https://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&feed=atom&action=historyTFPIE2012: Recursion Schemes of Scientific Models - Revision history2024-03-29T11:20:05ZRevision history for this page on the wikiMediaWiki 1.35.5https://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=60&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T09:29:49Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
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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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoI.png|thumb|alt=Demo: Direct Modelling with Intervals.|Direct Modelling with Intervals.]] The same six situations as above, but now with an interval of potential initial states. Note that periodic solutions are either self-focussing or self-dispersing, for one-sided and overshooting convergence, respectively. Note also how quickly the state becomes completely indeterminate when chaos emerges. The initial intervals are [0.3, 0.4] in all cases except the last, where it is [0.75, 0.7500000000001].</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoI.png|thumb|alt=Demo: Direct Modelling with Intervals.|Direct Modelling with Intervals.]] The same six situations as above, but now with an interval of potential initial states. Note that periodic solutions are either self-focussing or self-dispersing, for one-sided and overshooting convergence, respectively. Note also how quickly the state becomes completely indeterminate when chaos emerges. The initial intervals are [0.3, 0.4] in all cases except the last, where it is [0.75, 0.7500000000001].</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoD.png|thumb|alt=Demo: Direct Modelling with Distributions.|Direct Modelling with Distributions.]] Trajectories of continuous state distribution are plotted for 5 steps as differently coloured cdf graphs. Colours range over the rainbow, in order of increasing wavelength. The initial distribution is uniform on the whole domain in all cases except the last, where it is uniform on the middle third only. The distributions appear to converge in all cases. For <math>r = 4</math> (bottom row), the limiting distribution has been studied thoroughly: it is the ''Beta(1/2, 1/2)'' distribution. Note that the initial distribution hardly matters in this case; it converges quickly and almost surely.</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoD.png|thumb|alt=Demo: Direct Modelling with Distributions.|Direct Modelling with Distributions.]] Trajectories of continuous state distribution are plotted for 5 steps as differently coloured cdf graphs. Colours range over the rainbow, in order of increasing wavelength. The initial distribution is uniform on the whole domain in all cases except the last, where it is uniform on the middle third only. The distributions appear to converge in all cases. For <math>r = 4</math> (bottom row), the limiting distribution has been studied thoroughly: it is the ''Beta(1/2, 1/2)'' distribution. Note that the initial distribution hardly matters in this case; it converges quickly and almost surely.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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;">* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoR.png|thumb|alt=Demo: Inverse Modelling with Observations.|Inverse Modelling with Observations.]] Successively narrowed intervals are plotted. They are obtained by iterating the inference step for the observation ''the next state is in the upper half'' 20 times, starting with a completely indeterminate state. Four types of behaviour appear, from top-left to bottom-right:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 inference fails after one step, because there is no such state.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 inference stagnates, because the observation is not unique.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 inference converges slowly.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 inference converges quickly.</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;"><div>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</div></td></tr>
</table>Btranconhttps://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=58&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T09:26:05Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 09:26, 2 June 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l18" >Line 18:</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>*: The initial state is 0.35 in all cases except the last, where it is 0.7500000000001.</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 initial state is 0.35 in all cases except the last, where it is 0.7500000000001.</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoI.png|thumb|alt=Demo: Direct Modelling with Intervals.|Direct Modelling with Intervals.]] The same six situations as above, but now with an interval of potential initial states. Note that periodic solutions are either self-focussing or self-dispersing, for one-sided and overshooting convergence, respectively. Note also how quickly the state becomes completely indeterminate when chaos emerges. The initial intervals are [0.3, 0.4] in all cases except the last, where it is [0.75, 0.7500000000001].</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoI.png|thumb|alt=Demo: Direct Modelling with Intervals.|Direct Modelling with Intervals.]] The same six situations as above, but now with an interval of potential initial states. Note that periodic solutions are either self-focussing or self-dispersing, for one-sided and overshooting convergence, respectively. Note also how quickly the state becomes completely indeterminate when chaos emerges. The initial intervals are [0.3, 0.4] in all cases except the last, where it is [0.75, 0.7500000000001].</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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;">* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoD.png|thumb|alt=Demo: Direct Modelling with Distributions.|Direct Modelling with Distributions.]] Trajectories of continuous state distribution are plotted for 5 steps as differently coloured cdf graphs. Colours range over the rainbow, in order of increasing wavelength. The initial distribution is uniform on the whole domain in all cases except the last, where it is uniform on the middle third only. The distributions appear to converge in all cases. For <math>r = 4</math> (bottom row), the limiting distribution has been studied thoroughly: it is the ''Beta(1/2, 1/2)'' distribution. Note that the initial distribution hardly matters in this case; it converges quickly and almost surely.</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;"><div>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</div></td></tr>
