Fractals/Iterations of real numbers/r iterations

Dynamics: real analytic unimodal dynamicsFORTY YEARS OF UNIMODAL DYNAMICS: ON THE OCCASION OF ARTUR AVILA WINNING THE BRIN PRIZE by MIKHAIL LYUBICH Diagram types 2D diagrams parameter is a variable on horizontal axis bifurcation diagram : P-curves ( = periodic points) versus parameterBifurcation and Orbit Diagrams by Chip Ross orbit diagram : points of critical orbit versus parameter skeleton diagram ( critical curves = q-curves)math.stackexchange question: what-are-the-lines-on-a-bifurcation-diagram ? Lyapunow diagram : Lyapunov exponent versus parameterA revision of the Lyapunov exponent in 1D quadratic maps by Gerardo Pastor, Miguel Romera, Fausto Montoya Vitini. PHYSICA D NONLINEAR PHENOMENA 107(1):17 · AUGUST 1997 multiplier diagrams : multiplier of periodic orbit versus parameter constant parameter diagrams cobweb diagram or a Verhulst diagram wiki : Cobweb plot = Graphical iteration Iterates versus time diagramOne-Dimensional Dynamical Systems Part 3: Iteration by Hinke Osinga = connected scatter graph = time series invariant density diagram , histogram,wikipedia : histogram the distribution of the orbit,CHAOS THEORY: DEPENDENCE ON PARAMETER R frequency distribution bat-country by xian = power spectrum Power Spectrum of the Logistic Map from wolframmathematica Poincare plotPoincare plot in wikipediaphysics.stackexchange question : Poincaré plane and Logistic Map Transformation Exponential transformation of the parameter axis 3D diagrams Maps Due to numerical errors different implementations of the same equation can give different trajectories. For example: r*x*(1-x) and r*x - r*x*x Sensitivity to initial conditions: "small difference in the initial condition will produce large differences in the long-term behaviour of the system. This property is sometimes called the 'butterfly effect'."The logistic equation by Didier Gonze Map types doubling map Tent map Orbits of tent map: Logistic map names : logistic map May Robert M. 1976 Simple mathematical models with very complicated dynamics Nature 261 5560 459–467 10.1038/261459a0 1976Natur.261..459M 934280 10338.dmlcz/104555 free : Simple mathematical models with very complicated dynamics by R May logistic equation logistic difference equation discrete dynamical system The logistic mapThe Logistic Map and Chaos by Elmer G. WiensAusloos, Marcel, Dirickx, Michel (Eds.) : The Logistic Map and the Route to Chaos is defined by a recurrence relation ( difference equation) : where : is a given constant parameter is given the initial term is subsequent term determined by this relation Do not confuse it with : differential equation ( which gives continous version ) Bash code Logistic map by M.R. Titchener Javascript code from Khan Academy khanacademy ; logistic-map MATLAB code: r_values = (2:0.0002:4)'; iterations_per_value = 10; y = zeros(length(r_values), iterations_per_value); y0 = 0.5; y(:,1) = r_values.*y0*(1-y0); for i = 1:iterations_per_value-1 y(:,i+1) = r_values.*y(:,i).*(1-y(:,i)); end plot(r_values, y, '.', 'MarkerSize', 1); grid on; See also Lasin Lasin - Matlab code Maxima CAS code Logistic diagram by Mario Rodriguez Riotorto using Maxima CAS draw package /* Logistic diagram by Mario Rodriguez Riotorto using Maxima CAS draw packag */ pts:[]; for r:2.5 while r <= 4.0 step 0.001 do /* min r = 1 */ (x: 0.25, for k:1 thru 1000 do x: r * x * (1-x), /* to remove points from image compute and do not draw it */ for k:1 thru 500 do (x: r * x * (1-x), /* compute and draw it */ pts: cons([r,x], pts))); /* save points to draw it later, re=r, im=x */ load(draw); draw2d( terminal = 'png, file_name = "v", dimensions = [1900,1300], title = "Bifurcation diagram, x[i+1] = r*x[i]*(1 - x[i])", point_type = filled_circle, point_size = 0.2, color = black, points(pts)); explicit solutions The system with r=4 has the explicit solution for the nth iteration :Double precision errors in the logistic map: Statistical study and dynamical interpretation J. A. Oteo J. Ros Precision Numerical Precision in the Chaotic Regime : " the number of digits of