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". ------------------------------------------------------------- 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". -------------------------------------------------------------