</table>Btranconhttps://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=57&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T09:22:17Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 09:22, 2 June 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17" >Line 17:</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>*# Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories ''diverge exponentially''.</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>*# Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories ''diverge exponentially''.</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>*: The initial state is 0.35 in all cases except the last, where it is 0.7500000000001.</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 initial state is 0.35 in all cases except the last, where it is 0.7500000000001.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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;">* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoI.png|thumb|alt=Demo: Direct Modelling with Intervals.|Direct Modelling with Intervals.]] The same six situations as above, but now with an interval of potential initial states. Note that periodic solutions are either self-focussing or self-dispersing, for one-sided and overshooting convergence, respectively. Note also how quickly the state becomes completely indeterminate when chaos emerges. The initial intervals are [0.3, 0.4] in all cases except the last, where it is [0.75, 0.7500000000001].</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;"><div>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</div></td></tr>
</table>Btranconhttps://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=55&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T09:19:37Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 09:19, 2 June 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l16" >Line 16:</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>*# Chaotic behaviour at <math>r = 4</math>, ranging over the whole domain. Illustrates one of the defining properties of chaos: orbits (points connected by a trajectory) are ''dense''.</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>*# Chaotic behaviour at <math>r = 4</math>, ranging over the whole domain. Illustrates one of the defining properties of chaos: orbits (points connected by a trajectory) are ''dense''.</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>*# Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories ''diverge exponentially''.</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>*# Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories ''diverge exponentially''.</div></td></tr>
<tr><td class='diff-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 initial state is <del class="diffchange diffchange-inline">|</del>0.35<del class="diffchange diffchange-inline">| </del>in all cases except the last, where it is <del class="diffchange diffchange-inline">|</del>0.7500000000001<del class="diffchange diffchange-inline">|</del>.</div></td><td class='diff-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 initial state is 0.35 in all cases except the last, where it is 0.7500000000001.</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</div></td></tr>
</table>Btranconhttps://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=54&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T09:19:09Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
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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 09:19, 2 June 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10" >Line 10:</td>
<td colspan="2" class="diff-lineno">Line 10:</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;"></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;"></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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoT.png|thumb|alt=Demo: Direct Modelling with Trajectories.|Direct Modelling with Trajectories.]] Trajectories of the logistic map with different initial states and parameter values are plotted for 100 steps, showing six classes of behaviour. From top-left to bottom-right:</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoT.png|thumb|alt=Demo: Direct Modelling with Trajectories.|Direct Modelling with Trajectories.]] Trajectories of the logistic map with different initial states and parameter values are plotted for 100 steps, showing six classes of behaviour. From top-left to bottom-right:</div></td></tr>
<tr><td class='diff-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># Stable limit points that attract the initial state, either one-sided (<math>r<2</math>) or with overshooting (<math>r>2</math>). These correspond to the initial branch of the logistic map's bifurcation diagram.</div></td><td class='diff-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 class="diffchange diffchange-inline">*</ins># Stable limit points that attract the initial state, either one-sided (<math>r<2</math>) or with overshooting (<math>r>2</math>). These correspond to the initial branch of the logistic map's bifurcation diagram.</div></td></tr>
<tr><td class='diff-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># Stable periodic solution of period length 2 that attracts the initial state; corresponds to the first bifurcation at <math>r=3</math>.</div></td><td class='diff-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 class="diffchange diffchange-inline">*</ins># Stable periodic solution of period length 2 that attracts the initial state; corresponds to the first bifurcation at <math>r=3</math>.</div></td></tr>
<tr><td class='diff-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># Stable periodic solution of period length 2 that attracts the initial state; more difficult to find in the bifurcation diagram in a small window of orderly behaviour among chaos.</div></td><td class='diff-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 class="diffchange diffchange-inline">*</ins># Stable periodic solution of period length 2 that attracts the initial state; more difficult to find in the bifurcation diagram in a small window of orderly behaviour among chaos.</div></td></tr>
<tr><td class='diff-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># Stable periodic solution of period length 4 that attracts the initial state; corresponds to the second bifurcation at <math>r \approx 3.45</math>.</div></td><td class='diff-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 class="diffchange diffchange-inline">*</ins># Stable periodic solution of period length 4 that attracts the initial state; corresponds to the second bifurcation at <math>r \approx 3.45</math>.</div></td></tr>
<tr><td class='diff-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># Chaotic behaviour at <math>r = 4</math>, ranging over the whole domain. Illustrates one of the defining properties of chaos: orbits (points connected by a trajectory) are ''dense''.</div></td><td class='diff-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 class="diffchange diffchange-inline">*</ins># Chaotic behaviour at <math>r = 4</math>, ranging over the whole domain. Illustrates one of the defining properties of chaos: orbits (points connected by a trajectory) are ''dense''.</div></td></tr>
