MATHTOOLS User's Manual
Contents
1. e if the number of on neighbours is exactly three the cell will be on in the next generation This is regardless of the cell s current state e if the number of on neighbours is smaller or equal 1 or greater or equal 4 the cell will be off in the next generation 9 2 Disease This automat simulates the spread of a disease A cell can either be susceptiple 0 infected 1 ill 2 or immune 3 The automat is not fully deterministic 0 2 1 The rules e if a cell is susceptible it will be infected with a certain probability depending on the number of infected neighbours The cell will be infected if the number of neighbours multiplied with a random number between 0 and 1 exceeds 0 9 e infected cells become ill with a probability of 60 e ill cells stay ill 50 probability become immune 25 probability or susceptible 25 probability e immune cell become susceptible with a probability of 5 to represent deaths and births 9 3 Per Bak s sandpile model Per Bak s sandpile model is an example of self organized criticality It is a cellular automat whose configuration is determined by the number of sandgrains on a cell 0 3 1 The rules e a grain of sand is added at a randomly selected cell e a cell with more than 3 sand grains becomes unstable and topples by distributing one grain of sand to each of it s four neighbours This may cause unstable cells in the neighbourhood An avalanche is born e if any avalanche dies out an2. Tiger Graphics Equations parameter values and coordinates can be saved in a file These files get automati cally the extension tgp 1D or tgt 2D and are stored normally in lt Path to Mathtools gt fracfiles Before loading a file the mode 1D or 2D must be selected After loading the file the equations parameter values and the stored coordinate values are set The files can also be edited to set f e coordinate values not provided by default The editor must be set in Custom 5 13 Tiger Graphics MATHTOOLS 1 0 10 7 PS Print Postscript output is generated and stored in lt Path to Mathtools gt mftfiles plots The PS files are named automatically 14 Tiger Graphics Tiger Graphics MATHTOOLS 1 0 11 Bifurcation Tool The parabola equation Und T Tn 1 xp converges shows cycles or chaos depending on the parameter r 11 1 Calculate Shows the location of the limit points depending on the parameter r Clicking with the left mouse button into the picture shows the parabola of the selected value of r fla r 2 1 2 A left mouse click starts the iteration within the parabola initial value With further left mouse clicks the iteration can be continued right mouse click shows all iteration steps at once depending on the selected number of iterations If cut first values is switched on the initialization is cut off only the limit point resp cycle resp chaos is shown Remark If the parametri
3. In the LD mode up to three one dimen sional functions of the form can be plotted The function is only plotted if the checkbutton on the left of the defintion is switched on In the 2D mode a parametrized two dimensional function of the form x t can pe plotted The following math functions can be used within the formulas acos asin atan cos cosh exp log logi0 pow sin sinh sqrt tan tanh abs The formulas can be defined with up to 5 parameters a b c d e 10 1 Plot The defined function is plotted according to the settings of the independent variable x 1D or t 2D The axes can either be set by the sliders or set by Scale or AutoScale 10 2 Scale The dimension of the t x and y values can be set This values will modify the sliders and are treated as boundary values for the sliders Scale refreshes the plot 10 3 Autoscale The maximum values of y 1D resp x and y 2D are evalutaed and the plot is adapted to that values according to the settings of the independent variable 10 4 Clear Clear clears the output window 10 5 Coordinates If Show coor is switched on the coordinates at the mouse pointer are shown A left mouse click in the output windows shows a horizontal line within the output window at the actual y coordinate and its value a right mouse clickshows a vertical line within the output window at the actual x coordinate and its value 10 6 Load Save amp Edit
