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chaosbook:discrete [2009/02/12 15:12]
predrag
chaosbook:discrete [2010/02/02 07:55] (current)
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 (ChaosBook.org blog, chapter [[http://​chaosbook.org/​paper.shtml#​discrete|World in a mirror]]) ​ --- //​[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-12//​ (ChaosBook.org blog, chapter [[http://​chaosbook.org/​paper.shtml#​discrete|World in a mirror]]) ​ --- //​[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-12//​
  
-===== Kuramoto-Sivashinski ​desymmetrization =====+===== Discrete symmetry ​desymmetrization =====
  
 +==== Quotienting the discrete translation pCf isotropy subgroup ====
  
 +From Halcrow et al. paper on pCf equilibria:
 +
 +<​latex>​
 +\begin{equation}
 +        \label{subg4RR} ​
 + ​R_{xz} = \{e, \sigma_x \tau_{xz}, \sigma_z \tau_{xz}, \sigma_{xz}\}
 +        = \{e,​\sigma_{xz}\} \times \{e,​\sigma_{z}\tau_{xz}\}
 +        \simeq S \,.
 +\end{equation}
 +</​latex>​
 +
 +The <​latex>​R_{xz}</​latex>​ isotropy subgroup is particularly important, as the 
 +equilibria belong to this conjugacy class, as do
 +most of the solutions reported here. The //NBC// isotropy subgroup of
 +Schmiegel and our //S// are conjugate to <​latex>​R_{xz}</​latex>​ under
 +quarter-cell coordinate transformations. In keeping with previous literature,
 +we often represent this conjugacy class with
 +<​latex>​S = \{e, s_1, s_2, s_3\} = \{e, \sigma_z \tau_x, \sigma_x \tau_{xz},
 +\sigma_{xz} \tau_z\}</​latex>​ rather than the simpler conjugate group <​latex>​R_{xz}</​latex>​.
 +
 +{{gtspring2009:​gibson.png?​24}} Re. methods of visualizing the state-space portraits with the 
 +4th-order <​latex>​R_{xz}</​latex>​ isotropy subgroup quotiented out: the double-angle trick from Lorenz will not suffice here, since
 +we have mirror symmetry <​latex>​(x,​y,​z) \to (-x,​y,​z)</​latex>​ as well as the
 +rotation-about axis <​latex>​(x,​y,​z) \to (-x,​y,​-z)</​latex>​. The double-angle trick is
 +suitable only for the latter. It would reduce the four quadrants to
 +two, but unfortunately not in the way we would like: it would map
 +<​latex>​\tau_{xz} EQ2 to EQ2</​latex>​ and <​latex>​\tau_z EQ2 \to \tau_x EQ2</​latex>,​ leaving us with distinct
 +<​latex>​EQ2,​ \tau_x EQ2</​latex>​. And it's <​latex>​EQ2,​ \tau_x EQ2</​latex>​ we are most interested
 +in equating. -- // John F. Gibson 2009-03-19//​
chaosbook/discrete.1234480337.txt.gz ยท Last modified: 2009/02/12 15:12 by predrag