Basic-Stationary/Homotopy
This demo uses continuation in order to locate solutions of
a nonlinear equation. This is done by the introduction of a
"homotopy parameter" lambda into the equation, so that when
lambda = 0 there are known solutions, while taking lambda = 1
corresponds to the actual equation of interest.
The model equation is a simple one, namely g(u) = 0, where
g(u) = (u^2-1)(u^2-4) + u^2 e^{c u} ,
with c = 0.1. The homotopy parameter lambda is introduced as
follows:
g(u,lambda) = (u^2-1)(u^2-4) + lambda u^2 e^{cu} .
When lambda = 0, the equation g(u,0) = 0 has four solutions,
namely, u=-1,1,-2,2. By the Implicit Function Theorem, each
of these four solutions persists locally when lambda becomes
nonzero, because the derivative dg/du is nonzero for each of
them at lambda = 0.
To run this demo under linux or unix, type "@r hom" in the
shell window. This will continue one of the four solutions
at lambda = 0, namely, u=-1. Results are saved in the files
fort.7 - the bifurcation file
fort.8 - the solution file
fort.9 - the diagnostic file
To plot the results using pyplaut, type "@pp". To save the
files under the names "b.hom", "s.hom", and "b.hom", resp.,
type "@sv hom". To plot the (same) results in these files,
type "@pp hom".
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To run the demo for all four starting solutions using the
Python script "hom.auto", type "auto hom.auto". The results
are saved in files b.hom", "s.hom", "d.hom".
The "bifurcation diagram" can be plotted by typing "@pp hom",
which shows that only the starting solutions u=-1 and u=-2 at
lambda = 0 lead to solutions at lambda = 1. To save the plot,
type "sav" in the shell window, followed by the name by which
the plot is to be saved (e.g., "hom.eps", "hom.pdf", "hom.ps",
or "hom.png").
A plot can also be created by running the python script file
"plot.auto" by typing "auto plot.auto".
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