4. to Mathtools gt mftfiles After loading a file the equations parameter values and the stored coordinate values reso lution and endtime are set The files can also be edited to set f e coordinate values not provided by default This editing functionality does not work under windows at the moment Please use your usual editor to edit the files manually 8 9 PS Print Postscript output is generated and stored in lt Path to Mathtools gt mftfiles plots The PS files are named automatically 8 Tiger Graphics Tiger Graphics MATHTOOLS 1 0 9 Celluar Automata Cellular automata are discrete dynamical systems Space is represented by a uniform grid with each cell containing a value time advances in discrete steps and the rules of change of each cell is determined by the states of its closest neighbour cells Different celluar automata are implemented an can be simulated Conway sLife Disease Spread 2 different Per Bak sand piles Circular room Spreading Bugs The automat can be selected by the pull down menu The active one is diplayed 9 1 Life Life was devised in 1970 by John Horton Conway The automat knows two states dead 0 off or alive 1 on The neighbourhood of a cell is defined by the eight surrounding cells 9 1 1 The rules e if for a given cell the number of neighbours is exactly two the cell maintains its status quo into the next generation If the cell is on it stays on if it is off it stays off
5. B i 1 j amp o 1 67 B t j A AB i j 1 o B i j At AB i j 1 o B i j At i j denotes the cell in the i th row and j th column The differential equations are solved with Euler s scheme The time step for the calculations is At 0 01 Parameters Par value unit meaning r 54 75 1 y growth in old forest K 5000 bugs cell capacity of bugs u 36500 bugs cell y predation by birds M 500 bugs cell bug density controlling predation a 0 15 1 y aging factor of forest B 1 7 1 y damage of forest E0 1 0 1 y maximum part of bugs to diffuse Er 09 parameter for direction wind force 10 Tiger Graphics Tiger Graphics MATHTOOLS 1 0 The simulation shows the number of bugs per cell It takes several hundred generations until the dynamical state of the system appears Unfortunately this example is very slow due to the numerical solver of the differential equations 9 6 Diffusion Demonstration of diffusion The simulation should be started with a random filling or a small filled area circle or square The colortable is fixed to white black Filling can be done by a left mouse click 0 7 Advection Demonstration of advective transport The simulation should be started with a small filled area circle or square The colortable is fixed to white black Filling can be done by a left mouse click Three scenarios are implemented 1 1D advection 1 cell per time step simulation of advection without numerical diffusion The ad
6. Linux Make sure that at least gcc make tcl and tk are installed on your system 3 Linux Installation 1 Copy the file mathtools tgz to the directory where the software should be installed Extract the data from the tgz file by tar zxvf mathtools tgz The archive is unpacked to the directory MATHTOOLS 2 Adapting the Shell Go to the directory MATHTOOLS and edit the file mathtools sh and set the en vironmental variable MFTHOME to the directory where MATHTOOLS is installed f e export MFTHOME HOME MATHTOOLS 3 Add this path to the PATH variable of your system or copy the shell mathtools sh to the directory where your executables are in 4 Compiling The MathTools are running on 32 bit and 64 bit sytems Thus the binaries need to be built after unpacking Go to the directory MATHTOOLS srcby f e cd MATHTOOLS src and execute the script install all from an xterm or similar by typing install all All the Makefiles in the subdirs will be executed MATHTOOLS can then be started by executing mathtools sh 4 Windows Installation Execute the file setup exe and follow the instructions The program can be started from the menu in the execution group TigerGraphics 5 Custom The default editor and the default browser for the MATHTOOLS Help can be set Remark There are some defaults set Such it should not be necessary to customize the settings If an error message occurs while trying to edit or browse please select the exe cuta
7. Tiger Graphics MATHTOOLS Educational Tool for applied Maths Version 1 0 User s Manual C Kohlmeier amp F Hamberg Contents 8 9 Introduction Requirements Linux Installation Windows Installation Custom Fractals Mandelbrot Set Slope Celluar Automata 10 Plotter 11 Bifurcation Tool 12 IFS Iterative function systems Tiger Graphics MATHTOOLS 1 0 13 15 16 Tiger Graphics Tiger Graphics MATHTOOLS 1 0 1 Introduction License Tiger Graphics MATHTOOLS is an educational tool for the visualization of several nice mathematical things This program is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation either version 3 of the License or at your option any later version This program is distributed in the hope that it will be useful but WITHOUT ANY WAR RANTY without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU General Public License for more details You should have received a copy of the GNU General Public License along with this program If not see jhttp www gnu org licenses Tiger Graphics MATHTOOLS 1 0 2 Requirements MATHTOOLS is built on the base of Tc1 TK version 8 5 and Gnu C version 4 6 It is developed and tested on SuSE Linux Version 12 1 and runs also under Microsoft Windows Win95 or higher 2 1 Requirements for
