http://www.openpipeflow.org/index.php?action=history&feed=atom&title=Method_of_slicesMethod of slices - Revision history2026-10-10T08:49:56ZRevision history for this page on the wikiMediaWiki 1.39.1http://www.openpipeflow.org/index.php?title=Method_of_slices&diff=184&oldid=prevApwillis at 14:30, 5 November 20142014-11-05T14:30:56Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:30, 5 November 2014</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When a dynamical system has a homogeneous spatial dimension, a state can be shifted without altering its physical properties. The ''method of slices'' reduces all states that are identical up to a shift, to a unique state in a ''slice''. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When a dynamical system has a homogeneous spatial dimension, a state can be shifted without altering its physical properties. The ''method of slices'' reduces all states that are identical up to a shift, to a unique state in a ''slice''. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Patterns that repeat in time but reappear with a shift are known as relative periodic orbits (RPOs). The key property of the method of slices is that it closes these orbits, so that within the slice RPOs become periodic orbits (POs). </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Patterns that repeat in time but reappear with a shift are known as relative periodic orbits (RPOs). The key property of the method of slices is that it closes these orbits, so that within the slice RPOs become periodic orbits (POs). <ins style="font-weight: bold; text-decoration: none;">A special case is that of relative equilibria (travelling waves), which are steady in a moving frame. These are reduced to equilibria within the slice.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Searching for recurrences, i.e. for POs, in the slice is typically an order of magnitude less work than searching for RPOs in the full space. More importantly, to map out unstable manifolds, and to identify connections between orbits, this is possible with structurally-motivated coordinates only in the sliced space.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Searching for recurrences, i.e. for POs, in the slice is typically an order of magnitude less work than searching for RPOs in the full space. More importantly, to map out unstable manifolds, and to identify connections between orbits, this is possible with structurally-motivated coordinates only in the sliced space.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The method of slices reduces the continuous shift symmetry. The difficulty in symmetry reduction lies in the uniqueness of the shift for a given state, and in ensuring that <del style="font-weight: bold; text-decoration: none;">as </del>the state changes continuously the shift likewise changes continuously. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The method of slices reduces the continuous shift symmetry<ins style="font-weight: bold; text-decoration: none;">, and achieves this with the aid of a ''template'' or reference state</ins>. The difficulty in symmetry reduction lies in the uniqueness of the shift for a given state, and in ensuring that <ins style="font-weight: bold; text-decoration: none;">when </ins>the state changes continuously <ins style="font-weight: bold; text-decoration: none;">in time, </ins>the shift likewise changes continuously. Work on <ins style="font-weight: bold; text-decoration: none;">the </ins>method <ins style="font-weight: bold; text-decoration: none;">of slices </ins>has been funded by the EPSRC [http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/K03636X/1]. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Work on <del style="font-weight: bold; text-decoration: none;">this </del>method has been funded by the EPSRC [http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/K03636X/1]. </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Note </del>that <del style="font-weight: bold; text-decoration: none;">the symmetry-reduced space</del>, <del style="font-weight: bold; text-decoration: none;">i</del>.<del style="font-weight: bold; text-decoration: none;">e</del>. the <del style="font-weight: bold; text-decoration: none;">slice</del>, is not a <del style="font-weight: bold; text-decoration: none;">Poincaré section</del>. <del style="font-weight: bold; text-decoration: none;">Rather than having points </del>that <del style="font-weight: bold; text-decoration: none;">pierce the section, </del>a full trajectory is produced within the slice. Nor is <del style="font-weight: bold; text-decoration: none;">this </del>low dimensional modelling, as the original state can be fully reconstructed.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Other approaches </ins>that <ins style="font-weight: bold; text-decoration: none;">involve shifts include</ins>, <ins style="font-weight: bold; text-decoration: none;">e</ins>.<ins style="font-weight: bold; text-decoration: none;">g</ins>.<ins style="font-weight: bold; text-decoration: none;">, </ins>the <ins style="font-weight: bold; text-decoration: none;">selection of a moving Galilean frame. As this does not reduce states that differ by a shift to a unique state, however</ins>, <ins style="font-weight: bold; text-decoration: none;">it </ins>is not a <ins style="font-weight: bold; text-decoration: none;">symmetry reduction method</ins>. <ins style="font-weight: bold; text-decoration: none;">The moving frame will not reduce ''all'' relative equilibria to equilibria, for example.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Note </ins>that a full trajectory is produced within the slice<ins style="font-weight: bold; text-decoration: none;">, which is convenient for analysing stable and unstable manifolds, and connections between states. The slice is not the same as Poincaré section, where one has only points that pierce the section</ins>. Nor is <ins style="font-weight: bold; text-decoration: none;">the method of slices </ins>low dimensional modelling <ins style="font-weight: bold; text-decoration: none;">either</ins>, as the original state can be fully reconstructed.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The method of slices can be applied to any system with a homogeneous dimension. Examples where it have been applied include the </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The method of slices can be applied to any system with a homogeneous dimension. Examples where it have been applied include the </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Lorenz equations [http://arxiv.org/abs/1101.3037] and pipe flow [http://arxiv.org/abs/1203.3701].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Lorenz equations [http://arxiv.org/abs/1101.3037] and pipe flow [http://arxiv.org/abs/1203.3701].</div></td></tr>
