The left side can be resized by clicking on the vertical line forming the right hand boundary of the left regions, clicking, and dragging. The two right side graph/data windows can be resized by doing the same thing to the line between the graphs.

All these windows are linked together; that is, a change in one of the windows can affect each of the other windows. This will become clearer as we explore their uses.

Single Ordinary Differential Equations

Entering the equations

Here's an example of a first order ordinary differential equation which you can use to learn about the Tool. The logistic growth population model is used sometimes to model populations which have a limiting size ("carrying capacity"), but which initially grow somewhat exponentially. One form of the differential equation for this model is

y' = r * y * (1 - y/K)

where y is the population at time t, r is a proportionality constant, and K is the limiting size or carrying capacity of the population. Notice that while y is small, y' is approximately r y, which gives exponential growth, and when y approaches K, y' approaches zero. Also, if y > K, the population growth rate is negative.

To enter this equation into the ODE Architect Tool, click in the equation region of the left window and type the equation exactly as it is above. (You will need the * to indicate multiplication.) Place a semicolon (;) after the equation. This lets ODE Architect know that you have reached the end of the equation. (Equations can be spread out over multiple lines, if you wish, but put the semicolon only after the last line of the equation.) Important: Equations must be in the form y' = f(t,y). For instance, y' + 3*y = t is not an accepted form.

You will also need to enter values for the parameters r and K. You can (arbitrarily) choose r = 1 and K = 10. To enter these, type in the lines

r = 1;
K = 10;

Again, don't forget the semicolons at the end of the equations. (By the way, these do not have to be on separate lines. The semicolons tell the Tool where the equation ends.)

If you wish, you can type comments in this region. Any text or information you do not want the Tool to enter needs to be preceded by /* and followed by */. For instance, you might want to type

/*This is a logistic population growth model*/

somewhere in the region. (You can also "comment out" equations this way, if you don't want the solver to use them now, but might want the solver to use them later.) Again, these comments can be spread over multiple lines, as long as the first line starts with /* and the last line ends with */.

Once the equations are typed in, they need to be entered into the solver. You can do this by clicking the Enter button at the top of the solver region of the left window, by using the Equations menu item Enter, or by pressing the F4 key. Once you do this, the two windows at the right become active, and you can begin to solve the equation and draw graphs.

Drawing direction fields and solutions

Suppose you would like a direction field for this differential equation, together with a few solution curves. You can do this using either the t-y tab or the 2D tab at the bottom of either of the right regions. (The 2D tab allows you to choose any two variables to plot against one another, while the t-y tab was created based on the variables you typed into the differential equation.) Let's suppose you use the t-y tab in one of the right windows.

You can get the popup menu you need to set properties of the graph either by right-clicking in the window or by clicking on the Menu icon (top, looks like a list) from the four icons at the right of the graph window. Either way, you can get a direction field by clicking the Direction Fields item, or by choosing Edit. The latter gives you a few more options, so choose Edit this time. You will get a popup menu with four tabs: Plot, Dir Field, Equilibrium, and Titles.

The Plot tab allows you to choose solution curve colors, markers, background colors, whether or not to show a grid and colors for the grid, and markers for the graphs. The Equilibrium tab applies to system of equations, so we will discuss it later. The Title tab lets you give a title to your graph and to label the axes. The Dir Field tab lets you choose whether or not to show a direction field (do so now), what to use as the head of the line segments in the field, the density of the line segments, whether or not to show magnitudes, and what scales to use. This time choose arrows for the heads and accept the other defaults.

To get a solution curve in the window, we need to give the solver an initial condition. There are two ways to do this. If you need a curve for a precise set of initial conditions, go back to the solver region. Choose the IC tab (if it is not currently selected) and type in the initial condition you want to use. For instance, this time set t to 0 and y to 2. Click on the Solve (calculator) icon at the top of this region. (Alternatively, you could choose Solve from the Solutions menu item, or press the F5 key.) The solver does its thing, and a solution curve should appear in your t-y window on the right side of the screen.

