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Continuation Algorithms
In some problems with boundary conditions, often one has to plot the set of points lying in the parameters space that satifies these conditions.
If the set of point to be ploted is a curve, this can be done using the AUTO package. But if the set of point is of higher dimension AUTO does not do it. Nevertheless Michael Henderson's Multifario continuation algorithm can handle such cases.
Here are some examples of curves, surfaces or spaces, in up to 4D embedding spaces, plotted with own made routines developed in Mathematica:|
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n=2 |
n=3 |
n=4 |
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k=1 |
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k=2 |
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k=3 |
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Basins of attraction of the Shooting Method
Let us consider the following two points boundary value problem (BVP):
alpha'=M
M'=-Pi^2 H sin(alpha)
M(0)=0
M(1)=0
It is a well posed BVP : the number of boundary conditions is equal to
the number of differential equations.
But you have the freedom of varying H, so the solution manifold is
1D.
It corresponds to the pinned planar elastica (connexion path not
included).
Finding solutions of this BVP could be achieved by shooting. This means considering
the problem as an IVP: choosing alpha(0)=alpha0 and M(0)=0 and H (aiming part).
And then integrating up to s=1 (shooting part).
The final condition M(1)=0 can be understood
as an equation M(H,alpha0,s=1)=0.
This equation implicitely defines a 1D solution
manifold. So we use newton's method with H and alpha0 as unknowns and
M(H,alpha0,s=1)=0
and
x1 H + x2 alpha0 = x3
as the two equations to solve. We take 20000 points at random in a
rectangle (0,0) to (Pi,50) of the (alpha0,H) plane and use the
iterative Newton-Raphson method to find solutions.
Here is what we find :
| x1=1 x2=0 |
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| x1=1 x2=1 |
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| x1=0 x2=1 |
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You see few differents paths. So some initial guess converge to the
horizontal axis, some to the vertical one. And some other guesses to
the bifurcated paths. (Some 10% of initial points converge outside of the given
rectangle and very few do not
converge at all). Now a natural question arises: what converges to what ?
Can we part the (alpha0,H) plane into basins of attraction ?
It is known that Newton's method mainly give rise to
basins of attraction with fractal boundaries (see e.g. :
here or
here).
Here is what we find (colored points converge to the solution path of
the same color):
| x1=1 x2=0 |
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| x1=1 x2=1 |
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| x1=0 x2=1 |
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Then we zoom this last figure :
| x1=0 x2=1 |
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