{**************************************************************************
* FRACTALS: FEIGENBAUM DIAGRAM *
* ----------------------------------------------------------------------- *
* TUTORIAL: *
* *
* The dynamic process that allows to very simply simulate "chaos" has the *
* following mathematical form: *
* x = f (x , C ). *
* n+1 n *
* The only condition is that the relation between input x n and output *
* x n+1 be non linear. If this iterative process starts with an arbitrary *
* value x0, it will give a series of values: x1, x2,...,x n... The long *
* term behaviour of this series is interesting. *
* *
* Let us consider the classical example of the growth of a population over*
* several years. Let be an initial population x0 which after n years has *
* become x n. The growth ratio is: *
* r = ( x n+1 - x n ) / x n. *
* If this ratio remains constant, the dynamic law is: *
* *
* x n+1 = ( 1 + r ) x n *
* *
* After n years, the population will be: x n = ( 1 + r ) x0. *
* *
* To clarify the situation, let us consider the case where x0 = 0,001. *
* If the growth ratio equals 5%, the population will roughly double *
* every 15 years. As a matter of fact: *
* *
* x0 = 0,001 x1 = 0,00105 x2 = 0,0011025 *
* x3 = 0,00115763 x4 = 0,00121551 x5 = 0,00127628 *
* x6 = 0,00134010 x7 = 0,00140710 x8 = 0,00147746 *
* x9 = 0,00155133 x10 = 0,00162889 x11 = 0,00171034 *
* x12 = 0,00179586 x13 = 0,00188565 x14 = 0,00197993 *
* x15 = 0,00207893 ... *
* *
* This kind of growth is exponential. But this dynamic law is linear, *
* which is not judicious. Actually, the real growth of a poulation is *
* necessarily limited by the ressources of its habitat, which are not in *
* infinite quantity, such as food, for example. The belgian Verhulst was *
* one of the first to study this phenomenon in 1845. *
* He advised to consider a variable growth ratio r, taking the form *
* r = r - C x n. The dynamic law of growth then takes the form: *
* *
* x n+1 = ( 1 + r ) x n - C x² n *
* *
* By having C = r / X, the population increases up to the value X, then *
* stabilizes at this value. At least, this remains true so long as the *
* growth ratio is < 200 %. A human population has never such a high growth*
* ratio, but in the case of insects, for example, this can be quite *
* possible. For a growth ratio even higher, one can observe surprising *
* results (see verhulst.pas program). *
* *
* The calculation begins at x0 = 0,1 X. *
* *
* Case r = 1.8 *
* *
* The response curve climbs up rapidly and after some oscillations reaches*
* a limit that is an "attractor". *
* *
* Case r = 2.3 *
* *
* The curve oscilates rapidly between two levels that frame the value X. *
* The suite has two attractors. *
* *
* Case r = 2.5 *
* *
* The suite has four attractors. *
* *
* Case r = 3.0 *
* *
* The numbers x jump from one value to another one, without having twice *
* the same value. Such a behaviour can be qualified as "chaotic". *
* *
* The Feigenbaum diagram: *
* *
* To better observe the behaviour of such suites x when r varies, we now *
* only consider the "attractors" for each r value. The first 100 transient*
* values are skipped then at each r value 300 points are displayed. *
* For r < 2,57, the behaviour is non chaotic: the attractors are in *
* limited number. When r > 2,57, the attractors become queer and the dia- *
* gram has more and more ramifications until being fully inextricable: *
* now we have a chaotic bahaviour! *
* *
* The obtained picture is called the bifurcation diagram or Feigenbaum *
* tree. By an accurate analysis of the bifurcation points, the mathema- *
* tician Feigenbaum discovered a new universal constant. The length of the*
* r intervals for which a stable period is obtained, is shortened, when *
* the period is doubled, by a factor that tends toward the universal *
* constant k = 4,669201660910... *
* *
* This constant, called the Feigenbaum Constant, can be found in other *
* chaotic phenomenons, such as fluidic turbulences, chemical reactions, *
* and even in human heart! *
* ----------------------------------------------------------------------- *
* From "Graphisme dans le plan et dans l'espace avec Turbo Pascal 4.0 *
* By R. Dony - MASSON, Paris 1990, pages 189-192" [BIBLI 12]. *
* *
* TPW version by J-P Moreau, Paris *
* (www.jpmoreau.fr) *
**************************************************************************}
Program CHAOS; {WINCRT version with printing capability}
Uses WinCrtMy,WinTypes,WinProcs,WObjects,Strings,Type_def,CrtGr2D,Winprint;
Label 100;
Var dr,r,x : real_ar;
i,n : integer;
ch : char;
OldPen,CrtPen : HPen;
Pinfo : PPrinterInfo;
Procedure Draw_chaos(P:HDC; flag:boolean);
var margex : integer;
begin
if Not flag then
begin
CrtPen:=CreatePen(ps_Solid,1,RGB(127,0,255));
OldPen:=SelectObject(P,CrtPen); { plume bleue }
end;
Fenetre(1.9,3.0,0.0,1.6);
if MaxX > 1000 then margex:=200 else margex:=45;
Cloture(margex,MaxX-25,95,MaxY-5);
Axes(P);
Grille(P,0.1,0.1);
Gradue(P,0.2,0.3);
Bordure(P);
r:=1.9; dr:=0.0025;
{write titles}
if MaxX < 1000 then {screen}
begin
TextOut(P,75,30,'FEIGENBAUM DIAGRAM',18);
TextOut(P,50,10,'X',1);
TextOut(P,490,295,'R',1)
end
else {HP laser printer}
begin
TextOut(P,300,80,'FEIGENBAUM DIAGRAM',18);
TextOut(P,100,27,'X',1);
TextOut(P,2950,2050,'R',1)
end;
{main r loop}
repeat
x:=0.3;
for i:=1 to 200 do x:=(1+r)*x-r*x*x;
for i:=1 to 300 do
begin
x:=(1+r)*x-r*x*x;
if r>=1.95 then
begin
MoveXY(P,r,x);
LineXY(P,r+dr,x);
end
end;
r:=r+dr
until r > 3;
if Not flag then
begin
SelectObject(P,OldPen);
DeleteObject(CrtPen)
end
end;
{main program}
Begin
WinCrtInit('CHAOS');
Draw_chaos(CrtDc,FALSE);
Sortiegraphique;
if rep='i' then {send to printer}
begin
New(Pinfo,Init);
With Pinfo^ do
BEGIN
MaxX:=3200; MaxY:=2200; {HP laser landscape 300 dpi}
StartDoc('chaos');
Draw_chaos(PrintDc,TRUE);
NewFrame;
EndDoc
END;
Dispose(Pinfo,Done)
end;
DoneWinCrt
End.
{end of file chaos.pas}