build based on 1ad6a8c

dev/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.7","generation_timestamp":"2024-12-31T16:34:38","documenter_version":"1.8.0"}}
+{"documenter":{"julia_version":"1.10.7","generation_timestamp":"2024-12-31T16:49:15","documenter_version":"1.8.0"}}

dev/examples/gMAM_Maierstein/index.html

@@ -98,14 +98,14 @@
     markersize=10) for i in eachindex(fp)]
 
 lines!(ax2, reduce(hcat, tr), linewidth=1, color=(:black, 0.2))
-fig</code></pre><img src="66ea99e4.png" alt="Example block output"/><h2 id="Basins-of-attraction"><a class="docs-heading-anchor" href="#Basins-of-attraction">Basins of attraction</a><a id="Basins-of-attraction-1"></a><a class="docs-heading-anchor-permalink" href="#Basins-of-attraction" title="Permalink"></a></h2><p>Basins of attraction are the regions in the state space that lead to a particular attractor. We can find the basins of attraction using the function <code>basins</code>.</p><pre><code class="language-julia hljs">using Attractors
+fig</code></pre><img src="0a4574e4.png" alt="Example block output"/><h2 id="Basins-of-attraction"><a class="docs-heading-anchor" href="#Basins-of-attraction">Basins of attraction</a><a id="Basins-of-attraction-1"></a><a class="docs-heading-anchor-permalink" href="#Basins-of-attraction" title="Permalink"></a></h2><p>Basins of attraction are the regions in the state space that lead to a particular attractor. We can find the basins of attraction using the function <code>basins</code>.</p><pre><code class="language-julia hljs">using Attractors
 
 ba = basins(sys, [0.0, 0], [0.0, 1], [1.0, 0], intervals_to_box([-2, -2], [2, 2]), bstep=[0.01, 0.01], ϵ_mapper=0.001, Ttr=100)
 Ur, Vr, atr, M = ba
 heatmap(Ur, Vr, M)</code></pre><img src="a4971797.png" alt="Example block output"/><p>The basin boundaries can be quickly extracted using the function <code>basin_boundaries</code>.</p><pre><code class="language-julia hljs">bb = basinboundary(ba)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×401 Matrix{Float64}:
  -2.0  -1.99  -1.98  -1.97  -1.96  …  1.95  1.96  1.97  1.98  1.99  2.0
   0.0   0.0    0.0    0.0    0.0      0.0   0.0   0.0   0.0   0.0   0.0</code></pre><h2 id="Transitions"><a class="docs-heading-anchor" href="#Transitions">Transitions</a><a id="Transitions-1"></a><a class="docs-heading-anchor-permalink" href="#Transitions" title="Permalink"></a></h2><p>We can quickly find a path which computes a transition from one attractor to another using the function `transition.</p><pre><code class="language-julia hljs">paths_ends = (fp[stab][1], fp[stab][2])
-path, time, succes = transition(sys, paths_ends...);</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(2-dimensional StateSpaceSet{Float64} with 641 points, [517.7171629237718, 517.7376002302709, 517.7494329181949, 517.7599766401947, 517.7703452646747, 517.7792676865303, 517.788877834018, 517.7979336902946, 517.8064944742007, 517.8149114538829  …  523.8363302652367, 523.8493929443385, 523.8625686444242, 523.8765884429681, 523.8920703025415, 523.908016604301, 523.9248026164025, 523.9409609274759, 523.957922755368, 523.957922755368], true)</code></pre><pre><code class="language-julia hljs">fig = Figure(size=(600, 400), fontsize=13)
+path, time, succes = transition(sys, paths_ends...);</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(2-dimensional StateSpaceSet{Float64} with 900 points, [106.48141489813125, 106.49780026691688, 106.5119462440886, 106.52693095484169, 106.53908156437723, 106.55011244791457, 106.56099909907199, 106.57074227742457, 106.58044624767932, 106.59005455391035  …  115.17352926435885, 115.18420243831719, 115.19534784999497, 115.20647170494696, 115.21768471454715, 115.22925062199255, 115.24101942028322, 115.25378758660449, 115.2681517737159, 115.2681517737159], true)</code></pre><pre><code class="language-julia hljs">fig = Figure(size=(600, 400), fontsize=13)
 ax = Axis(fig[1, 1], xlabel=&quot;u&quot;, ylabel=&quot;v&quot;, aspect=1.4,
     xgridcolor=:transparent, ygridcolor=:transparent,
     ylabelrotation=0)
@@ -125,11 +125,11 @@
 fig
 