Notice that there are two arrows at the upper right of the solver region. The one pointing to the right is probably the one that is in the "down" position. This means that the solution will be drawn for increasing values of t. To get a "backwards" solution instead, you would press the left pointing arrow before pressing Solve. Do so now, and a pop-up menu will appear. Choose Extend to add the new data points to the original data point "run."

Anytime you do another run with the same initial conditions (including generating both forward and backward solutions) the solver will prompt you to choose among three things. Choose Continue to generate a new set of data from the starting point of the old data, but to leave the old data intact. The new data will be kept in a separate run from the old data (see the next section), even though its graph might simply lie on top of the preceding one. Choose Extend to generate more data from the ending point of the old data (if you use the same direction) or from the beginning (if you reverse directions). Choose Replace to remove the most recent data run and re-do the solution. Notice that you can bypass this prompt and automatically choose Extend by pressing the Extend (calculator with a + sign) icon instead of the Solve icon.

Another way to get a solution curve is to choose the Solve icon (third from the top, with a graph and calculator) from the four icons at the right of the graph window. You can then click anywhere within the graph window, and a solution will be calculated and plotted. The point where you click is entered as the initial conditions in the solver region of the left window, the solver runs, and a curve is drawn. Once again the direction arrows in the solver region choose whether to solve for increasing t or decreasing t. If you want both directions, make sure to click the other direction arrow and press Extend (or Solve) while these initial conditions are still active (i.e. before clicking a different point in the graph window).

One interesting point about graph windows - if your cursor is in the graph window, the coordinates of the point where the cursor is located show up at the lower right corner of the entire ODE Architect Tool window. You can use these coordinates to help in deciding where to click in the graph. Draw a second solution on your graph using the method of the preceding paragraph, starting at a point near (2.5,3.5). Don't forget to go in both directions.

Multiple solutions/data runs

Each time you solve with new initial conditions, or choose Continue when you solve with the same initial conditions as before, a new data run is generated. The easiest way to see what this means is to make one of your right side windows a data window. To do so, click the Data tab at the bottom of that window. There will be numbered tabs at the lower part of the window, one for each data run you have generated. Make sure you have generated at least two runs (by setting initial conditions or by clicking in the graph window, as described above).

Each of the numbered tabs gives you the list of all data calculated during that run, including values of the independent and dependent variables and all parameters. Which columns appear is adjustable by either right-clicking in the window or clicking on the Menu icon at the right side of the window, and choosing the columns from the pop-up menu. You scroll up and down through a particular data run by using the scroll bar at the right hand side of the window. The initial conditions are given at the top of the data run. Notice that when you choose a tab for a different data run, the values of the initial conditions in the solver region change to those of that run. This is one way to be able to choose a particular data run to extend, or to obtain both a forward and backward solution for a particular set of initial conditions.

You can also choose which data run to view using the graph window. Choose the Select icon (second from the top, graph with a pointer). If you then click on any solution curve, the data tab for that curve will be chosen in the Data window, and its initial conditions will show up in the lower region of the left window. The curve itself will also be highlighted. Notice that clicking on a data tab for a particular run will cause its solution curve to be highlighted as long as the Select icon in the graph window is "down."

Exploring and animating the graph

You can trace (explore) any solution curve. To do so, click the Explore icon (graph with vertical and horizontal lines) just below the menus, choose the Explore item from the Solutions menu, or press the F10 key. An Explore window will appear on your screen. The scroll bar at the bottom of that window allows you to move through the data points on your currently chosen run. You will see dashed horizontal and vertical lines on your graph showing where the data point is located, and the data point itself will be highlighted in your data window. You can switch to a different data run by clicking on its tab in the data window, or by using the Select icon to highlight a different curve in the graph window.

Another thing you can do is to animate a data run to trace a solution curve. To do this, first select one of your data runs. To animate it, press the Animate icon (pendulum) just below the menus, choose the Animate item from the Solutions menu, or press the F9 key. You can adjust the animation settings using either of the last two methods, or by pressing the Animate Dialog icon (pendulum with list) next to the Animate icon instead of pressing the Animate icon. You will see the solution curve traced from left to right in your graph window, and you will see the data points scroll in your Data window.