<tr><td class='diff-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># Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories ''diverge exponentially''.</div></td><td class='diff-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 class="diffchange diffchange-inline">*</ins># Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories ''diverge exponentially''<ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 class="diffchange diffchange-inline">*: The initial state is |0.35| in all cases except the last, where it is |0.7500000000001|</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;"><div>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|thumb|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</div></td></tr>
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</table>Btranconhttps://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=53&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T09:15:54Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
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<tr class="diff-title" lang="en">
<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 09:15, 2 June 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td>
<td colspan="2" class="diff-lineno">Line 9:</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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</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;"></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;"></td></tr>
<tr><td class='diff-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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoT.png|<del class="diffchange diffchange-inline">frame</del>|alt=Demo: Direct Modelling with Trajectories.|Direct Modelling with Trajectories.]] Trajectories of the logistic map with different initial states and parameter values are plotted for 100 steps, showing six classes of behaviour. From top-left to bottom-right:</div></td><td class='diff-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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoT.png|<ins class="diffchange diffchange-inline">thumb</ins>|alt=Demo: Direct Modelling with Trajectories.|Direct Modelling with Trajectories.]] Trajectories of the logistic map with different initial states and parameter values are plotted for 100 steps, showing six classes of behaviour. From top-left to bottom-right:</div></td></tr>
<tr><td class='diff-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 class="diffchange diffchange-inline">** </del>Stable limit points that attract the initial state, either one-sided (<math>r<2</math>) or with overshooting (<math>r>2</math>). These correspond to the initial branch of the logistic map's bifurcation diagram.</div></td><td class='diff-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 class="diffchange diffchange-inline"># </ins>Stable limit points that attract the initial state, either one-sided (<math>r<2</math>) or with overshooting (<math>r>2</math>). These correspond to the initial branch of the logistic map's bifurcation diagram.</div></td></tr>
<tr><td class='diff-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 class="diffchange diffchange-inline">** </del>Stable periodic solution of period length 2 that attracts the initial state; corresponds to the first bifurcation at <math>r=3</math>.</div></td><td class='diff-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 class="diffchange diffchange-inline"># </ins>Stable periodic solution of period length 2 that attracts the initial state; corresponds to the first bifurcation at <math>r=3</math>.</div></td></tr>
<tr><td class='diff-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 class="diffchange diffchange-inline">** </del>Stable periodic solution of period length 2 that attracts the initial state; more difficult to find in the bifurcation diagram in a small window of orderly behaviour among chaos.</div></td><td class='diff-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 class="diffchange diffchange-inline"># </ins>Stable periodic solution of period length 2 that attracts the initial state; more difficult to find in the bifurcation diagram in a small window of orderly behaviour among chaos.</div></td></tr>
<tr><td class='diff-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 class="diffchange diffchange-inline">** </del>Stable periodic solution of period length 4 that attracts the initial state; corresponds to the second bifurcation at <math>r \approx 3.45</math>.</div></td><td class='diff-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 class="diffchange diffchange-inline"># </ins>Stable periodic solution of period length 4 that attracts the initial state; corresponds to the second bifurcation at <math>r \approx 3.45</math>.</div></td></tr>
<tr><td class='diff-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 class="diffchange diffchange-inline">** </del>Chaotic behaviour at <math>r = 4</math>, ranging over the whole domain. Illustrates one of the defining properties of chaos: orbits (points connected by a trajectory) are ''dense''.</div></td><td class='diff-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 class="diffchange diffchange-inline"># </ins>Chaotic behaviour at <math>r = 4</math>, ranging over the whole domain. Illustrates one of the defining properties of chaos: orbits (points connected by a trajectory) are ''dense''.</div></td></tr>
<tr><td class='diff-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 class="diffchange diffchange-inline">** </del>Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories ''diverge exponentially''.</div></td><td class='diff-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 class="diffchange diffchange-inline"># </ins>Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories ''diverge exponentially''.</div></td></tr>
<tr><td class='diff-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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|<del class="diffchange diffchange-inline">frame</del>|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</div></td><td class='diff-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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|<ins class="diffchange diffchange-inline">thumb</ins>|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <math>r=4</math> are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</div></td></tr>
</table>Btranconhttps://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=51&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T09:14:14Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
<col class="diff-marker" />
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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 09:14, 2 June 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td>
<td colspan="2" class="diff-lineno">Line 9:</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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</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;"></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;"></td></tr>