 lines!(ax, path, color=:black)
-fig</code></pre><img src="5f4d0b05.png" alt="Example block output"/><p>If we want to compute many: <code>transitions</code> is the function to use.</p><pre><code class="language-julia hljs">tt = transitions(sys, paths_ends..., 3; tmax=1e3);</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Transition path ensemble of 3 samples
+fig</code></pre><img src="cf5c0022.png" alt="Example block output"/><p>If we want to compute many: <code>transitions</code> is the function to use.</p><pre><code class="language-julia hljs">tt = transitions(sys, paths_ends..., 3; tmax=1e3);</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Transition path ensemble of 3 samples
            - sampling success rate:      1.0
-           - mean residence time:        470.887
-           - mean transition time:       8.359
-           - normalized transition rate: 56.3</code></pre><pre><code class="language-julia hljs">fig = Figure(size=(600, 400), fontsize=13)
+           - mean residence time:        63.288
+           - mean transition time:       10.049
+           - normalized transition rate: 6.3</code></pre><pre><code class="language-julia hljs">fig = Figure(size=(600, 400), fontsize=13)
 ax = Axis(fig[1, 1], xlabel=&quot;u&quot;, ylabel=&quot;v&quot;, aspect=1.4,
     xgridcolor=:transparent, ygridcolor=:transparent,
     ylabelrotation=0)
@@ -150,7 +150,7 @@
 for i in 1:length(tt.paths)
     lines!(ax, tt.paths[i])
 end
-fig</code></pre><img src="7eb9dc25.png" alt="Example block output"/><h2 id="Large-deviation-theory"><a class="docs-heading-anchor" href="#Large-deviation-theory">Large deviation theory</a><a id="Large-deviation-theory-1"></a><a class="docs-heading-anchor-permalink" href="#Large-deviation-theory" title="Permalink"></a></h2><p>In the context of nonlinear dynamics, Large Deviation Theory provides tools to quantify the probability of rare events that deviate significantly from the system&#39;s typical behavior. These rare events might be extreme values of a system&#39;s output, sudden transitions between different states, or other phenomena that occur with very low probability but can have significant implications for the system&#39;s overall behavior.</p><p>Large deviation theory applies principles from probability theory and statistical mechanics to develop a rigorous mathematical description of these rare events. It uses the concept of a rate function, which measures the exponential decay rate of the probability of large deviations from the mean or typical behavior. This rate function plays a crucial role in quantifying the likelihood of rare events and understanding their impact on the system.</p><p>For example, in a system exhibiting chaotic behavior, LDT can help quantify the probability of sudden large shifts in the system&#39;s trajectory. Similarly, in a system with multiple stable states, it can provide insight into the likelihood and pathways of transitions between these states under fluctuations. In the context of the Minimum Action Method (MAM) and the Geometric Minimum Action Method (gMAM), Large Deviation Theory is used to handle the large deviations action functional on the space of curves. This is a key part of how these methods analyze dynamical systems.</p><p>The Maier-Stein model is a typical benchmark to test such LDT techniques. Let us try to reproduce the following figure from <a href="https://homepages.warwick.ac.uk/staff/T.Grafke/simplified-geometric-minimum-action-method-for-the-computation-of-instantons.html">Tobias Grafke&#39;s blog post</a>:</p><p><img src="../maierstein-dynamics.png" alt="maier_stein"/></p><p>Let us first make an initial path:</p><pre><code class="language-julia hljs">xx = range(-1.0, 1.0, length=100)
+fig</code></pre><img src="672fe833.png" alt="Example block output"/><h2 id="Large-deviation-theory"><a class="docs-heading-anchor" href="#Large-deviation-theory">Large deviation theory</a><a id="Large-deviation-theory-1"></a><a class="docs-heading-anchor-permalink" href="#Large-deviation-theory" title="Permalink"></a></h2><p>In the context of nonlinear dynamics, Large Deviation Theory provides tools to quantify the probability of rare events that deviate significantly from the system&#39;s typical behavior. These rare events might be extreme values of a system&#39;s output, sudden transitions between different states, or other phenomena that occur with very low probability but can have significant implications for the system&#39;s overall behavior.</p><p>Large deviation theory applies principles from probability theory and statistical mechanics to develop a rigorous mathematical description of these rare events. It uses the concept of a rate function, which measures the exponential decay rate of the probability of large deviations from the mean or typical behavior. This rate function plays a crucial role in quantifying the likelihood of rare events and understanding their impact on the system.</p><p>For example, in a system exhibiting chaotic behavior, LDT can help quantify the probability of sudden large shifts in the system&#39;s trajectory. Similarly, in a system with multiple stable states, it can provide insight into the likelihood and pathways of transitions between these states under fluctuations. In the context of the Minimum Action Method (MAM) and the Geometric Minimum Action Method (gMAM), Large Deviation Theory is used to handle the large deviations action functional on the space of curves. This is a key part of how these methods analyze dynamical systems.</p><p>The Maier-Stein model is a typical benchmark to test such LDT techniques. Let us try to reproduce the following figure from <a href="https://homepages.warwick.ac.uk/staff/T.Grafke/simplified-geometric-minimum-action-method-for-the-computation-of-instantons.html">Tobias Grafke&#39;s blog post</a>:</p><p><img src="../maierstein-dynamics.png" alt="maier_stein"/></p><p>Let us first make an initial path:</p><pre><code class="language-julia hljs">xx = range(-1.0, 1.0, length=100)
 yy = 0.3 .* (-xx .^ 2 .+ 1)
 init = Matrix([xx yy]&#39;)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×100 Matrix{Float64}:
  -1.0  -0.979798   -0.959596   -0.939394   …  0.959596   0.979798   1.0
@@ -195,4 +195,4 @@
 
