{************************************************************************** * 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}