Traj User`s Manual
Contents
1. To toggle back and forth between the two colors choose the Toggle Back 33 34 CHAPTER 5 THE VIEW MENU ground from the View menu 5 4 Color Another option is to change the way that the color is applied to the function All of the examples that we have seen have applied the color with respect to the z coordinate The part of a function with the lowest z value are red while the parts with the largest value on the z axis are blue The colors in between range from red to green to blue There are two other options You can color the function in the order that the data points are generated or you can color the function in a single color black or white To color the function in order first click on the View menu and click on the color option In the menu that drops down choose Color by Order An example of what the Lorenz attractor looks like is shown in Figure 5 1 This is not a very good example but there are times when you are not sure where the function starts and ends When the function is colored by order the first numbers are colored red and the last numbers are colored blue The numbers in the middle go from red to green to blue If you choose the option Single Color the function is plotted in either black or white If the background is white the function is plotted in black otherwise the background is black and the function is white This option is provided in case you want to print the viewport in black and white2. 5 5 Format Another option for the way a function can be plotted is whether or not each data point is plotted as a point in space or the data points are connected Usually it is more convenient to examine the data points connected So far all of the examples seen have been for connected points If you are viewing data generated from difference equations or there are an extremely large number of points then it may be better to view the plot as a collection of plots To choose between the two options click on the View menu and click on Format There are two options If you click on Draw Points then points will be drawn If you choose Draw Lines then lines connecting the data points will be drawn All of the figure shown so far have been drawn using lines The plot in Figure 5 2 is the Lorenz attractor drawn using the points 5 6 Axis The program allows you to look at the data in different ways The most useful feature is that it allows you to look at the data along dif ferent axis Unfortunately we are three dimensional animals and we cannot Trajectory User s Manual 5 6 AXIS 35 traj File select View Help Figure 5 1 The Lorenz attractor plotted by coloring the points in the order that they were originally generated visualize beyond three dimensions without a whole lot of meditation and probably a couple of medications The solutions to systems of ODEs represent the values of a number of functions The do
3. To move an object this will be called translate here move the mouse cursor into the viewing window press and hold the middle button down and move the cursor On the far right of the window there is a set of buttons used to control how objects are viewed in the main window The button in the middle marked with a c is used to center the objects in the window For example if you move the cross hair away from the center using the middle mouse button you can press the c button and it will be automatically centered in the viewing window The top button is used to zoom in on the object in the window and the bottom button is used to zoom out Pressing the button makes things look bigger and pressing the button makes things look smaller These buttons will not do anything unless some object is in the window The rotation point is not considered to be an object It is simply a reference point and always looks the same size no matter how you zoom in and out on the window 1 3 Differential Equations Differential equations are an important tool They use used to develop mathematical models to describe physical phenom ena In many physical systems we do not know the equation describing the physical quantity we are interested in but by applying a physical principle a relationship that includes the physical quantity and its rate of change can be found For example if we are interested in finding the position of so
4. do this move the mouse pointer over the viewing window Press and hold down the far left button on your mouse and then move the mouse This is a long winded way to say click and drag the pointer across the window You should be able to see the view change as you drag the pointer around the window The images in Figures 2 3 through 2 5 show different views of the function It is difficult to visualize what the function looks like from looking at just one view The program allows you to move and examine the function from a wide variety of view points Given a function you should rotate and view the function from all possible angles You can move translate the whole function by dragging the mouse with the middle button held down It works the same way as the rotations de scribed above The difference is that the whole image moves up and down as well as left and right Notice that if you translate the image the rotations appear to be different The rotations always occur with respect to the center of the window Trajectory User s Manual 14 CHAPTER 2 THE FILE MENU traj PHE File Select View Help Figure 2 2 Initial view of the function defined in circle dat If you want to center the object in the window click on the button marked c on the far right of the window This will reset the view to the same as it was when you first read in the file This allows you to make whatever changes you want and easily reset the view 2 4 2
5. Reading the data file lorenz dat There is another example file included with the program called lorenz dat If you import the file in the same way described in the previous subsection you should see the top of the Lorenz attractor see Figure 2 6 This is an approximation to a well known equation but the specifics are not discussed here Try reading both the circle dat file and the lorenz dat file to see what it looks like to have two approximations at once If you import the file after reading circle dat and you look real closely Trajectory User s Manual 2 4 IMPORTING DATA 15 traj File select View Help Figure 2 3 View of the function defined in circle dat after a small rotation you should see that the function defined in circle dat is still in the picture as well A better view of the attractor can be seen by rotating the picture see Figure 2 7 and viewing it from different angles The different views should make it easier to see the much smaller spiral in the center of the other plot If you are feeling particularly adventurous try hitting the tab key sev eral times Fach time you hit the key a box should appear around the different plots I am assuming that you have loaded both circle dat and lorenz dat This is the selector box and is used to let you know which ap proximation is selected You can change the way the selected function is plotted There is more information on how to use the select featur
