chaosbook:discrete
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chaosbook:discrete [2009/03/19 09:03] predrag John's remarks on visualization of R_{xz} quotiented state space |
chaosbook:discrete [2010/02/02 07:55] |
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| - | <- [[:chaosbook]] | ||
| - | ====== Chapter: World in a mirror ====== | ||
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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// | ||
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| - | ===== Discrete symmetry desymmetrization ===== | ||
| - | |||
| - | ==== Quotienting the discrete translation pCF symmetry ==== | ||
| - | |||
| - | From Halcrow et al. paper on pCf equilibria: | ||
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| - | <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.txt ยท Last modified: 2010/02/02 07:55 (external edit)
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