Created: 2014-05-19 13:36
Updated: 2014-10-25 22:31

Computer Algebra: Verhulst Process, Mandelbrot Set and Beyond

Yubin Ng -
Xin Chen -
CH Bryan Liu -

Introduction/ Abstract

The project aims to investigate various examples and properties of the Verhulst Process and the Mandelbrot Set through mathematical and graphical means, with the aid of graph plotting programs.

A copy of all figures are available in ./Documents/Graphs/

Verhulst Process (/Verhulst)

  • Figure 2.1 (/Verhulst/Diagrams/iteration_plot.eps) is generated by MatLab function: /Verhulst/Functions/graph.m

  • Bifurcation Diagrams, including:

    • (/Verhulst/Diagrams/bgraph.eps)
    • Figure 2.2 (/Verhulst/Diagrams/bgraph2.eps)
    • (/Verhulst/Diagrams/bgraph3.eps)
    • Figure 3.1 (/Verhulst/Diagrams/logisticBgraph.eps)
    • (/Verhulst/Diagrams/logBgraph2.eps)
    • Figure 3.3a (/Verhulst/Diagrams/sinBgraph1.eps)
    • Figure 3.3b (/Verhulst/Diagrams/sinBgraph2.eps) is generated by MatLab function: /Verhulst/Functions/bgraph.m. To obtain corresponding maps, subsitute suitable function arguments.
  • x_k Iteration Diagrams, including:

    • Figure 2.3a (/Verhulst/Diagrams/2.2xk.eps)
    • Figure 2.3b (/Verhulst/Diagrams/2.8xk.eps)
    • Figure 3.2a (/Verhulst/Diagrams/log3.2Xk.eps)
    • Figure 3.2b (/Verhulst/Diagrams/log3.7Xk.eps)
    • Figure 3.4a (/Verhulst/Diagrams/sin2.65Xk.eps)
    • Figure 3.4b (/Verhulst/Diagrams/sin3Xk.eps) is generated by MatLab function: /Verhulst/Functions/xk.m. Replace line 2 (matlab x=map(1:n)) by the corresponding mapping function to obtain various iteration diagrams.

Verhulst Process - Gaussian Map (/Verhulst_Gaussian)

  • All diagrams in /Verhulst_Gaussian/Diagrams, including

    • Figure 3.5a (/Verhulst_Gaussian/Diagrams/Gaussian_Map_3D_View_1.eps)
    • Figure 3.5b (/Verhulst_Gaussian/Diagrams/Gaussian_Map_3D_View_2.eps)
    • Figure 3.6 (/Verhulst_Gaussian/Diagrams/Gaussian_Map_Bifurcation(a=7).eps)
    • Figure 3.7a (/Verhulst_Gaussian/Diagrams/Gaussian_Map_Cobweb(a=7 b=0.5).eps)
    • Figure 3.7b (/Verhulst_Gaussian/Diagrams/Gaussian_Map_Cobweb(a=7 b=-0.275).eps)
    • Figure 3.7c (/Verhulst_Gaussian/Diagrams/Gaussian_Map_Cobweb(a=7 b=-0.51).eps)
    • Figure 3.7d (/Verhulst_Gaussian/Diagrams/Gaussian_Map_Cobweb(a=7 b=-0.675).eps) is generated by MatLab file: Verhulst_Gaussian/Functions/Plot_Gaussian_Maps.m which parameters were documented in the .m file accordingly

    .eps files are exported via the script print2eps provided by Oliver Woodford from Matlab Central (

Mandelbrot Set (/Mandelbrot)

  • Diagram of Mandelbrot set (Figure 4.1 - /Mandelbrot/mandelbrot.eps) is generated by MatLab file: Mandelbrot/mandelbrot.m with parameters (20000 iterations, 12800 pixels width, 7200 pixels height)

  • Diagrams of Julia Set (Figures 4.2 - 4.11 in /Mandelbrot/Julia/) is generated by MatLab file: Mandelbrot/Julia/julia.m with function parameter c specified in section 4.4 of the report:

    julia1.eps: c = +i 
    julia2.eps: c = -0.12 + 0.74i
    julia3.eps: c = -0.11 + 0.6557i
    julia4.eps: c = -0.194 + 0.6557i
    julia5.eps: c = -0.74543 + 0.11301i
    julia6.eps: c = -1.25
    julia7.eps: c = -0.481762 - 0.531657i
    julia8.eps: c = -0.39054 - 0.58679i
    julia9.eps: c = -0.15652 - 1.03225i
    julia10.eps: c = -0.11031 - 0.67037i
    julia11.eps: c = 0.27334 + 0.00742i
    julia12.eps: c = 0.31 + 0.4i

    Except julia4.eps and julia10.eps all figures in this section are attached in the appendix of the report

  • Figure 4.13 is a combination of the above figures via graphic editing software

  • Figure 4.14 is downloaded and converted to .eps format from wikimedia where picture is declared to be relased into public domain, the use of this figure is acknowledged in the reference section

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