6. a function For example the function can be plotted using different color schemes and dif ferent values can be plotted on different axes The program allows you to make changes to one function or all of the functions If a function is selected and changes to the view only affect the function selected If no functions are selected then any change affects all of the functions The things that you can change include whether or not the cross hairs are displayed It also includes the background color as well as the color scheme used to generate the plot You can also toggle between plotting a function using dots at the data points or lines connecting the data points Finally you can change which variable is plotted on the x y and z axis 5 2 Show Rotation Point The rotation point is misnamed and is an artifact of a lack of imagination of the programmer The rotations take place relative to the center of the viewing window The cross hair that is displayed simply shows the center of all of the functions You can turn this off or on depending on what you think looks best To do this select the option Show Rotation Point from the View menu The cross hair will be toggled on and off as you choose this option 5 3 Toggle Background The color of the background can be toggled between white and black Printing the plot prints the background in the color on the screen If you are printing onto a sheet of paper it may be best to use a white background
7. called examples The file has six different functions within it The first three functions are an approximation that gives the Lorenz attractor The next three functions are an approximation of a set of equations that gives what is Trajectory User s Manual 5 6 AXIS 37 called the Butterfly Attractor In this example clear all of the plots and import the file six dat The approximation that was made generated an extremely large number of points The resulting plot does not look very good It looks a little bit better plotting in points mode see Section 5 5 A view of the function is shown in Figure 5 3 EF File select View Help Figure 5 3 The Lorenz attractor plotted using points Data file is from six dat The first three functions are approximations of the Lorenz attractor If you want to see what the other functions look like you need to change which functions are plotted on the x y and z axes Each axis will have to be defined separately The windows that are used to do this are shown in Figures 5 4 through 5 6 Trajectory User s Manual 38 CHAPTER 5 THE VIEW MENU First to choose to reset the x axis click on the View Menu and click on Axis In the menu that appears click on Set X Axis The window shown in Figure 5 4 should appear The functions in the file have to be labeled in some manner The convention is that the first column is called Time and the rest of the functions are numbered
8. phase space This system of equations entered in the equation window is shown in Figure 3 2 First the number of dependent variables must be specified In this case we have a system of two equations so at the top left of the window a 2 is entered next to the text N Next the two equations must be entered In particular the first equation d Yo t volt grl Yo2 t is entered as y2 Trajectory User s Manual 24 CHAPTER 3 THE EQUATION WINDOW v traj Equation Entry File Integration Help N 2 d dt y01 y2 Initial 1 000000 d dt yO2 y1 Initial 1 000000 ddt y03 Initial ddt y04 Initial ddt y05 Initial ddt y06 initial d dt y07 Initial d dt yos Initial Start Time 0 End Time 100 0 Time Step 0 1 File Name Figure 3 2 Equation window with two dimensional ODE that results in a circle in the phase space The second equation gy sort yo1 is entered as yl To the right of each equation the initial condition for the coresponding dependent variable is specified Finally the initial time the end time and the time step must be specified Here the initial time is set to 0 and the final time is set to 100 The time step is set to 0 1 Once these numbers and the equations are set a plot can be displayed by clicking on the Plot button When the button is clicked an a
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10. starting at zero The six functions that are found in the file six dat are named in the following way the first function column one is called the variable Time This is just a convention and not always followed The second function column two is called the variable y00 The third function column 3 is called the variable y01 The final function column 6 is called the variable y04 Variable to gr h E Choose which variable to plot along the x axis This only changes items that are selected Close Window Variable Time Variable y00 Variable y0 Variable yO2 Variable Variable Figure 5 4 Choose the X axis to be y02 To view the butterfly attractor first set the x axis to be plotted using the variable y02 Then set the y axis to be plotted using the variable y03 and finally set the z axis to be plotted using the variable y04 The steps to do this are shown in Figures 5 4 through 5 6 Note that as you do this the intermediate plots are incoherent Sometimes this happens Often times you have to look at the different Trajectory User s Manual 5 6 AXIS 39 Variable to gr h on the Y axis Nl x Choose which variable to plot along the asis This only changes items that are selected Close Window Variable Time Variable y00 variable y0 Variable yOz variable variable Figure 5 5 Choose the Y axis to be y03 phase portraits until you find something that displays a pat
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12. Traj User s Manual Kelly Black E MEE File Select View Help Copyright c 2004 Kelly Black Permission is granted to copy distribute and or modify this document under the terms of the GNU Free Documentation License Version 1 2 or any later version published by the Free Software Foundation with no Invari ant Sections no Front Cover Texts and no Back Cover Texts A copy of the license is included in the appendix entitled GNU Free Documentation License Trajectory User s Manual Contents 1 1 12 1 3 2 The 2 1 2 2 2 3 2 4 2 5 2 6 3 The 3 1 3 2 3 3 4 The 4 1 4 2 4 3 4 4 4 5 4 6 Introduction OVETVIEWE e ch Pak O ts A aA r Gout IN ORAN ZANA The Interface 2 02406 4 hb be w do GR ee 4 3 Differential Equations 8 2 0 do E it a Sy File Menu Introduction AA 0 zato eT LTR U 1400 Bh Clearing the Window w oe weta RO e AI Bee es Data Pile mewa pia u 2 R W AS A A TA a YZ Importing Date 2 Edek kB e oie APA ewe e EE 2 4 1 Reading the data file circle dat 2 4 2 Reading the data file lorenz dat 2 4 3 Data File Format us ufa wali ad SGO ARKO kg Paliwo Ro ze sa ye e ia BEA ee oe Y QUINE a a e a Mop ee oe la ee i Equation Window Entering An Equation Darle alcala e Saving and Recalling Equations Operations and Functions 004 Select Menu ntroduction sal E ed eee ton oe Gye ee a yli ZZ Using the Tab K