8. bles according to your systems installation Note For Windows systems please select the explorer exe as browser It will start the internet explorer 4 Tiger Graphics Tiger Graphics MATHTOOLS 1 0 6 Fractals Several self similar fractals can be constructed Highway dragon Koch Curve Cantor Set Weierstrass Monster Sierpinski triangle Pythagoras Tree Sunflower The number of iteration steps can be set by the slider The construction laws are self explanatory if a low number of iteration steps is selected 6 1 Calculate The number of selected iterations is calculated and displayed 6 1 1 Parameter settings Highway dragon The direction of the construction can be selected The effect can be seen if only a few iterations are selected Weierstrass monster The limit object of the Weierstrass Monster is a function which is continuous in R but not differentiable at any point in R It is defined by N f t ACD gin AM k 0 where N is the number of iterations The values for and s can be selected Note not every parameter combination leads to nice monsters The fractal dimension of the monster is estimated and displayed This is only a very rough estimation because the calculation is done on the base of the Nth iteration by the curve length Sunflower The configuration of seeds in a sunflower is simulated Starting from a central point the radius of every step is increased by s while the angle is increased by giv
9. en in degree 6 2 PS Print A Postscript output is generated and stored in lt Path to Mathtools gt mftfiles plots The PS files are named automatically Tiger Graphics 5 Tiger Graphics MATHTOOLS 1 0 7 Mandelbrot Set The Mandelbrot set is constructed and displayed The Mandelbrot set is defined as the following subset of the complex plane M c C zn remains bounded tn41 22 c o c Such in the picture the black object is the Mandelbrot set The nice colors are determined by the speed of divergence Within the picture a rectangle can be defined by clicking the left mouse button and dragging the mouse while the button is pressed Calculate starts the calculation of the new cut out The number of iterations must be set as higher the more you go into detail The Julia set of a point is calculated if clicked with the right mouse button The Julia set of a given point c is defined as the following set of points in the complex plane Je x C an remains bounded n41 42 c o 2 7 1 Calculate The part of the Mandelbrot Set is calculated depending on the cut out The number of iterations determines the accuracy 7 2 Save amp Print A Postscript output and a gif output is generated and stored in lt Path to Mathtools gt mftfiles plots The files are named automatically 7 3 Newton roots The tool allows also the visualization of the basins of the roots z of the complex equation x 1 calcu
10. ess If Color is selected each step is plotted in a different color If Clear between is selected only the quadrangles of the last step are printed 12 2 Reset Resets the iteration of the copy machine see 12 1 12 3 Clear screen All printed stuff is deleted The maps still exist 12 4 PS Print Postscript output is generated and stored in lt Path to Mathtools gt mftfiles plots The PS files are named automatically 12 5 Set IFS reference rectangle occurs in the picture window By left mouse click up to four affine maps can be defined The first click defines the image point of the upper left corner the second of the lower left corner and the third the image point of the lower right corner The resulting quadrangle is plotted This process can be repeated three times The last map is additionally stored as fourth map It is only evaluated if e four different maps are defined or e Condensation is switched on 1The Chessboard is defined by 5 affine maps The last one is treated as condensation 16 Tiger Graphics Tiger Graphics MATHTOOLS 1 0 12 6 Set Parameters The parameters of the actual IFS are shown and can be modified by e selecting another predefined IFS or e loading an IFS which has been saved further or e defining a new IFS by Set IFS Every row represent a map The parameters are defined as follows x Qi bi x ei Ja r 3 MN i 1 4 The values w in the last column give the probab
