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editDescriptionLogphi comp.PNG |
English: Diagram of real potential coputed with 2 methods : simple and full |
Source | Own work |
Author | Adam majewski |
Long description
editThis diagram is related with discrete dynamical system based on complex quadratic polynomials when c=0
What program does :
- computes points of external ray for angle t=0 . See instruction zz:GiveRay(t,c)
- for each point in zz computes potential using simple method and saves to list pair of real numbers : distance to boundary and potential
- for each point in zz computes potential using full method and saves to list pair of real numbers : distance to boundary and potential
- draws above 2 sets of points using distance on horizontal axis and potential on vertical axis
Maxima src code
edit/* Maxima CAS batch file Distributed under the GNU Public License. draws 2 diagrams of potential in basin of attraction in infinity for fc(z)=z*z+c tested on : wxMaxima 0.7.6 http://wxmaxima.sourceforge.net Maxima 5.16.3 http://maxima.sourceforge.net Using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL) */ /* gives a list of points ( complex numbers) of external ray for angle 0 of unit circle ( or f0(z) ) */ GiveRay(t,c):= block( [r], /* range for drawing R=2^r ; as r tends to 0 R tends to 1 */ rMin:1E-10, /* 1E-4; rMin > 0 ; if rMin=0 then program has infinity loop !!!!! */ rMax:1, caution:0.9330329915368074, /* r:r*caution ; it gives smaller r */ /* */ R_max:200, /* */ zz:[], /* array for z points of ray in fc plane */ /* some w-points of external ray in f0 plane */ r:rMax, while 2^r<R_max do r:2*r, /* find point w on ray near infinity (R>=R_max) in f0 plane */ R:2^r, w:rectform(ev(R*exp(2*%pi*%i*t))), z:w, /* near infinity z=w */ zz:cons(z,zz), unless r<rMin do ( /* new smaller R */ r:r*caution, R:2^r, /* */ z:rectform(ev(R*exp(2*%pi*%i*t))), zz:cons(z,zz) ), return(zz) )$ /* gives distancve between 2 complex points */ GiveDistance(c1,c2):=sqrt((realpart(c2)-realpart(c1))^2 +(imagpart(c2)-imagpart(c1))^2)$ /* log(x) in Maxima is a natural (base e) logarithm of x */ log2(x) := log(x) / log(2); /* returns floating point number full definition */ GiveFLogPhi(z0,c,e_r,i_max):= block( [z:z0, logphi:log(cabs(z)), fac:1/2, i:0], while i<i_max and cabs(z)<e_r do (z:z*z+c, logphi:logphi+fac*log(cabs(1+c/(z*z))), i:i+1 ), if i=iMax then logphi:0, return(float(logphi)) )$ /* returns floating point number simple definition */ GiveSLogPhi(z0,c,e_r,i_max):= block( [z:z0, logphi, fac:1/2, i:0], while i<i_max and cabs(z)<e_r do (z:z*z+c, fac:fac/2, i:i+1 ), if i=iMax then logphi:0 else logphi:fac*log(cabs(z)), return(float(logphi)) )$ /* ----------------------- main ----------------------------------------------------*/ start:elapsed_run_time (); /* root point of period 2 component connected with period 1 component */ c:0; /* hyperbolic */ t:0; ER:3; iMax:1000; /* compute ray points & save to zz lists */ zz:GiveRay(t,c)$ /* landing points for external ray */ /* landing point of ray */ zh:%e^(%i*t*2*%pi); /* compute potential = log(Phi)*/ xy_FLogPhi:[]; for z in zz do xy_FLogPhi:cons([GiveDistance(z,zh),GiveFLogPhi(z,c,ER,iMax)],xy_FLogPhi); xy_SLogPhi:[]; for z in zz do xy_SLogPhi:cons([GiveDistance(z,zh),GiveSLogPhi(z,c,ER,iMax)],xy_SLogPhi); /* draw it using draw package by */ x_max:1.0; y_max:20; load(draw); draw2d( terminal = 'screen, /* */ user_preamble = "set grid;set key right bottom ", points_joined =true, point_size = 0.6, point_type = filled_circle, xlabel = "distance to boundary", ylabel = "potential", /* */ color = green, key = " FullLogPhi", points(xy_FLogPhi), /* */ color = blue, key = " SimpleLogPhi", points(xy_SLogPhi) )$
Licensing
editI, the copyright holder of this work, hereby publish it under the following licenses:
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current | 09:00, 25 October 2009 | 1,072 × 621 (24 KB) | Soul windsurfer (talk | contribs) | {{Information |Description={{en|1=Diagram of real potential coputed with 2 methods : simple and full}} |Source={{own}} |Author=Adam majewski |Date= |Permission= |other_versions= }} |
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