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14. at and lorenz dat 1 0000000e 00 0 0000000e 00 0 0000000e 00 9 9875026e 01 4 9979169e 02 5 0000000e 02 9 9500417e 01 9 9833417e 02 1 0000000e 01 9 8877108e 01 1 4943813e 01 1 5000000e 01 8 8919115e 01 4 5 53589e 01 9 9000000e 00 8 6521263e 01 5 0140513e 01 9 9500000e 00 8 3907153e 01 5 4402111e 01 1 0000000e 01 Table 2 1 First four lines and last three lines of the file circle dat Trajectory Users Manual 20 Trajectory CHAPTER 2 THE FILE MENU User s Manual Chapter 3 The Equation Window The two ways to obtain equations to plot is to read a data file or to specify a system of differential equations Here we focus on how to specify a system of differential equations A system of equations can be entered through the Equation Entry window This window is created when you choose File from the main window and then choose Add From Equation The window that appears is shown in Figure 3 1 The labels for each function in the system of equations are assumed to be of the form yo1 t yoo t yos t up to Yos t Once a system of equations is specified the program will approximate the system using one of the built in solvers A system is specified once the following is provided e The number of equations in the system The differential equation for each function The initial condition for each function e The start time e The end time e The time step to be used for the solver Once each o
15. bels 21 Functions 25 Importing Data 11 12 14 Integration 24 Interface Overview 5 Labels 21 Lines 34 Lorenz Equation 25 INDEX lorenz dat 14 Main menu 7 Modeling 8 9 Move 7 13 Numerical Approximation 9 25 ODE Acronym 5 Operations 25 Overview 5 The Interface 5 Phase space 36 Points 34 Quit 18 Quitting 11 RK4 Acronym 9 RKF Acronym 9 Rotate 7 12 Rotation Point 6 33 Runge Kutta scheme 9 25 Runge Kutta Fehlberg scheme 9 25 Saving Equations 25 Select All menu option 31 Select Menu 29 Deselect All 31 Select All 31 Select Next 31 Tab Key 15 29 Select Next menu option 31 Selection 15 29 32 Delete 15 30 Trajectory 55 Tab Key 15 29 Translate 7 View Background 33 Color 34 Format 34 Lines 34 Points 34 Rotation Point 33 View Menu 33 Viewing Format 34 Viewing Window 7 Zoom In 7 Out 7 Rotation Point 7 User s Manual
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18. e on page 29 and more information on what to do with a selected function throughout this manual but especially in the chapter on the View menu option Chapter 5 Trajectory User s Manual 16 CHAPTER 2 THE FILE MENU traj PHE File select View Help Figure 2 4 View of the function defined in circle dat after another small rotation You can also delete a selected function To delete a function first select and then hit the delete key Be careful about this because there is no warning Unlike the clear plot option the function is simply deleted with little fanfare 2 4 3 Data File Format The data file is kept as simple as possible Each row should contain the same number of points as the dimension number and the number of rows is equal to the number points you want to plot For example the function defined in the file circle dat has three dimensions There are also a large number of points used to describe the function Each row in the data file has three numbers one for each dimension and each data point is in a separate row Trajectory User s Manual 2 5 EQUATIONS 17 E File select View Help Figure 2 5 View of the function defined in circle dat after yet another small rotation The first four lines and last three lines are shown in Table 2 1 The file was created using matlab and saving the array in ASCII format The first row defines the first data point 1 0 0 The second row defines the seco
19. ear the viewing window to add new approximations print and to quit the program Each of the options is described in separate sections The options to clear the window and to quit are relatively straightforward You should be very careful about choosing the quit button At present the application quits and does not ask for confirmation It is possible to lose your work if you are not careful 2 2 Clearing the Window The program allows you to display the ap proximation to more than one equation at a time You can clear the viewport of all of the approximations by choosing the Clear Plot option from within the File menu A window will pop up asking if are sure If you click the button that says yes the whole window will be cleared If you click the button that says no then no action will take place 2 3 Data File There are two ways to get a solution or an approximation into the viewing window One way is to read a text file that contains the numbers representing the approximation The other way is to enter in the equations and let the program construct an approximation The program can read in the data much quicker than it can produce an approximation It can be more troublesome to generate the data using another program and save the data In some circumstances the data files either already exist or someone cre ated a file for a specific purpose In either case the program has the ability to import a file The way to import a
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22. ey to Make Selections Selecting All of the functions Stepping Through the Functions Deselecting all of the Functions cs els Deleting functions lt 4a ai ead ERE EES RS 11 11 11 11 12 12 14 16 17 18 21 22 25 25 4 CONTENTS 5 The View Menu 33 Bide Introduction e shana p A Me A 33 5 2 Show Rotation Point 33 5 3 Toggle Background hs li es ir 33 A A A AA 34 5 0 A A A a II BARA 34 BO ASS pto GE aa O ta A RA Gd e ot 2002 Seats te 34 A GNU Free Documentation License 43 1 APPLICABILITY AND DEFINITIONS 44 2 VERBATIM COPYING 20570650 Pa tran a A 6 44 46 3 COPYING IN QUANTITY BS 005 e Belo 46 4 MODIFICATIONS 40 ace dhs ie eae tas RAW E he AA 47 5 COMBINING DOCUMENTS 49 6 COLLECTIONS OF DOCUMENTS as eas e 50 7 AGGREGATION WITH INDEPENDENT WORKS 50 Oe TRANSLATION 0 as bn ty amp abe aoe Fw Gaia 51 9 TERMINATION aca ee Pada Aa Ea ole wr 51 10 FUTURE REVISIONS OF THIS LICENSE 51 ADDENDUM How to use this License for your documents 52 Bibliography 53 Index 54 Trajectory User s Manual Chapter 1 Introduction 1 1 Overview The program traj was written to help students visualize the solution to systems of ordinary differential equations ODEs It allows you to read a data file or specify a system of equations which can be approxi mated using several different
