Lyapunov exponent - order of magnitude

[dudy]

Hello,
Analyzing data from a chaotic pendulum, I calculated the Lyapunov exponent to be somewhere around 10^5 . While my gut tells me something is wrong with this number , i failed to find any information regarding the order of magnitude of Lyapunov exponents and their meaning.
Can someone give me a "feeling" for different magnitudes of Lyapunov exponents, or a place I can get a less theoretical and more practical read about them?
Thank you

 

[waofy]

That value does seem far too large, unless the typical frequency of the oscillations are in the 100kHz range. The Lyapunov exponent just tells you the rate at which two close trajectories diverge (e.g. two simulations of the pendulum with slightly different initial conditions). Due to the exponential divergence in chaos, the effect doesn't usually become noticable until after several cycles of the system.

If the oscillating frequency is ~1Hz and the Lyapunov exponent is ~10⁵s^-1 then trajectories would diverge well before a single cycle. In this case it becomes very hard to define the attractor of the system. I imagine you'd need a huge number of dimensions so that trajectories wouldn't come back on themselves too quickly. A more "normal" value for the Lyapunov exponent would be ~0.1s^-1.

By the way, what method are you using to calculate the exponent? For a thorough analysis you'll typically have to calculate the Lyapunov spectrum i.e. the exponent for each of the dimensions. The reason is that trajectories will diverge at different rates or even converge depending on the direction in phase space.

Hope that helps. If you need any more info then have a read of the book Nonlinear dynamics and chaos by Steven Strogatz; it explains lots of things with pictures and nice non-mathematical descriptions.

 

[dudy]

First of all thank you very much for the reply- it is extremely helpful.
About the method I'm using- What I did is to take two points from my data that are far apart on the time-line, but have very close values, and watch how the difference (of values) between them grows with time. I repeated this process with different pairs of points.
In light of your reply I'm guessing this is not the right way to go about it, but- why? I mean, I thought I was pretty much calculating the Lyapunov exponent "by definition" this way.

 

[waofy]

Yes, you're right, that is exactly how the Lyapunov exponent is defined. The reason why it's not usually calculated this way is because the exponent only defines local behaviour for bounded systems (i.e. ones that come back on themselves to form cycles). If the divergence due to the Lyapunov exponent held true for all time then the system wouldn't have an attractor, there would just be a series of exponential trajectories moving towards +/- infinity.

The standard method for calculating the Lyapunov spectrum is to plot a single trajectory of the system in phase space. This assumes you have outputs of the system for all dimensions (for chaos there should be at least 3) - if not then you can just use Takens embedding theorem on a 1-D time series to get the extra dimensions. You can then use the algorithm by Eckmann and Ruelle (http://pra.aps.org/abstract/PRA/v34/i6/p4971_1" ).

 

[sardar]

.

The Lyapunov exponent is a measure of chaos and sensitivity to initial conditions in a system. It represents the rate of divergence of nearby trajectories in phase space, and can range from negative values (indicating stability) to positive values (indicating chaos). The order of magnitude of the Lyapunov exponent is an indication of the strength of chaotic behavior in the system.

In general, a Lyapunov exponent of 10^5 is considered very high and indicates a highly chaotic system. This means that even small changes in initial conditions can lead to drastically different outcomes in the system. It is important to note that the magnitude of the Lyapunov exponent is dependent on the specific system being studied, so there is no universal "feeling" for different magnitudes.

To gain a better understanding of the practical implications of different Lyapunov exponent magnitudes, I suggest looking into case studies or real-world applications in fields such as weather forecasting, economics, or biology. These examples can provide a more hands-on and practical perspective on the impact of chaotic systems and the role of the Lyapunov exponent in understanding and predicting their behavior.

Additionally, consulting with experts in the field of chaos theory or nonlinear dynamics may also be helpful in gaining a deeper understanding of the significance of different Lyapunov exponent magnitudes. They can provide insights and context specific to your research and help guide your interpretation of the results.