. i actually need to know if there is a measure for chaos. for example, i am working on simulating chaotic oscillator, and i get chaos when i vary different components within some value. i would like to know how the chaos produced with different conditions are different, how i can quantify the degree of chaos, or dimension of chaos, whichever applies. Thanks. –  Vinaya Shrestha Aug 1 '11 at 22:42
    
I read in some paper that multiple positive Lyapunov exponents mean high-dimensional chaos, i am just trying to explore more into this area. –  Vinaya Shrestha Aug 1 '11 at 22:44
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Okay, perhaps if you can tell us what having a "degree of chaos" might accomplish for you, someone here might be able to tell you if there is a precise definition that would be useful for your application. For example, do you want some knowledge of "how much" orbits mix, or "how sensitive" the dynamics are on the initial condition or perhaps you want to measure something else? –  Ryan Budney Aug 1 '11 at 23:20
    
With further reading, I was able to find some papers, where chaos is characterized as low dimensional and high-dimensional chaos. Low dimensional chaos is characterized by a single positive Lyapunov exponent. High-dimensional or hyperchaos, is characterized by a system with multiple positive Lyapunov exponent. Basically, this is was i was trying to find. Thanks for your response. –  Vinaya Shrestha Aug 3 '11 at 20:29
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2 Answers

up vote 4 down vote accepted

Consider the map $T: \mathbb R^4 \to \mathbb R^4$ given by $$ T x = \begin{pmatrix} 1 & 1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \\\ 0 & 0 & \cos(\theta) & \sin(\theta) \\\ 0 & 0 & -\sin(\theta) & \cos(\theta) \end{pmatrix} $$ Then two Lyapunov exponents are $\neq 0$, and two are zero. Furthermore, one sees that the dynamic splits into a chaotic part given by $\begin{pmatrix} 1 & 1 \\\ 1 & 0 \end{pmatrix}$, the cat map, and by a completely regular one given by a rotation by $\theta$.

Hence, you want all Lyapunov exponent non-zero to get fully chaotic dynamics.

Of course, one might interpret all these things differently depending on what one means by chaos.

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can you elaborate this a little, i could barely understand –  Vinaya Shrestha Aug 1 '11 at 22:37
    
What part of my answer do you want me to elaborate on? It contains quite a few (completely standard) but non-trivial claims. –  Helge Aug 2 '11 at 0:18
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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 http://www.scholarpedia.org/article/Pesin_entropy_formula

Quoting this article,

The content of this formula is that the entropy of a measure is given exactly by the total expansion in the system.

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I found that what i actually was looking for is referred to as hyperchaos, I have my answer. Thanks. –  Vinaya Shrestha Aug 3 '11 at 20:31
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