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24. f these items is provided the program will display an approxima tion once the Plot button is clicked 21 22 CHAPTER 3 THE EQUATION WINDOW hd traj Equation Entry A File Integration Help d dt y01 Initial d dt y02 Initial d dt y03 Initial d dtyd initiat d dt y05 Initial ddt y06 Initial ddt yO Initial d dt y08 Initial Start Time End Time Time Step File Name Piot Figure 3 1 Equation window with no equations 3 1 Entering An Equation To enter a system of equations you need to first specify the number of equations in the system This number is specified at the top left text box in the equation window Once this number is specified you must then enter each differential equation in the entries below The dependent variables are labeled from yo t to Yos t and the independent variable is assumed to be time t For each of these functions a differential equation must be specified A number of standard functions are recognized and are listed in subsection 3 3 A function is composed of constants the independent variable the independent variables the basic operations and pre defined functions listed in subsection 3 3 The basic operations include addition subtraction including unary minus multiplication division and exponentiation The symbol used for exponentiation is Some examples will be given below The order of operations is the usual convention a
25. file will be briefly described and the 11 12 CHAPTER 2 THE FILE MENU format of the file will then be discussed 2 4 Importing Data To read a file choose the option Add from data file from the file menu A window should immediately pop up You need to enter a file name in the box You can also choose a file using a point and click on the file name displayed in the window There should be three examples installed on your system If you do not know where the traj package was installed you should ask the person who installed it on your system There should be a subdirectory called examples in the directory where traj was installed There are three files that can be imported circle dat lorenz dat and six dat The file circle dat contains the solution that looks like a spiral The file lorenz dat contains an approximation to a system of ODEs that gives the Lorenz Attractor The file six dat contains the Lorenz Attractor as well as the Butterfly Attractor It is used for as example in the section that describes how to change the axis page 34 2 4 1 Reading the data file circle dat The first example focuses on the function defined in the data file circle dat This function defines a spiral and has three parts The domain is the times from zero to ten As the time increases the other two parts define a circle of radius one The three parts can be plotted as a three dimensional figure using an x a y and a z axis T
26. he data file has been created so that the times are initially plotted on the z axis The format of the data file is described on page 16 and the options for changing the axes is described on page 34 The other two parts are initially plotted on the x and y axes The first step to read the data file and plot it is to choose Add from data file from the File menu When this option is chosen a window should appear asking for a file name see Figure 2 1 It is assumed that you are familiar with the idea of directories and how to use a file dialog box Find the directory where the example files are installed on your system you will have to ask the person who installed the files and choose the file called circle dat Assuming that you can find the file and it has not been corrupted then you should see the image shown in Figure 2 2 This is a head on view of the function The time axis is pointing straight out of the window and cannot be seen The effect is that it looks like a two dimensional plot of a circle To view how the whole function appears you have to rotate the view To Trajectory User s Manual 2 4 IMPORTING DATA 13 Import Data File Create Dir Delete File Rename File home black c 0penglitraj examples al circle dat circle eqn lorenz dat six dat trial eqn wobblyCircle eqn Selection home black c openglitraj examples circle dat OK Cancel Figure 2 1 Importing the file circle dat
27. ly look at ODEs Given Equation 1 1 the goal is to figure out what the unknown function y t is It turns out we also need to know at least one more thing about the function In equation 1 1 we are given an initial condition which is specified in the second line Equation 1 1 is usually examined because it is has the least amount of notation possible and it can be solved In fact it is one of the few equations that we can actually solve Unfortunately most ODES that arise in the real world are far more com plex than the one given in equation 1 1 In fact the vast majority of ODEs do not have known closed form solutions This means we do not yet know how to find an expression for the solution in convenient terms One way around this problem is to use the computer to build an approximation to 20ne of the dirty little secrets of the mathematics department is that there are relatively few ODEs that can be solved analytically The vast majority of the equations that we can solve analytically are those found in the ODE class given at most colleges This leads far too many students to think that mathematics is just a bunch of techniques and most of it is known Nothing could be further from the truth Trajectory User s Manual 1 3 DIFFERENTIAL EQUATIONS 9 the solution There are many different ways to construct an approximation Each method has its own advantages and disadvantages The program traj has three different methods built i
28. main of the functions is some specific time interval It is useful to look at each function by itself as a function of time At the same time there is no law that says we have to view each of these functions by themselves We can also plot them against each other to see how they are related Each function is represented on the computer as a list of numbers For a system of equations the computer stores the same number of points for each function Given a list of three functions we get three lists of numbers The values of the functions at a specific time is stored as three numbers We can Trajectory User s Manual 36 CHAPTER 5 THE VIEW MENU I x select View Help Figure 5 2 The Lorenz Attractor from lorenz dat plotted using points treat those three numbers as a point in three dimensional space This is the basic idea behind the phase space Given a list of functions we pick any three of the functions and treat the functions as points in three dimensional space The program allows you to choose whichever functions you would like and plot them along whichever axis you would like to see As you change which function is plotted along which axis you can view the different phase portraits of the function As an example there is a data file called six dat included in the examples If you do not know where the file is located ask the person who installed the package on your system The file should be located in the subdirectory