<tr><td class='diff-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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|frame|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with r=4 are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</div></td><td class='diff-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 class="diffchange diffchange-inline">* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoT.png|frame|alt=Demo: Direct Modelling with Trajectories.|Direct Modelling with Trajectories.]] Trajectories of the logistic map with different initial states and parameter values are plotted for 100 steps, showing six classes of behaviour. From top-left to bottom-right:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 class="diffchange diffchange-inline">** Stable limit points that attract the initial state, either one-sided (<math>r<2</math>) or with overshooting (<math>r>2</math>). These correspond to the initial branch of the logistic map's bifurcation diagram.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 class="diffchange diffchange-inline">** Stable periodic solution of period length 2 that attracts the initial state; corresponds to the first bifurcation at <math>r=3</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 class="diffchange diffchange-inline">** Stable periodic solution of period length 2 that attracts the initial state; more difficult to find in the bifurcation diagram in a small window of orderly behaviour among chaos.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 class="diffchange diffchange-inline">** Stable periodic solution of period length 4 that attracts the initial state; corresponds to the second bifurcation at <math>r \approx 3.45</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 class="diffchange diffchange-inline">** Chaotic behaviour at <math>r = 4</math>, ranging over the whole domain. Illustrates one of the defining properties of chaos: orbits (points connected by a trajectory) are ''dense''.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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 class="diffchange diffchange-inline">** Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories ''diverge exponentially''.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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>* [[File:TFPIE2012-_Recursion_Schemes_of_Scientific_Models-demoA.png|frame|alt=Demo: Inverse Modelling with Input.|Inverse Modelling with Input.]] Backwards trajectories for the logistic map with <ins class="diffchange diffchange-inline"><math></ins>r=4<ins class="diffchange diffchange-inline"></math> </ins>are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</div></td></tr>
</table>Btranconhttps://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=50&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T09:05:16Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
<col class="diff-marker" />
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<tr class="diff-title" lang="en">
<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 09:05, 2 June 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td>
<td colspan="2" class="diff-lineno">Line 9:</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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</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;"></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;"></td></tr>
<tr><td class='diff-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>[[File:TFPIE2012<del class="diffchange diffchange-inline">:</del>_Recursion_Schemes_of_Scientific_Models-demoA.png|<del class="diffchange diffchange-inline">thumb</del>|alt=<del class="diffchange diffchange-inline">Example alt text</del>|<del class="diffchange diffchange-inline">Example caption</del>]]</div></td><td class='diff-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 class="diffchange diffchange-inline">* </ins>[[File:TFPIE2012<ins class="diffchange diffchange-inline">-</ins>_Recursion_Schemes_of_Scientific_Models-demoA.png|<ins class="diffchange diffchange-inline">frame</ins>|alt=<ins class="diffchange diffchange-inline">Demo: Inverse Modelling with Input.</ins>|<ins class="diffchange diffchange-inline">Inverse Modelling with Input.</ins>]] <ins class="diffchange diffchange-inline">Backwards trajectories for the logistic map with r=4 are plotted for 100 steps. In the forward mode, the map is completely chaotic for this parameter value, i.e., periodic of all lengths exist, but are unstable. In the backwards mode, these solutions become stable such that most initial states are attracted to them. By the simple expedient of driving the reverse map with a periodic stream of binary input of the desired period length, convergence towards a solution of that period length is effected. Running forwards from a state obtained by this procedure, the appearance of a periodic trajectory is produced. Of course, it will break down eventually because of chaos, but the longer the backwards search procedure, the longer the appearance can be maintained. Simple cases of length 1 and 3 are shown, as well as for 5 and 17. Those are virtually impossible to construct in the forward mode.</ins></div></td></tr>
</table>Btranconhttps://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=47&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T08:58:50Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
<col class="diff-marker" />
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<tr class="diff-title" lang="en">
<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 08:58, 2 June 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td>
<td colspan="2" class="diff-lineno">Line 9:</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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</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;"></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;"></td></tr>
<tr><td class='diff-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>[[File:.png|thumb|alt=Example alt text|Example caption]]</div></td><td class='diff-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>[[File:<ins class="diffchange diffchange-inline">TFPIE2012:_Recursion_Schemes_of_Scientific_Models-demoA</ins>.png|thumb|alt=Example alt text|Example caption]]</div></td></tr>
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</table>Btranconhttps://wiki.tfpie.science.ru.nl/index.php?title=TFPIE2012:_Recursion_Schemes_of_Scientific_Models&diff=46&oldid=prevBtrancon: /* Visual Demonstrations */2012-06-02T08:58:11Z<p><span dir="auto"><span class="autocomment">Visual Demonstrations</span></span></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
<col class="diff-marker" />
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<tr class="diff-title" lang="en">
<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 08:58, 2 June 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8" >Line 8:</td>
<td colspan="2" class="diff-lineno">Line 8:</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;"></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;"></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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</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 following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-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;">[[File:.png|thumb|alt=Example alt text|Example caption]]</ins></div></td></tr>
</table>Btrancon