 lines!(ax, init, linewidth=3, color=:black, linestyle=:dash)
 lines!(ax, MLP, linewidth=3, color=:orange)
-fig</code></pre><img src="75107219.png" alt="Example block output"/><hr/><p>Author: Orjan Ameye ([email protected]) Date: 13 Feb 2024</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tutorial/">« Tutorial</a><a class="docs-footer-nextpage" href="../sgMAM_KPO/">sgMAM for the Kerr Parametric Oscillator »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Tuesday 31 December 2024 16:34">Tuesday 31 December 2024</span>. Using Julia version 1.10.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+fig</code></pre><img src="75107219.png" alt="Example block output"/><hr/><p>Author: Orjan Ameye ([email protected]) Date: 13 Feb 2024</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tutorial/">« Tutorial</a><a class="docs-footer-nextpage" href="../sgMAM_KPO/">sgMAM for the Kerr Parametric Oscillator »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Tuesday 31 December 2024 16:49">Tuesday 31 December 2024</span>. Using Julia version 1.10.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/examples/sgMAM_KPO/index.html

@@ -35,7 +35,7 @@
     return Matrix([H_pu H_pv]')
 end
 
-sys = SgmamSystem(H_x, H_p)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">SgmamSystem(Main.H_x, Main.H_p)</code></pre><p>We saved this function in the <code>SgmamSystem</code> struct. We want to find the optimal path between two attractors in the phase space. We define the initial trajectory as <code>wiggle</code> between the two attractors.</p><pre><code class="language-julia hljs"># setup
+sys = SgmamSystem{false, 2}(H_x, H_p)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Doubled 2-dimensional phase space containing out-of-place functions</code></pre><p>We saved this function in the <code>SgmamSystem</code> struct. We want to find the optimal path between two attractors in the phase space. We define the initial trajectory as <code>wiggle</code> between the two attractors.</p><pre><code class="language-julia hljs"># setup
 Nt = 500  # number of discrete time steps
 s = collect(range(0; stop=1, length=Nt))
 
@@ -78,4 +78,4 @@
     colormap=[:gray, :gray]
 )
 axislegend(ax)
-fig</code></pre><img src="34f50c09.png" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../gMAM_Maierstein/">« Anlyses for the Maier-Stein system</a><a class="docs-footer-nextpage" href="../../man/CoupledSDEs/">Define a CoupledSDEs system »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Tuesday 31 December 2024 16:34">Tuesday 31 December 2024</span>. Using Julia version 1.10.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+fig</code></pre><img src="34f50c09.png" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../gMAM_Maierstein/">« Anlyses for the Maier-Stein system</a><a class="docs-footer-nextpage" href="../../man/CoupledSDEs/">Define a CoupledSDEs system »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Tuesday 31 December 2024 16:49">Tuesday 31 December 2024</span>. Using Julia version 1.10.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>