Modifying the graph

You may wish to change the way the graphs are displayed, including color, labels, and scales. These can be adjusted from the popup menu you get by either right-clicking in the graph window or clicking on the top icon at the right side of the graph menu. Use Edit to change the appearance of the graph, including color, labels, style, thickness (width), markers, etc. and, as we saw before, display direction fields. You also can print your graphs, copy them, and maximize them to completely fill the computer screen.

The other thing you can do from this popup menu is zoom and scale your graphs. The default is to autoscale both axes. As you do additional runs, you might see the scales (and ranges) of the graphs change from time to time to adjust to the new calculated data. If you do not want this to happen, or if you want to have a specific range on one or both of the axes, you need to turn off autoscale. You can do this using the Auto Scales item, or from within the Scales item. To manually change the scales and range, use the Scales item. You will get a popup window with tabs for each axis. The axes are adjusted individually using these tabs, including whether or not to autoscale that particular axis. Set your X-scale to a range of 0 to 10 and your Y-scale to a range of -5 to 15, and click OK. (For each axis, you will have to turn off Auto Scale before setting the range.) The direction field for the graph should become a bit more interesting.

For practice, draw an additional solution curve by choosing the solution icon (third from top, with a graph and calculator) and clicking on any point where the y coordinate is between 0 and 10. Make your curve "two directional" by clicking on the opposite direction arrow (in the lower region of the left window), and clicking the Extend button. If you wish, you could repeat this for a couple more solution curves.

Next, while the forward solution arrow is down, click on a point where the y coordinate is less than 0. You should get the message "Warning. Solution can not proceed with required accuracy." Click OK, and the forward solution curve will be drawn. The same message will appear if you use the backward solution arrow and click on a point where the y coordinate is greater than 10. (Why do you think the message appears?) While not a problem at the moment, in the next section we will see that this can be a major annoyance.

Sweeps

Sweeps are used to generate several solution curves at once. These curves can correspond to varying an initial condition, varying a parameter, or even varying two of these at a time.

First, clear the runs you have already calculated. To do this, click the Clear button (pencil eraser icon) at the lower right of the solver region, and choose Clear All Runs. Click Yes when prompted. Make sure the forward solution arrow is "down." Also reset the initial condition value of t to 0. (The y value can be whatever you want.)

Next, do a sweep on initial conditions. To do so, click the Sweep tab at the bottom of the solver region, and choose single sweep. Use the drop down box under Sweep 1 to choose to sweep on the variable y. Make 0 the Start, 12 the Stop, and make # Points 7. (This will correspond to y coordinate initial conditions of 0, 2, 4, 6, 8, 10, and 12, all with t coordinate 0.) Click the Sweep button (three calculators icon) at the top of the region. Seven data runs will be calculated and listed in the data window, and the solution curves will appear in the graph window.

One of the annoyances of the ODE Architect Tool happens with sweeps. To see it, click the IC tab and clear all runs. Click the Sweep tab, and change the Start for the sweep from 0 to -2. Click the Sweep button. The sweep bombs out with the "Warning. Solution can not proceed with required accuracy." message. The rest of the sweep is not performed, even though for those initial conditions the required accuracy is obtainable. Hopefully this will be fixed in future versions of the software.

Finally, do a sweep on the parameter K. Once again, click the IC tab, clear all runs, but this time set the initial conditions to t = 0 and y = 2. Click the Sweep tab, choose K as the sweep variable, make 4 the Start, 10 the Stop, and make # Points 4. Click the Sweep button. The result shows how the population size would depend on the carrying capacity. This ability of ODE Architect to sweep on parameters is a nice feature missing from many other ODE packages. (Note that the direction field makes no sense with this sweep, as it depends on K.)

Other 2D graphs

While you have the last sweep from the preceding section available, it might be nice to see the effect of varying K on the rate of population growth. Choose the 2D tab in the window currently displaying the data. Popup the Edit menu for that window (right-click, or click the top icon at the right of the window), and choose the Plots tab. Keep t as the variable for the X-axis, and use one of the drop down boxes to choose y' (not y) as the variable for the Y-axis. (Note that you can plot up to five variables against t, and can choose the color of the graphs for each variable.) You should have a plot of four graphs of y' vs. t (for four different values of K). If you choose the Selector icon (second from top) in each window, the currently chosen data run graph will be highlighted in each. Clicking on a different graph in one window will also highlight the corresponding graph in the other window.