precision which must be specified is about 0.6 of the number of iterations. Hence, to determine is x10 000, we need about 6000 digits." "Hence it is not possible to predict the value of xn for very large n in the chaotic regime." The Logistic Map by A. Peter Young Better image thumb|right|Bifurcation diagram of the logistic map To show more detaile use tips by User:PAR: "The horizontal axis is the r parameter, the vertical axis is the x variable. The image was created by forming a 1601 x 1001 array representing increments of 0.001 in r and x. A starting value of x=0.25 was used, and the map was iterated 1000 times in order to stabilize the values of x. 100,000 x -values were then calculated for each value of r and for each x value, the corresponding (x,r) pixel in the image was incremented by one. All values in a column (corresponding to a particular value of r) were then multiplied by the number of non-zero pixels in that column, in order to even out the intensities. Values above 250,000 were set to 250,000, and then the entire image was normalized to 0-255. Finally, pixels for values of r below 3.57 were darkened to increase visibility." See also : tips from learner.org earner.org textbook "The "problem" with pretty much all fractal-type systems, is, that, what happens at the beginning of an actually infinite iterational scheme, is often not descriptive of the long-time limit behaviour. And that's happening with the perturbed Lyapunov exponent from Mario Markus' algorithm as well. If I recall correctly, he states in his 90's article something like "let the x-value settle in". It's similar to the statement "for sufficiently large N" in math papers or in general with convergent series. The actual value of how many skipping iterations one performs, is, however, (at least afaik) just a guess till you're satisfied with the quality of the image. In my images, I could not get rid of those artifacts in full, one example being the UFO image, where vasyan did a great job removing those spots: vasyan's: https://fractalforums.org/index.php?action=gallery;sa=view;id=2388 my version (with spots): https://fractalforums.org/index.php?action=gallery;sa=view;id=1960" marcm200fractalforums.org : what-could-these-artifacts-be-on-my-lyapunov-fractal-renders Lyapunov exponent code in Matlabcalculate lyapunov of the logistic map -LASIN G. Pastor, M. Romera and F. Montoya, "A revision of the Lyapunov exponent in 1D quadratic maps", Physica D, 107 (1997), 17-22. Preprint Invariant Measure An invariant measure or probability density in state space Invariant Measure Exercise by James P. Sethna, Christopher R. Myers. Video on YoutubeInvariant measure of logistic map by todo314 Real quadratic map right|thumb|Lyapunov exponent - image and Maxima CAS code Great images by Chip RossChip Ross : Bifurcation and Orbit diagrams For , the code in MATLAB can be written as: c = (0:0.001:2)'; iterations_per_value = 100; y = zeros(length(c), iterations_per_value); y0 = 0; y(:,1) = y0.^2 - c; for i = 1:iterations_per_value-1 y(:,i+1) = y(:,i).^2 - c; end plot(c, y, '.', 'MarkerSize', 1, 'MarkerEdgeColor', 'black'); Maxima CAS code for drawing real quadratic map : : /* based on the code by by Mario Rodriguez Riotorto */ pts:[]; for c:-2.0 while c <= 0.25 step 0.001 do (x: 0.0, for k:1 thru 1000 do x: x * x+c, /* to remove points from image compute and do not draw it */ for k:1 thru 500 do (x: x * x+c, /* compute and draw it */ pts: cons([c,x], pts))); /* save points to draw it later, re=r, im=x */ load(draw); draw2d( terminal = 'svg, file_name = "b", dimensions = [1900,1300], title = "Bifurcation diagram, x[i+1] = x[i]*x[i] +c", point_type = filled_circle, point_size = 0.2, color = black, points(pts)); Lyapunov exponent program lapunow; { program draws bifurcation diagram y[n+1]=y[n]*y[n]+x,} { blue} { x: -2 < x < 0.25 } { y: -2 < y < 2 } { and Lyapunov exponet for each x { white} uses crt,graph, { modul niestandardowy } bmpM, {screenCopy} Grafm; var xe,xemax,xe0,yemax,i1,i2:integer; yer,y,x,w,dx,lap:real; const xmin=-2; { wspolczynnik funkcji fx(y) } xmax=0.25; ymax=2; ymin=-2; i1max=100; { liczba