maple动力系统.doc

Bifurcation Logistic Equation The original algorithm was developed by P. F. Verhulst in 1845 to describe the increase in population of animals (including humans as well ...). It uses the Logistic Equation x(n+1) = r*x(n)*(1-x(n)) Starting with any value for x = [0 .. 1] the curve splits in two at r = 3 (hence the name bifurcation and behaves chaotically with r 3.56994 ... Note the empty spaces in the graph. Author of the following procedures is Alain Schauber, Forbach, France. restart: with(plots): feigenbaum:=proc(debut,fin,pas) # written by Alain Schauber, Forbach, France, aschau@worldnet.fr local k, itere, a, b, s; s := {}; a := debut; # la variable s va repr_senter lensemble des points _ construire while a = fin do itere := 0.1; for k to 50 do itere := a*itere*(1-itere) od; for k to 100 do itere := a*itere*(1-itere); s:=s union {[a, evalf(itere, 4)]}; od; a := a+pas od; plot([op(s)], a=debut .. fin, style=POINT, symbol=POINT) end: feigenbaum(3, 4, 0.01); # overview lignes_critiques := proc(n) # written by Alain Schauber, Forbach, France local f, l, k; f := (a,x) - a*x*(1-x); l := NULL; for k to n do l := l, plot((unapply(a*x*(1-x),x)@@k)(.5), a=3.55 .. 4, y=0 .. 1, color=black) od; end: display({lignes_critiques(4)}); display({lignes_critiques(10)}); display({lignes_critiques(8), feigenbaum(3.55, 4, 0.01)}); Escape-time Fractal A function originally designed by me for FRACTINT. As shown by the iteration command z:=cos(z)*z^2+c it is a variant of the Mandelbrot set. # BIOMORPH2 # Based on a Maple V algorithm taken from the book # Maple V - Programming Guide by M.B. Monagan, # K.O. Geddes, G. Labahn and S. Vorkoetter, # Springer Verlag, modified by John Oprea # # modification of initialization and