/************************************************************************** * FRACTALS: THE VERHULST 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". * * * * See also program chaos.pas (Feigenbaum diagram). * * * * ----------------------------------------------------------------------- * * From "Graphisme dans le plan et dans l'espace avec Turbo Pascal 4.0 * * By R. Dony - MASSON, Paris 1990, page 189" [BIBLI 12]. * * * * Microsoft C++ version in API style by J-P Moreau, Paris * * (Program to use with Verhulst.mak and Gr2D.cpp) * * (www.jpmoreau.fr) * **************************************************************************/ #include #include #include HDC hdc; RECT rect; //Functions of module Gr2D.cpp void Fenetre(double,double,double,double); void Cloture(int,int,int,int); void Bordure(); void Axes(); void Gradue(double,double); void Grille(double,double); void MoveXY(double,double); void LineXY(double,double); //"home made" graphic commands for hdc environment used by above functions void DrawPixel(int ix,int iy) { //sorry, no other available command found Rectangle(hdc,rect.left+ix,rect.top+iy, rect.left+ix+2,rect.top+iy+1); } void Swap(int *i1,int *i2) { int it; it=*i1; *i1=*i2; *i2=it; } void DrawLine(int ix1,int iy1,int ix2,int iy2) { int i,il,ix,iy; float dx,dy; if (ix2