29. me object we probably do not know in advance what its location is Otherwise there would not be any problem To find the position we have to apply some physical Trajectory User s Manual 8 CHAPTER 1 INTRODUCTION principle and try to work the resulting equation backward For the movement of objects we often apply Newton s Second Law which states that the mass times the acceleration of an object is equal to the sum of all of the forces on the object Another way to state Newton s Second Law is that momentum is conserved The momentum of some object does not change unless a force is applied This focus on momentum is one of the great break throughs in modern science 3 It turns out that the acceleration velocity and position of an object are all related The way we express this relationship mathematically is through the derivative or instantaneous rate of change The acceleration is the derivative of the velocity and the velocity is the derivative of the position For more information on how this relationship is derived see the books by Ostebee and Zorn 2 Hughes Hallet and Gleason 1 or any other text book An example of an ODE is the equation ZIO ult 1 1 y 0 w The equation is called an ordinary differential equation because it has a derivative in it The word ordinary is used because the function only depends on one variable t In higher dimensions the word ordinary is replaced by the word partial We will on
30. must be enclosed in parenthesis Example sin t y6 cos Cosine The argument must be enclosed in parenthesis Example cos yl y2 t tan Tangent The argument must be enclosed in parenthesis Example tan t y2 asin Inverse Sine arcsine The argument must be enclosed in parenthesis Example asin y3 acos Inverse Cosine arccosine The argument must be enclosed in paren thesis Example acos t yl 1 atan Inverse Tangent arctangent The argument must be enclosed in parenthesis Example atan yl y4 abs Absolute Value The argument must be enclosed in parenthesis Example abs y2 sqrt Square Root The argument must be enclosed in parenthesis Example sqrt y1 1 floor Floor The largest integer that is smaller than the argument passed The argument must be enclosed in parenthesis Example floor yl y2 ceil Ceiling The smallest integer that is bigger than the argument passed The argument must be enclosed in parenthesis Example ceil y1 trunc Truncation Remove the fraction from the argument passed The argument must be enclosed in parenthesis Example trunc y3 y2 Trajectory User s Manual 28 Trajectory CHAPTER 3 THE EQUATION WINDOW User s Manual Chapter 4 The Select Menu 4 1 Introduction There are many ways to change the way a function is viewed The different options are discussed throughout this manual and especially in chapter 5 You can change the options o
31. n a fourth order Runge Kutta method a Runge Kutta Fehlberg method and Euler s method The fourth order Runge Kutta RK4 method is a good general purpose scheme for building an approximation The Runge Kutta Fehlberg RKF method is based on the RK4 method but it is an adaptive method It changes the way it steps in time if the function appears to show some wide variations Euler s method gives an awful approximation The only reason it is included is because it is easiest to explain and many teachers like to have it included so that it can be used as a learning tool The important thing to keep in mind here is that there are three things There is a physical system The physical system is described using a mathe matical model Often times the mathematical model is approximated using some sort of numerical scheme You should always keep this in mind Some times the mathematical model is not very good so it does not matter how good the numerical approximation is Some times the mathematical model is good but the numerical scheme is not very good To be useful all the steps have to be good 3The definition of the word good is something that both scientists and mathemati cians have to agree on It is best left for more advanced studies Trajectory User s Manual 10 Trajectory CHAPTER 1 INTRODUCTION User s Manual Chapter 2 The File Menu 2 1 Introduction The file menu contains the options that allow you cl
32. n all of the functions or on individual functions that are in the current viewport In order to change the options for a particular function you must first select it and there are a couple of different ways to do this If no functions are selected and an option is changed the change will effect all of the functions being viewed For all of the examples here it is assumed that you have at least two functions plotted in the main viewing window In particular the figures given will use the two functions defined by the example data files lorenz dat and circle dat Information on how to import the two files is given on page 12 and page 14 The different ways to select an object include using the tab key to select one function at a time Using the options from within the Select option on the main menu you can select all of the functions step through the functions or turn off all of your selections Each of these different options is discussed in the sections that follow 4 2 Using the Tab Key to Make Selections The program keeps track of the functions that you add to the viewport and keeps them in a list You can select one function at a time and step through each function by hitting the tab key Each time you hit the tab key it will turn off the current selection move to the next function in the list and select it When a function is selected a box around the function is drawn In Figures 4 1 and 4 2 the sequence is shown In this partic
33. n or of any later version that has been published not as a draft by the Free Software Foundation If the Document does not specify a version number of this License you may choose any version ever published not as a draft by the Free Software Foundation Trajectory User s Manual 52 APPENDIX A GNU FREE DOCUMENTATTON LICENSE ADDENDUM How to use this License for your documents To use this License in a document you have written include a copy of the License in the document and put the following copyright and license notices just after the title page Copyright YEAR YOUR NAME Permission is granted to copy distribute and or modify this document under the terms of the GNU Free Documentation License Version 1 2 or any later ver sion published by the Free Software Foundation with no Invari ant Sections no Front Cover Texts and no Back Cover Texts A copy of the license is included in the section entitled GNU Free Documentation License If you have Invariant Sections Front Cover Texts and Back Cover Texts replace the with Texts line with this with the Invariant Sections being LIST THEIR TITLES with the Front Cover Texts being LIST and with the Back Cover Texts being LIST If you have Invariant Sections without Cover Texts or some other com bination of the three merge those two alternatives to suit the situation If your document contains nontrivial examples of program code we rec ommend releasing the
