Practice round

In order to help grasp the concepts and pitfalls with dynamical-system data analysis a ‘practive round’ is introduced before starting to deal with the field-campaign data.

The objective

Say there exists a list of dynamical variables (\(a\)) whoose evolution (\(a(t)\)) depends on the a larger list of variables (\(a_1\)) according to,

\[\frac{\mathrm{d}a}{\mathrm{d}t} = \mathcal{L}\left( a_1 \right),\]

where \(a \in a_1\) and \(\mathcal{L}\) represents some operator.

Because the central assumption here is that the physical processes and their interactions are hard to understand, we need to assume that the field data may only loosely represent \(a\) and/or \(a_1\). For generality we introduce \(a_2\) with \(a_2 \in a_1\) and the actial meassurement data available for anlysis (\(b\)) is,

\[b = \mathcal{f}\left(a_2\right)\]

At this point is becomes obvious that the original dynamical paramters (\(a\)) are lickely obfuscated by the meassurement set \((b)\), especially if we do not know what either \(a, a_1, \mathcal{L}\left(a_1 \right)\) nor \(a_2\) is. So retrieving the original system for \(a(t)\) is challenging, and may require a proper process-based understanding.

A starting point

A first sanity check could be to investigate if the meassurement data is complete, which we define as:

\[\frac{\partial b}{\partial t} = \mathcal{B}(b).\]

Where \(\mathcal{B}\) is some functional operator. Consider an example:

A favourable scenario

The classical predator-prey model for \(F\)oxes and \(B\)unnies, \(a = \{F, B\}\):

\[\frac{\mathrm{d}F}{\mathrm{d}t} = FB - F,\] \[\frac{\mathrm{d}B}{\mathrm{d}t} = \frac{2}{3}B - \frac{4}{3}FB,\]

which has periodic solitions that we make more more interesting by adding a random perturbation each time-integration step (details are else where and not really relevant). A plot of the solution in \(F--B\) space is plotted below:

Parametric plot of a(t)

\(b\) is defines as a linear transformation of \(a\) with small random noise.

\[b = Ba\]

More specifically, \(b\) has 3 components according to:

\[b = \begin{pmatrix} 1 & -2\\ -3 & 4 \\ -5 & 6 \end{pmatrix} a + \mathrm{noise}.\]

The data entries (by design) lie on a 2D plane which warrants the usage of principal component analysis. Altough the reduction in data dimensionality is nice, it may further convolute the interpretation of the data.

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