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Julia programming

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Latest revision as of 21:51, 15 March 2021

Here is the same but animated:

Playing with the parameter space:

And the bottom case, animated:

It is interesting to focus on the surviving spot. We can zoom on this area and see what is going on there. The cells have locked into this stable pattern, involving, apparently, two cells only:

excited = @animate for i ∈ 1:30
           excite()
           plotspace(145, 220, 225, 300)
       end

a configuration by locking two neighbours with a period a+b=30.">

It is interesting as well to change parameters during the simulation. This locks the patterns into some dislocated sequences which give rise to apparent cycles of stationary-evolution in space, which, however, are due to structures that formed under previous conditions of evolution. For instance, the following was obtained by iterating 50 iterations with parameters a=50 and g=25 then changing g=5 for 50 iterations and coming back to the initial g=25 and cycling for ever:

There clearly appears to have two types of oscillations, one seemingly stationary in space whereas the over travels as wavefronts, that get periodically frozen. This is, however, merely an optical artifact, that can be well understood by looking at the state of the cell's evolution in time. Here is such a cut obtained in a similar configuration as the one displayed as a full density plot:

for i=1:75
   println(global iteration+=1); excite();
   Plots.display(plot(space[[100],:]', ylims=(0,a+g), lw=2, legend=false))
end

One can see how the dislocations cause this alternances of drifting or collapsing evolution of the cell states.

Clearly there are many variations and configurations to explore, this is barely scratching the surface. Instead of delving further into this model, we turn to the most famous and also more impressive Hodgepodge machine.

Table of Contents

Crash Course on Scientific Computing

— • 21:51, 15 March 2021

@@ Line 1: / Line 1: @@
-One can generate an array from a range <tt>1:10</tt> using <tt>collect()</tt>:
+<center><wz tip="Still frames for an excitable medium with a=11 and g=2">[[File:Screenshot_20210315_164359.png|400px]]</wz></center>
-<syntaxhighlight lang="python">
-collect(1:10)
-</syntaxhighlight>
-<pre>
--element Array{Int64,1}:
-</pre>
-Let us now turn to a typical numerical problem, solving differential equations. We'll start with Ordinary Differential Equations (ODE). Namely, we consider equations of the type:
+Here is the same but animated:
-$$\label{eq:ODE1} y'(t)=f(t,y(t))$$
+<center><wz tip="250 iterations of the a=11 g=2 excitable model.">[[File:excitable-medium-a2-b11.gif|400px]]</wz></center>
-The most famous algorithm is Euler's method and the most popular the RK4 variant of the Runge–Kutta methods. Methods can be explicit (the solution only depends on quantities already known) or implicit (the solution involves other unknowns which requires solving intermediate equations). The backward Euler method is an implicit method. Euler's method is the simplest one and the typical starting point to introduce the basic idea. Not surprisingly, it is also one of the least efficient one. We discretize time into steps $t_1$, $t_2$, $\cdots$, $t_n$ separated by the same interval $h$. The solutions at these times are denoted $y_n=y(t_n)$. Back to $\eqref{eq:ODE1}$, approximating $y'(t)\approx (y(t+h)-y(t))/h$, we have:
+Playing with the parameter space:
-$$y_{n+1}=y_n+hf(t_n,y_n)$$
+<center><wz tip="Two (out of the many) configurations:">[[File:Screenshot_20210315_181551.png|400px]]</wz></center>
-This is very simple since from the solution $y_n$ at time $t_n$ we can compute the solution at $t_{n+1}$. Explicitly.
+And the bottom case, animated:
-Let us solve, for instance, $y'=-y$, whose solution is the [[exponential]] function.
+<center><wz tip="250 iterations of the a=10 g=20 excitable model with a lot of initially infected cells in various stages, preventing extinction.">[[File:excitable-medium-a10-b20.gif|400px]]</wz></center>
+It is interesting to focus on the surviving spot. We can zoom on this area and see what is going on there. The cells have locked into this stable pattern, involving, apparently, two cells only:
 <syntaxhighlight lang="python">
-npts = 100; # number of points
+excited = @animate for i ∈ 1:30
-y=0.0*collect(1:npts); # predefine the array
+           excite()
-h=.1; # timesteps, so total time = npts*h=10
+           plotspace(145, 220, 225, 300)
-# The differential equation to solve
+       end
-function f(t,y)
-    -y
-end
-y[1]=1; # initial condition
-# The numerical method (Euler's)
-for i=1:npts-1
-    y[i+1]=y[i]+h*f((i-1)*h,y[i])
-end
 </syntaxhighlight>
-And we can now compare the numerical solution to the real one:
+<center><wz tip="A stable 2-cells pattern that survives in the g>a configuration by locking two neighbours with a period a+b=30.">[[File:stable-2cells-pattern.gif|400px]]</wz></center>
+It is interesting as well to change parameters during the simulation. This locks the patterns into some dislocated sequences which give rise to apparent cycles of stationary-evolution in space, which, however, are due to structures that formed under previous conditions of evolution. For instance, the following was obtained by iterating 50 iterations with parameters <tt>a=50 and g=25</tt> then changing <tt>g=5</tt> for 50 iterations and coming back to the initial <tt>g=25</tt> and cycling for ever:
+<center><wz tip="Strange oscillations in the wake of changing parameters in mid-flight. This is an interesting optical effect due to dephasing of the wavefronts caused by dislocations in the structure following various types of evolutions.">[[File:strange-oscillations.gif|400px]]</wz></center>
+There clearly appears to have two types of oscillations, one seemingly stationary in space whereas the over travels as wavefronts, that get periodically frozen. This is, however, merely an optical artifact, that can be well understood by looking at the state of the cell's evolution in time. Here is such a cut obtained in a similar configuration as the one displayed as a full density plot:
 <syntaxhighlight lang="python">
-plot(((1:npts).-1)*h, [y[i] for i=1:npts], label="Euler", xlabel="t", ylabel="y'=-y")
+for i=1:75
-plot!(x->exp(-x),0, 10, label="Exact")
+   println(global iteration+=1); excite();
+   Plots.display(plot(space[[100],:]', ylims=(0,a+g), lw=2, legend=false))
+end
 </syntaxhighlight>
-<center><wz tip="Euler's method vs exact solution. The right one had the further option yscale=:log10">[[File:Screenshot_04-03-2020_083840.jpg|600px]]</wz></center>
+<center><wz tip="Slice through the cellular automaton showing two types of oscillations, as a result of dislocations: drifting and collapsing.">[[File:wavefronts-excitable-medium-cut.gif|400px]]</wz></center>
-Let us try the more difficult example:
-$$y'={y\over 2}+2\sin(3t)$$
+One can see how the dislocations cause this alternances of drifting or collapsing evolution of the cell states.
-whose solution can be found with an integrating factor [http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx]
+Clearly there are many variations and configurations to explore, this is barely scratching the surface. Instead of delving further into this model, we turn to the most famous and also more impressive Hodgepodge machine.
-$$y\left( t \right) =  - \frac{{24}}{{37}}\cos \left( {3t} \right)\, - \frac{4}{{37}}\sin \left( {3t} \right)\, + \left( {{y_0} + \frac{{24}}{{37}}} \right){{\bf{e}}^{\frac{t}{2}}}$$
+{{WLP6}}