January 29th, 2018, 01:49 AM  #1 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,591 Thanks: 546 Math Focus: Yet to find out.  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 02:08 AM. Reason: bracket 
January 29th, 2018, 06:25 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 376 Thanks: 205 Math Focus: Dynamical systems, analytic function theory, numerics 
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. 
January 29th, 2018, 01:11 PM  #3  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,591 Thanks: 546 Math Focus: Yet to find out. 
Thanks. Quote:
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}^{n1} \log \left \dfrac{\delta_{n+1}}{\delta_n} \right $. Of course, in practice we may need to rescale the trajectories and some other fiddly things. Please correct me if I've misunderstood something. Quote:
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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!  
January 29th, 2018, 07:55 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 376 Thanks: 205 Math Focus: Dynamical systems, analytic function theory, numerics 
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 finitetime 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 noneuclidean 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 noneuclidean manifolds. It is this one: https://vtechworks.lib.vt.edu/bitstr...pdf?sequence=1 Last edited by SDK; January 29th, 2018 at 08:18 PM. 
January 29th, 2018, 09:27 PM  #5  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,591 Thanks: 546 Math Focus: Yet to find out.  Quote:
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January 30th, 2018, 09:33 PM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,591 Thanks: 546 Math Focus: Yet to find out.  

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