34. nd data point The last line defines the last data point that will be plotted The way the data file was created the first two dimension are data points describing a circle in two dimensions The third dimension given in the third column can be thought of as the time and is equally spaced each time step is 0 05 in whatever units you happen to be using 2 5 Equations Choosing the option Add from equation from the File menu brings up the equation window This window has the most options and is examined in its own chapter Chapter 3 If you want to look at some Trajectory User s Manual 18 CHAPTER 2 THE FILE MENU traj PHE File select View Help Figure 2 6 Initial view of the functions in circle dat and lorenz dat examples read in some of the example equation files These files have the extension eqn and should be installed in the example directory If you do not know where this directory is ask the person who installed the package The example subdirectory should be in the directory where the package was originally installed but may have been moved by the person who installed the package 2 6 Quitting To quit choose the quit option from the file menu Be careful about doing this No confirmation message will be displayed The program will simply exit with no fanfare what so ever Trajectory User s Manual 2 6 QUITTING 19 File select View Help Figure 2 7 Rotated view of the functions in circle d
35. nd parenthesis can be used to group operations When typing in an equation the independent variables are referred to as Trajectory User s Manual 3 1 ENTERING AN EQUATION 23 YO1 Y02 through Y08 The variables do not have to be typed in upper case and the 0 is optional For example to refer to the function Yoa t you could also type either y08 or y8 The independent variable is typed as t or T Constants are typed as a string of numbers Once the equations for each dependent variable are entered an initial condition for each variable must be specified This is a number which spec ifies the value of the dependent variable at the initial time The initial or starting time and the end time is specified at the end of the list of equa tions Additionally an initial time step must be specified which is used by the numerical approximation that is chosen Once all of these things are entered a plot of the approximation is dis played once the plot button is clicked As an example we examine two different systems of equations The first is a simpler linear problem ot Yoa t 3 1a alt yolt 3 1b where the intial conditions are yo1 0 1 3 2a Yyo2 0 1 3 2b Here we will plot the equation for time between 0 and 100 with an initial time step of 0 1 A list of the available approximation schemes is given on page 9 The solution to this equation is a circle in
36. numerical techniques The program allows you to view the solution in 3D space and you can choose the phase space yourself This manual is designed to offer an overview of how to use the program and is not designed to be an introduction to ODEs It is assumed that you have some basic knowledge of ODEs The program was designed for students learning the topic however so the examples are relatively straight forward and some background info is included The organization of this manual is not intuitive if you are focused on the ODEs It is designed to offer a reference for using the program I have chosen to organize this manual in terms of the program interface Hopefully this will make it easier to find information while you are using the program It is probably quite maddening if you are reading this manual hoping to learn about the equations themselves 1 2 The Interface The name of the program is called traj To start the program from the command line in linux simply type in the name of the program user traj If the program does not start make sure that it is installed and that the program is in your path The path is a list of directories that are searched 6 CHAPTER 1 INTRODUCTION whenever you enter a command The first file that is found in the list of directories that matches the name typed in is executed A window like the one shown in Figure 1 1 should have appeared assum ing that the program is installed correc
37. o copy and distribute verbatim copies of this license document but changing it is not allowed Preamble The purpose of this License is to make a manual textbook or other func tional and useful document free in the sense of freedom to assure everyone the effective freedom to copy and redistribute it with or without modifying it either commercially or noncommercially Secondarily this License pre serves for the author and publisher a way to get credit for their work while not being considered responsible for modifications made by others This License is a kind of copyleft which means that derivative works of the document must themselves be free in the same sense It complements 43 44 APPENDIX A GNU FREE DOCUMENTATION LICENSE the GNU General Public License which is a copyleft license designed for free software We have designed this License in order to use it for manuals for free software because free software needs free documentation a free program should come with manuals providing the same freedoms that the software does But this License is not limited to software manuals it can be used for any textual work regardless of subject matter or whether it is published as a printed book We recommend this License principally for works whose purpose is instruction or reference 1 APPLICABILITY AND DEFINITIONS This License applies to any manual or other work in any medium that contains a notice placed by the cop
38. pproximation to the equation is found and then plotted The method used to construct the approximation is specified in the Integration menu To check or set the approximation method click on Trajectory User s Manual 3 2 SAVING AND RECALLING EQUATIONS 25 the Integration menu at the top of the window The current approximation method is marked To change the approximation method choose a differ ent option from the menu The default method is a Runge Kutta Fehlberg method using a fourth order approximation with a fifth order check The second example demonstrates the Lorenz equation d gal 12 y01 yor 3 3a d qre 28Y01 Yoo Yo1Y03 3 3b d qos 2 7 Yyo1Yo2 Yo3 3 36 Here we use the initial condition yoi 0 3 3 4a yo2 0 4 3 4b 0 18 3 40 The way to enter this equation is shown in Figure 3 3 Once the equations the initial conditions and the times are specified you can click on the plot button and an approximation is found and plotted 3 2 Saving and Recalling Equations Once a set of equations are en tered they can be saved and recalled for later use To save a set set of equa tions clicke on the File menu and choose Save You will be prompted to supply a file name It is assumed that the file will end with an eqn You can then recall the file by clicking on the File menu and choosing Open You must then specify the file name of a p