iteracji } i2max=20; lapmax=10; lapmin=-10; function wielomian2st(y,x:real) :real; begin wielomian2st:=y*y+x; end; { wielomian2st } procedure wstep; begin opengraf; randomize; { przygotowanie generatora liczb losowych } xemax:=getmaxx; { liczba pixeli } yemax:=getmaxy; w:=(yemax+1)/(ymax-ymin); dx:=(xmax-xmin)/(xemax+1); end; begin {cialo} wstep; for xe:=xemax downTo 0 do begin {xe} x:=xmin+xe*dx; { liniowe skalowanie x=a*xe+b } i1:=0; i2:=0; lap:=0; y:=random; { losowy wybor y0 : 0<y0<1 } while (abs(y)<ymax) and (i1<i1max) do begin {while i1} y:=wielomian2st(y,x); i1:=i1+1; lap:=lap+ln(abs(2*y)+0.1); if keypressed then halt; end; {while i1} while (i2<i2max) and (abs(y)<ymax) do begin {while i2} y:=wielomian2st(y,x); yer:=(y-ymin)*w; { skalowanie } putpixel(xe,yemax-round(yer),blue); { diagram bifurkacyjny } i2:=i2+1; lap:=lap+ln(abs(2*y)+0.1); if keypressed then halt; end; {while i2} lap:=lap/(i1max+i2max); yer:=(lap-lapmin)*(yemax+1)/(lapmax-lapmin); putpixel(xe,yemax-round(yer),white); { wsp Lapunowa } putpixel(xe,yemax-round(-ymin*w),red); { y=0 } putpixel(xe,yemax-round((1-ymin)*w),green); { y=1} end; {xe} {..... os 0Y .......................................................} setcolor(red); xe0:=round((0-Xmin)/dx); {xe0= xe : x=0 } line(xe0,0,xe0,yemax); SetColor(red); OutTextXY(XeMax-50,yemax-round((0-ymin)*w)+10,'y=0 '); SetColor(blue); OutTextXY(XeMax-50,yemax-round((1-ymin)*w)+10,'y=1'); {....................................................................} screenCopy('screen',640,480); {} repeat until keypressed; closegraph; end. { Adam Majewski Turbo Pascal 7.0 Borland MS-Dos / Microsoft} Zoom fractal forums: rendering-bifurcation-diagrams Cycles n-cycles of polynomial maps by Cheng Zhang and paper: Cycles of the logistic map Points Misiurewicz Points of the Logistic Map by the J. C. Sprott M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535 G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Misiurewicz point patterns generation in one-dimensional quadratic maps", Physica A, 292 (2001), 207-230 G. Pastor, M. Romera and F. Montoya, "On the calculation of Misiurewicz patterns in one-dimensional quadratic maps" Physica A, 232 (1996), 536-553. Preprint Accumulation points bifurcation points Constants Feigenbaum constants in wikipedia Properities self-similarity, scaling and renormalization Feigenbaums Scaling Law For TheLogistic Map demonstrations from wolfram : FeigenbaumsScalingLawForTheLogisticMap See also Geogebra: The Feigenbaum loci Author:a.zampa oeis search: logistic map Logistic Map by B. L. Badger Analytical properties of horizontal visibility graphs in the Feigenbaum scenario Bartolo Luque, Lucas Lacasa, Fernando J. Ballesteros, Alberto Robledo The Quadratic Map is Topologically Conjugate to the Shift Map - Gareth Roberts Numerical Errors in Logistic Map Calculation J. C. Sprott Numbers and Computers by Ronald T. Kneusel, Reviewed by David S. Mazel , on 01/27/2016 Chaotic Modelling and Simulation Analysis of Chaotic Models, Attractors and Forms Christos H. Skiadas Charilaos Skiadas delayed_logistic The logistic map revisited Jerzy Ombach, Cracow, Poland Structure in the Parameter Dependence of Order and Chaos for the Quadratic Map Brian R. Hunt and Edward Ott Windows of periodicity scaling by Evgeny Demidov in Webgl by Ricky Reusser high-precision-feigenbaum-alpha-calc-using-julia by Stuart Brorson Feigenbaum Tree by 3DXM Consortium Pascal Yang Hui triangles and power laws_in_the_logistic_map by Carlos Velarde Alberto Robledo demonstrations.wolfram : EstimatingTheFeigenbaumConstantFromAOneParameterScalingLaw Egwald Mathematics: Nonlinear Dynamics: The Logistic Map and Chaos by Elmer G. Wiens software pynamical - Python package for modeling, simulating, visualizing, and animating discrete nonlinear dynamical systems and chaos Chaos. Visualizations connecting chaos theory, fractals, and the logistic map! Written by Jonny Hyman, 2020 Videos: Illustration of Chaotic Systems Using Logistic Map by Zylasable Bifurcation Diagram for the Logistic Map Using R by Oxlajuj N'oj References