User:Pokipsy76/sandbox

Features of subtle edit warring:

• The hostile revert. One editor add information and this is automatically reverted by another without explanation. If the first editor protests in a way that is at all intemperate, the hostile editor invokes all the Wiki rules about incivility.
• Pony express excuses for removing information and links. This is the determined removal of information on any pretext whatever. If one reason fails to stand up the hostile editor moves on to the next reason, and the next and the next. If someone protests in a way that is at all intemperate, the hostile editor invokes all the Wiki rules about incivility.
• Mushroom edits. Editor A makes a point with a properly cited link. A hostile editor then removes the direct link and leaves an indirect link. The indirect link preserves the appearance of a link but makes it hard for the general reader to find the specific web page. The reason? It helps to conceal information that the hostile editor doesn't like.
• Tunnel Vision edits. Editor A puts a point in a wider context. Editor B removes the point and the links on the pretext that the wider context is 'off topic'.
• Original Research abuse. Claims that something would be original research in article A but not original research in article B.

(to be developed, starting draft taken from here)

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Frame

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Iterated linear maps are important in dynamical systems because the local behaviour of nonlinear maps near non-singular fixed points is connected to the behaviour of the iterated linear maps given by their linear approximation (by Hartman-Grobman theorem and Kolmogorov–Arnold–Moser theorem).

Linear maps of the plane are classified according to their eigenvalues which in the 2-dimensional case are given by:

$2\lambda =-\mathrm {tr} (A)\pm {\sqrt {(\mathrm {tr} (A))^{2}-4\mathrm {det} (A)}}$

Distinct real eigenvalues

If the map has two distinct real eigenvalues then it is conjugated (by a linear transformation) to the linear map

$B={\begin{bmatrix}\lambda &0\\0&\mu \end{bmatrix}}$

and

$B^{n}={\begin{bmatrix}\lambda ^{n}&0\\0&\mu ^{n}\end{bmatrix}}$

therefore

• if both eigenvalues have absolute value greater than 1 the map is expansive
• if both eigenvalues have absolute value smaller than 1 the map is contractive
• if one eigenvalue is greater and the other smaller than 1 the map is called hyperbolic and it has a contracting and an expanding direction given by the eigenspaces

Single real eigenvalue

A linear map with equal eigenvalues is conjugated (by a linear transformation) to the linear map

$B={\begin{bmatrix}\lambda &a\\0&\lambda \end{bmatrix}}$

if $a=0$ then $B$ is a homothety and $B^{n}=\lambda ^{n}\mathrm {Id}$, otherwise

$B^{n}={\begin{bmatrix}\lambda ^{n}&na\lambda ^{n-1}\\0&\lambda ^{n}\end{bmatrix}}=\lambda ^{n}{\begin{bmatrix}1&{\frac {na}{\lambda }}\\0&1\end{bmatrix}}$

and when $\lambda =1$ the map is called parabolic.