11. ilities for selecting this map by the chaos game If the sum of the first probabilities exceed 1 the following maps are not evaluated If Condensation is active the last map is evaluated as condensation independent of the probabilities The probability of the predefined maps are defined by the determinant of the matrix If an IFS is defined by Set IFS the probabilities are equally distributed The values can be saved in a file The files holding IFS data get the extension ifs and are stored in lt Path to Mathtools gt mftfiles by default References Barnsley 2000 BARNSLEY M Fractals everywhere Morgan Kaufmann 2000 534 S Tiger Graphics 17
12. lated by Newton s method The three basins B are defined as the following sets of points f x ID f z 23 Fay fe 2 1 B x Cltn gt zi o T Un41 In 6 Tiger Graphics Tiger Graphics MATHTOOLS 1 0 8 Slope This tool allows the visualization of simple one or two dimensional differential equations 8 1 One Dimensional differential equation Set xz 1 This means that x t t to 0 Set y to the differential equation which should be considered Slope field shows the slope field of the differential equation in the t y plane 8 2 Two Dimensional differential equation Set 2 and y to your differential equation system which should be considered Slope field shows the slope field of the system in the x y phase plane 8 3 Parameters and Math Built in functions Up to five parameters a b c d e can be used within the formulas The following math functions can be used within the formulas acos asin atan cos cosh exp log logi0 pow sin sinh sqrt tan tanh abs 8 4 Linear If the parameters a b c d are set linear makes a linear system from these parameters ax Ny fab x yj Ned y The Eigenvalues und Eigenvectors the trace and determinant are displayed in the Info window 8 4 1 Ejigenrichtung The directions of the Eigenvectors are plotted 8 5 Coordinates The coordinates of the x y plane can be set by the two cordinate sliders If these settings are too small for the act
13. other grain of sand is added to a randomly selected cell There are two models implemented Tiger Graphics 9 Tiger Graphics MATHTOOLS 1 0 e 4 neighbours cell with more than 3 sand grains becomes unstable and topples by distributing one grain of sand to each of it s eight neighbours e the same model but with 8 neighbours A cell with more than 7 sand grains becomes unstable and topples by distributing one grain of sand to each of it s eight neighbours 9 4 Circular Room The circular room is a nice tool to demonstrate self organization It is most impressive to start from a randomized initial state 9 4 1 The rules e The N states are circulary arranged The state N is identified with state 0 Each cell has 4 neighbours e The state of a cell is increased by one if at least one neighbour has the state of the cell plus 1 The Circular Room is implemented for 6 and 20 states 9 5 Bug Spread This is an example of an automat combined with a differential equation model The are two state variables the number of bugs B within the part of the forest cell and the mean age A of the trees in this part of the forest The differential equations for the bug growth the aging of the forest and the bug spread are given by 2 2 LEE at K B M gx OA BN The diffusive spread is realized by an automat scheme with an additional wind direction from west to east ABli j ia Blid At AB i 1 j e amp 0 1 er B i j At A
14. te faster without displaying 9 12 Run Run starts the simulation The generation is displayed in the info window The simulation can be stopped by Interrupt Tiger Graphics 11 Tiger Graphics MATHTOOLS 1 0 9 13 9 14 9 15 9 16 9 16 1 9 16 2 9 17 12 Step Step calculates the next generation Interrupt Interrupt stops the simulation Clear Clear sets all cells to zero Setting the initial states The initial state of the automat can either be set by mouse within the automat field De pending on the number of states the following settings are possible left mouse click sets cell to 1 lt Shift gt left mouse click sets the cell to 2 lt Strg gt left mouse click resp lt ctrl gt left mouse click sets the cell to 3 e e e e middle or right mouse click resets the cell to 0 It is possible to save a configuration by Save in a file These files get automatically the extension lif and are stored normally in lt Path to Mathtools gt mftfiles The configuration can be loaded by Load Randomfill Randomfill fills the field at random depending on the number of possible states Fill All cells are filled with the selected fill value PS Print Postscript output is generated and stored in lt Path to Mathtools gt mftfiles plots The PS files are named automatically Tiger Graphics Tiger Graphics MATHTOOLS 1 0 10 Plotter This tool allows the visualization of functions