39. reviously saved set of equations 3 3 Operations and Functions The following functions are recognized within an equation entry The sum of two numbers Unary plus is not supported Example yl 5 Either the difference of two numbers are the negative of a given number Unary minus is supported Example y2 8 or y3 t Trajectory User s Manual 26 CHAPTER 3 THE EQUATION WINDOW YEMAS EIA File Integration Help NE d dt y01 12 y01 y02 Initial 3 ddtyQ2 26 y01 y02 y01 y03 _ Initial la d dt yO3 2 7 y01 y02 y03 Initial 18 d dt y04 Initial ddt y05 Initial d dt y06 Initial d dt y07 Initial ddt y08 Initial statTmejo 8 a End Time 100000 0 Time Step 0 1 File Name home black c opengtra Figure 3 3 Equation window with the system of equations for the Lorenz equation The product of two numbers Example yl x y3 The quotient of two numbers Example y2 8 Exponentiation of two numbers Example 12 5 y1 Left parenthesis Right parenthesis Example y1 t y4 1 3 In Natural Logarithm The argument must be enclosed in parenthesis Example In yl 4 Trajectory User s Manual 3 3 OPERATIONS AND FUNCTIONS 21 exp Exponential The argument must be enclosed in parenthesis Example exp y2 4 sin Sine The argument
40. s on the front cover and Back Cover Texts on the back cover Both covers must also clearly and legibly identify you as the publisher of these copies The front cover must present the full title with all words of the title equally prominent and visible You may add other material on the covers in addition Copying with changes limited to the covers as long as they pre serve the title of the Document and satisfy these conditions can be treated as verbatim copying in other respects If the required texts for either cover are too voluminous to fit legibly you should put the first ones listed as many as fit reasonably on the actual cover and continue the rest onto adjacent pages If you publish or distribute Opaque copies of the Document number ing more than 100 you must either include a machine readable Transparent Trajectory User s Manual 47 copy along with each Opaque copy or state in or with each Opaque copy a computer network location from which the general network using public has access to download using public standard network protocols a complete Transparent copy of the Document free of added material If you use the latter option you must take reasonably prudent steps when you begin dis tribution of Opaque copies in quantity to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy directly or through your agents or retail
41. se examples in parallel under your choice of free soft ware license such as the GNU General Public License to permit their use in free software Trajectory User s Manual Bibliography 1 D HuGHES HALLET AND A M GLEASON Calculus John Wiley amp Sons Inc New York 1 ed 1994 2 A OSTEBEE AND P ZORN Calculus from Graphical Numerical and Symbolic Points of View vol 1 Saunders College Publishing New York 1 ed 1997 3 R S WESTFALL Never at Rest A Biography of Isaac Newton Cam bridge University Press Cambridge UK 1998 93 Index Acronyms ODE 5 RK4 9 RKF 9 Approximation 9 24 Axis 34 Changing 34 Background 33 Button 7 13 Center View 7 13 circle dat 12 14 Clearing the Window 11 32 Color 34 Data File Format 16 Data Files 11 Delete 15 30 32 Deselect All menu option 31 Differential Equations Example 8 Introduction 7 Modeling 8 9 Entering Equations 21 Equation Labels 21 Equation Window 17 21 Approximation 24 Entering Equations 21 Equation 22 Equation Labels 21 54 Equations 25 Function Labels 21 Functions 25 Integration 24 Labels 21 Operations 25 Saving Equations 25 Equations 11 17 21 22 25 Euler s Scheme 9 File Dialog Box 12 File Menu 11 32 Clearing the Window 11 32 Data Files 11 Equation Window 17 21 Equations 17 21 Importing Data 11 12 14 Quit 18 Format 34 Data File 16 Function La
42. tern That is why this program was written It allows you to search through the different phase portraits to try to find any patterns that might be hidden in the data Note that in Figure 5 7 if you look at the plot without moving it is extremely difficult to see any structure in the data To see the structure you should view it as a set of points and rotate the plot around to look at it from all different angles Trajectory User s Manual 40 CHAPTER 5 THE VIEW MENU Variable to gr h on the Z axis Choose which variable to plot along the axis This only changes items that are selected Close Window Variable Time Variable yOo variable y0 Variable yOz Variable yOa e yi Oe Figure 5 6 Choose the Z axis to be y04 Trajectory User s Manual 5 6 AXIS 41 traj File Select View Help Figure 5 7 The Butterfly attractor plotted using points Data file is from six dat Trajectory User s Manual 42 Trajectory CHAPTER 5 THE VIEW MENU User s Manual Appendix A GNU Free Documentation License The following description of the GNU Free Documentation License was found at the URL http www gnu org licenses licenses html FDL This is a ver batim copy of the license found on 29 April 2004 and the full text immedi ately follows Version 1 2 November 2002 Copyright 2000 2001 2002 Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA Everyone is permitted t
43. tly and is running on your system This window is refereed to as the Main Window The large window may be blank or have a small cross hair in the middle of it depending on how the defaults are set up on your system The cross hair can be toggled on and off by clicking on the View menu and choosing the option Show Rotation Point File select View Help Figure 1 1 Blank screen from start up 1This is an operating system question If this does not make sense you should contact someone familiar with your system Trajectory User s Manual 1 3 DIFFERENTIAL EQUATIONS T Across the top of the window is the main menu By moving your mouse over the different items in the main window and clicking the left button more menus drop down and can be seen Clicking on the items in the drop down menus allows you to choose the various options described in this manual It is assumed that you are familiar with using the mouse to choose the different menus and the options themselves will be the main topic of this document Directly below the main menu is the viewing window This is where the solutions to the differential equations will appear The different menu options will allow you to change how the solutions appear A number of other keys allow you to rotate and move the solutions in the window To rotate an object in the viewing window move the mouse cursor into the viewing window press and hold the left button down and move the cursor
