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January 29th, 2018, 02:49 AM   #1
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Rate of separation

How do I track the rate of separation of nearby orbits in a phase space defined by points $r \in \mathbb{R}^2$ and angles $\phi \in [0, 2\pi)$?

edit: Suppose the map or function which defines the evolution of these points has no explicit definition (no obvious derivative).

It's a straight forward procedure if the phase space is $\mathbb{R}^n$ for example: we just consider the rate at which the distance between the two trajectories change.

Last edited by Joppy; January 29th, 2018 at 03:08 AM. Reason: bracket
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January 29th, 2018, 07:25 AM   #2
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It seems you are talking about Lyapunov exponents but I'm not completely sure since you say this is straightforward in $\mathbb{R}^n$ which is typically not the case.

Are you interested in computing these explicitly or only symbolically? In other words, do you have to actually determine these values for some system or do you need to write down the formulas for some computation?

In either case, one usually computes a derivative for this. In the case of a map this is straightforward and for continuous dynamical systems this amounts to computing the first variation (at least for $C^1$ vector fields).

I am unsure why the phase space plays a role in this. Please elaborate.
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January 29th, 2018, 02:11 PM   #3
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Thanks.

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Originally Posted by SDK View Post
It seems you are talking about Lyapunov exponents but I'm not completely sure since you say this is straightforward in $\mathbb{R}^n$ which is typically not the case.
Yes I'm talking about the Lyapunov exponents. I was intentionally being vague in the chance that my question made some sort of sense standalone.

Ok maybe I was generalizing a bit too much when saying it's straightforward in $\mathbb{R}^n$. What I'm getting at here is that at the very least, on each application of our map we can record the distance between our two orbits. With this distance function, say $\delta_n = ||F^n(\mathbf{x}) - F^n(\mathbf{x}+\delta_0)||, \mathbf{x} \in \mathbb{R}^n$ where $\delta_0$ is some initial separation vector.

If there is exponential divergence between these two orbits, then a straightforward computational formula for the LE is given by $\lambda = \dfrac{1}{n} \sum_{i = 0}^{n-1} \log \left| \dfrac{\delta_{n+1}}{\delta_n} \right| $. Of course, in practice we may need to re-scale the trajectories and some other fiddly things. Please correct me if I've misunderstood something.


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Originally Posted by SDK View Post
Are you interested in computing these explicitly or only symbolically? In other words, do you have to actually determine these values for some system or do you need to write down the formulas for some computation?
I need to determine the values explicitly yes.

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Originally Posted by SDK View Post
In either case, one usually computes a derivative for this. In the case of a map this is straightforward and for continuous dynamical systems this amounts to computing the first variation (at least for $C^1$ vector fields).

I am unsure why the phase space plays a role in this. Please elaborate.
I don't think it is straightforward if there is no explicit expression for the map itself. However as you say, it does amount to computing the derivative which I'm sure can be defined in terms of the phase space variables.

I think I can narrow it down to this: How can we define an iterative procedure for calculating the derivative of a system whose states depend on both position and angle?

Thanks again!
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January 29th, 2018, 08:55 PM   #4
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Your formula for the Lyapunov exponent seems fine except you are only looking at a finite number of iterates. Typically, one takes two limits $|\delta| \to 0$ and $n \to 0$ but if you are only interested in separation over a fixed number of iterates then $\delta \to 0$ limit for the sum you wrote is preceisely what I would look at. This is a discrete version of the finite-time Lyapunov exponent and you would expect this to take the average value of the largest eigenvalue along the linearized orbit.

If you aren't able to take a derivative then how are you computing the map? Should I assume you have this map as some sort of time series data? If so, then I would use finite differences or interpolation depending on how dense your data is.

It may be helpful to know that if you are looking for strong attraction/repulsion such as maximally hyperbolic NHIM, then finite differences will ALWAYS identify such features though the resolution may be less than ideal. There are even algorithms for extracting this on non-euclidean manifolds see this paper for instance.
http://www.cds.caltech.edu/~marsden/...ory/LCS3BP.pdf

The corresponding formula for the finite iteration Lyapunov exponent (FILE) is given in eq: 2.1 in the linked paper.


If I'm misunderstanding something let me know.

Edit: Sorry the paper above is not the one with the algorithm for non-euclidean manifolds. It is this one:
https://vtechworks.lib.vt.edu/bitstr...pdf?sequence=1
Thanks from Joppy

Last edited by SDK; January 29th, 2018 at 09:18 PM.
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January 29th, 2018, 10:27 PM   #5
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Quote:
Originally Posted by SDK View Post
Your formula for the Lyapunov exponent seems fine except you are only looking at a finite number of iterates. Typically, one takes two limits $|\delta| \to 0$ and $n \to 0$ but if you are only interested in separation over a fixed number of iterates then $\delta \to 0$ limit for the sum you wrote is preceisely what I would look at. This is a discrete version of the finite-time Lyapunov exponent and you would expect this to take the average value of the largest eigenvalue along the linearized orbit.
Yes certainly, I forgot the limit there .

Quote:
Originally Posted by SDK View Post
If you aren't able to take a derivative then how are you computing the map? Should I assume you have this map as some sort of time series data? If so, then I would use finite differences or interpolation depending on how dense your data is.
I lied.. I do have the derivative, and have calculated the LE using this. However I want to be able to check the results using more elementary methods. Judging from your comments, I think I must be making some silly mistakes somewhere, I'll keep at it.


Quote:
Originally Posted by SDK View Post
It may be helpful to know that if you are looking for strong attraction/repulsion such as maximally hyperbolic NHIM, then finite differences will ALWAYS identify such features though the resolution may be less than ideal. There are even algorithms for extracting this on non-euclidean manifolds see this paper for instance.
http://www.cds.caltech.edu/~marsden/...ory/LCS3BP.pdf

The corresponding formula for the finite iteration Lyapunov exponent (FILE) is given in eq: 2.1 in the linked paper.


If I'm misunderstanding something let me know.

Edit: Sorry the paper above is not the one with the algorithm for non-euclidean manifolds. It is this one:
https://vtechworks.lib.vt.edu/bitstr...pdf?sequence=1
That's very interesting, thanks for the paper.
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January 30th, 2018, 10:33 PM   #6
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Quote:
Originally Posted by SDK View Post

I am unsure why the phase space plays a role in this. Please elaborate.
How can it not? How am I supposed to measure distance (in the most general sense of the word) in a space where the coordinates are defined not by just points, but angles also?
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