</table>Apwillishttp://www.openpipeflow.org/index.php?title=Method_of_slices&diff=183&oldid=prevApwillis at 14:00, 5 November 20142014-11-05T14:00:58Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-content" />
<col class="diff-marker" />
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<tr class="diff-title" lang="en-GB">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:00, 5 November 2014</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>When a dynamical system has a homogeneous spatial dimension, <del style="font-weight: bold; text-decoration: none;">patterns </del>can <del style="font-weight: bold; text-decoration: none;">arise at any location in this dimension</del>. The ''method of slices'' reduces all states that are identical <del style="font-weight: bold; text-decoration: none;">but for </del>a shift to a unique state in a ''slice''. Patterns that repeat in time but with a shift are relative periodic orbits (RPOs). The key property of the method of slices is that it closes these orbits<del style="font-weight: bold; text-decoration: none;">. Within </del>the slice RPOs <del style="font-weight: bold; text-decoration: none;"> </del>become periodic orbits (POs). </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>When a dynamical system has a homogeneous spatial dimension, <ins style="font-weight: bold; text-decoration: none;">a state </ins>can <ins style="font-weight: bold; text-decoration: none;">be shifted without altering its physical properties</ins>. The ''method of slices'' reduces all states that are identical <ins style="font-weight: bold; text-decoration: none;">up to </ins>a shift<ins style="font-weight: bold; text-decoration: none;">, </ins>to a unique state in a ''slice''. </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Patterns that repeat in time but <ins style="font-weight: bold; text-decoration: none;">reappear </ins>with a shift are <ins style="font-weight: bold; text-decoration: none;">known as </ins>relative periodic orbits (RPOs). The key property of the method of slices is that it closes these orbits<ins style="font-weight: bold; text-decoration: none;">, so that within </ins>the slice RPOs become periodic orbits (POs). </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Searching for recurrences, i.e. for POs, in the slice is typically an order of magnitude less work than searching for RPOs in the full space. More importantly, to map out unstable manifolds, and to identify connections between orbits, this is possible with structurally-motivated coordinates only in the sliced space.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Searching for recurrences, i.e. for POs, in the slice is typically an order of magnitude less work than searching for RPOs in the full space. More importantly, to map out unstable manifolds, and to identify connections between orbits, this is possible with structurally-motivated coordinates only in the sliced space.</div></td></tr>
</table>Apwillishttp://www.openpipeflow.org/index.php?title=Method_of_slices&diff=175&oldid=prevApwillis at 10:51, 5 November 20142014-11-05T10:51:07Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:51, 5 November 2014</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">TO DO!</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">When a dynamical system has a homogeneous spatial dimension, patterns can arise at any location in this dimension. The ''method of slices'' reduces all states that are identical but for a shift to a unique state in a ''slice''. Patterns that repeat in time but with a shift are relative periodic orbits (RPOs). The key property of the method of slices is that it closes these orbits. Within the slice RPOs become periodic orbits (POs). </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Searching for recurrences, i.e. for POs, in the slice is typically an order of magnitude less work than searching for RPOs in the full space. More importantly, to map out unstable manifolds, and to identify connections between orbits, this is possible with structurally-motivated coordinates only in the sliced space.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The method of slices reduces the continuous shift symmetry. The difficulty in symmetry reduction lies in the uniqueness of the shift for a given state, and in ensuring that as the state changes continuously the shift likewise changes continuously. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Work on this method has been funded by the EPSRC [http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/K03636X/1]. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Note that the symmetry-reduced space, i.e. the slice, is not a Poincaré section. Rather than having points that pierce the section, a full trajectory is produced within the slice. Nor is this low dimensional modelling, as the original state can be fully reconstructed.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The method of slices can be applied to any system with a homogeneous dimension. Examples where it have been applied include the </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Lorenz equations [http://arxiv.org/abs/1101.3037] and pipe flow [http://arxiv.org/abs/1203.3701].</ins></div></td></tr>
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