44. ular example 29 30 CHAPTER 4 THE SELECT MENU traj PHE File select View Help Figure 4 1 The spiral is selected and the Lorenz attractor is not selected the Lorenz attractor was imported first followed by the spiral The first time the tab key is struck the spiral is selected The list is kept in reverse order This is shown in Figure 4 1 The second time the tab key is struck the selection box moves from the spiral to the Lorenz attractor This is shown in Figure 4 2 Once you have stepped through and have chosen each of the functions the next time you hit the tab key then no selection is made In this example the third time you hit the tab key no selection is in effect To start the selection sequence again you have to hit tab again and the whole process begins anew Be careful about hitting the delete key while any of the functions are selected A selected function will be deleted The program will not ask for confirmation it will just delete the function Trajectory User s Manual 4 3 SELECTING ALL OF THE FUNCTIONS 31 File Select View Help Figure 4 2 The spiral is not selected and the Lorenz attractor is selected 4 3 Selecting All of the functions You do not have to select only one function at a time It is possible to select all of the functions at one time To do this choose the option Select All from the Select menu This will select all of the functions in the viewport and a box will be dra
45. ut format LaTeX input format SGML or XML using a publicly available DTD and standard conforming simple HTML PostScript or PDF designed for human modification Examples of trans parent image formats include PNG XCF and JPG Opaque formats include proprietary formats that can be read and edited only by proprietary word processors SGML or XML for which the DTD and or processing tools are not generally available and the machine generated HTML PostScript or PDF produced by some word processors for output purposes only The Title Page means for a printed book the title page itself plus such following pages as are needed to hold legibly the material this License requires to appear in the title page For works in formats which do not have any title page as such Title Page means the text near the most prominent appearance of the work s title preceding the beginning of the body of the text A section Entitled XYZ means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses follow ing text that translates XYZ in another language Here XYZ stands for a specific section name mentioned below such as Acknowledgements Dedications Endorsements or History To Preserve the Title of such a section when you modify the Document means that it re mains a section Entitled XYZ according to this definition Trajectory User s Manual 46 APPENDI
46. wn around each function An example of this is shown in Figure 4 3 4 4 Stepping Through the Functions Another way to step through and select one function at a time is using the Select Next option in the Select menu This works exactly like the tab key as discussed in section 4 2 This option is in place in case the tab key does not work on your system or you are more comfortable using the menu Trajectory User s Manual 32 CHAPTER 4 THE SELECT MENU E PHE File select View Help Figure 4 3 Both the spiral is selected and the Lorenz attractor is selected 4 5 Deselecting all of the Functions If you would like to turn off all of the selections choose the Deselect All option from the Select menu This simply turns off all of the selections and all of the bounding boxes are turned off 4 6 Deleting functions There are two different ways to delete functions To delete a specific function you simply select it and hit the delete key The function will be deleted but the program will not ask for confirmation It is assumed that when you hit the delete key you really mean it The other way to delete functions is to clear the whole menu To do this choose Clear Plot from the File menu You will be given a chance to confirm your decision If you answer yes in the box that pops up all of the functions will be deleted Trajectory User s Manual Chapter 5 The View Menu 5 1 Introduction You can change the way you view
47. works in or on a volume of a storage or distribution medium is called an aggregate if the copyright resulting from the compilation is not used to limit the legal rights of the compilation s users beyond what the individual works permit When the Document is included in an aggregate this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document If the Cover Text requirement of section 3 is applicable to these copies of the Document then if the Document is less than one half of the entire aggregate the Document s Cover Texts may be placed on covers that bracket the Document within the aggregate or the electronic equivalent of covers if the Document is in electronic form Otherwise they must appear on printed covers that bracket the whole aggregate Trajectory User s Manual 51 8 TRANSLATION Translation is considered a kind of modification so you may distribute translations of the Document under the terms of section 4 Replacing Invari ant Sections with translations requires special permission from their copy right holders but you may include translations of some or all Invariant Sec tions in addition to the original versions of these Invariant Sections You may include a translation of this License and all the license notices in the Document and any Warranty Disclaimers provided that you also include the original English version of this License and the original v
48. yright holder saying it can be distributed under the terms of this License Such a notice grants a world wide royalty free license unlimited in duration to use that work under the conditions stated herein The Document below refers to any such manual or work Any member of the public is a licensee and is addressed as you You accept the license if you copy modify or distribute the work in a way requiring permission under copyright law A Modified Version of the Document means any work containing the Document or a portion of it either copied verbatim or with modifications and or translated into another language A Secondary Section is a named appendix or a front matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document s overall subject or to related matters and contains nothing that could fall directly within that overall subject Thus if the Document is in part a textbook of mathematics a Sec ondary Section may not explain any mathematics The relationship could be a matter of historical connection with the subject or with related matters or of legal commercial philosophical ethical or political position regarding them The Invariant Sections are certain Secondary Sections whose titles are designated as being those of Invariant Sections in the notice that says that the Document is released under this License If a se
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