15. ual problem they can be reset by Reset Coor to the interval 10 10 x 10 10 Larger intervals must be set by editing the function file See 8 8 8 6 Trajectories Trajectories to given initial values can be plotted in two ways 1 Setting initial values for x and y in 2 and yo and pressing Initial value or 2 Setting an initial value by a left mouse click within the slope field In this case it is helpful to switch on Show coor to see the coordinates of the mouse cursor The calculation is done by evaluating the differential equation with a second order method By middle mouse click the evaluation is done with Euler s method The time step for the evaluation is At 0 01 The simulation stops either at Endtime or can be stopped by a right mouse click in the slope field Tiger Graphics 7 Tiger Graphics MATHTOOLS 1 0 8 6 1 Time Plot In the case of a two dimensional system the time information cannot be seen in the phase diagram If Time Plot is switched on a second plot window is opened showing the temporal evolution of the trajectory 8 7 Isoclines The isoclines of the system x 0 y 0 can be plotted This functionality is still under work The recent results must be taken with carell 8 8 Load Save amp Edit Equations parameter values coordinates resolution and endtime can be saved in a file These files get automatically the extension tgf and are stored normally in lt Path
16. vection velocity is set to one cell per time step in x direction 2 1D advection 0 1 cell per time step simulation of advection demonstrating the effect of numerical diffusion The advection velocity is set to 0 1 cell per time step in x direction 3 2D advection 0 1 cell per time step simulation of advection demonstrating the effect of numerical diffusion The advection velocity is set to 0 1 cell per time step in north east direction 9 8 Color Table The different automata look nicest with their default color table but the colortable can be changed for some automata and is displayed in the color bar The automata with discrete states are best visualized by the discrete color table The number of colors displayed in the color bar corresponds to the number of states of the active automat 9 9 Cells per row The number of cells per row can be selcted The automat field is then built quadratic The higher the number the slower the simulation runs 9 10 View of the world The world can either be assumed as closed Torus in this case the first row is neighbour of the last one and the first column is neighbour of the last column If the case of a plain world boundary cells have less neighbours 9 11 Show If Show is switched on every generation is shown if switched off the picture is not actualized until Show is switched on again This is useful if only higher generation numbers are of interest because the simulation becomes qui
17. zation leads to a limit point switching cut first values on is not senseful because the limit point cannot be seen 11 2 PS Print A Postscript output is generated and stored in lt Path to Mathtools gt mftfiles plots The PS files are named automatically 11 3 Misc The tool allows also the visualization of n41 In Hr Xn 1 an which is the discretization of the logistic differential equation g a 1 2 Tiger Graphics 15 Tiger Graphics MATHTOOLS 1 0 12 IFS Iterative function systems Iterative function systems are generators for self similar images Further information on IFS can be found in Barnsley 2000 This tool allows to define up to 4 affine maps per image Predefined sets of maps are available for Fern leaf Highway dragon Maple leaf Simple tree Phytagoras tree Sierpinski s triangle Chess board 12 1 Calculate 12 1 1 Copy machine switched off Starting from an initial point the chaos game is started The number of points to be drawn can be selected Due to the accuracy in the random function the algorithm runs into a cycle Such only a few number of points can be distinguished 12 1 2 Copy machine switched on A quadrangle is mapped by the given affine maps Pressing calculate gain maps the new created quadrangles again and so on Because of the increasing number of quadrangles this process become slower and slower Such only 8 iterations are possible by the machine Reset resets this proc
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