If you have several variables chosen in the Y-axis portion of the Edit 2D Graph popup menu, you can decide for which ones to display the corresponding plots by using the box just after the number for that variable. An "x" in the box means the curves corresponding to that variable will be displayed, while if the box is cleared those curves will be hidden. In this way you can use one window to switch among plots of many different variables against a common X-axis variable.

You might also wish to look at one other 2D plot, namely that of y' against y. To create this plot, in the 2D tab window popup the Edit menu, choose y for the X-axis variable name, and keep y' as the Y-axis variable. Notice that the plots begin with y = 2, as that was the initial condition on y in the last sweep you did. You might wish to extend the sweep in the backwards direction to get a more complete graph. Do this by clicking on the backward arrow in the solver region, and clicking on the Extend (calculator with a +) icon. What are the shapes of the four curves that appear? Do these shapes make sense, considering the differential equation itself?

Equilibrium

The last thing to do for now is to find equilibria for the differential equation. First, using the IC tab in the solver region, clear all runs. Click the Equilibrium tab in this region, and type in an initial guess of 8 for y. Press the Calculate EQ button. A horizontal line will appear in your t-y graph at y = 10 (with a number 1 at its right end), the other graph window will change to the Equil tab, and the result of the search (y = 10, Successful) will appear as row 1.

There is another way to find equilibria, using a graph window. In the t-y graph window, first choose the Equilibrium icon (last of the four, with a graph and EQ). Then click on any point with y-coordinate approximately 2. (Remember, the coordinates of the cursor show up at the lower right corner of the entire ODE Architect Tool window.) The equilibrium value of y = 0 should be found, and another horizontal line will appear on your graph (with the number 2 at its right end, indicating the row of the table containing that value).

The numbers on the graph correspond to the numbers in the table of equilibrium values appearing when the Equil tab is chosen. You can delete one of these values (if you wish) by clicking the one you wish to delete in the table, right-clicking anywhere in the Equil tab window, choosing Delete from the resulting popup menu, and answering Yes to the prompt.

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Systems of Ordinary Differential Equations

Much of how to use the ODE Architect Tool for systems of ordinary differential equations is the same as for single equations. You should read that section above before going through this one, as there are fewer details below.

Entering the equations

Here is a predator-prey model with satiation to use in this section. x represents the number of prey while y represents the number of predators. Use File New to clear the old equations (if any are present). Then type the equations and parameter values exactly as below (except they do not need to be centered), including the semicolons (;) at the end of each equation. As in the single equation case, the derivative (or parameter) must be alone on the left side of the equation.

x'=x*(a-b*x-c*y/(k+x));
y'=y*(-d+e*x/(k+x));
a=1;
b=1;
c=1;
d=0.5;
e=1;
k=0.2;

Of course, you could simply insert all the parameter values in the top two equations, but you should at least keep the parameter k separate, as you will do a sweep on it later.

The model's first equation, giving the growth rate of the prey, has three terms. The first (ax) is the usual (exponential) reproduction term. The second (bx²) is the term causing a logistic limiting population. The third (-cxy/(k+x)) represents x as the prey for the predator y, but with the twist that an abundance of prey (large x) will reduce the percentage of prey eaten by the predators. (In a model without satiation, the denominator is 1 instead of k+x.)

The second equation, giving the growth rate of the predators, has two terms. The first (-dy) represents the fact that the predators will die off in the absence of prey, and the second (exy/(k+x)) gives the growth rate for the predators from eating the prey. (Once again, in a model without satiation, the denominator is 1 instead of k+x.)

Once you have typed in all the equations, press the Enter key in the upper left of the solver region. Two graph windows will become active, each with t-x, t-y, x-y, 2D, 3D, Equil, Data, and Model tabs. Choose the x-y tab for the top window, and the Data tab for the bottom one.

Phase portraits

To make a phase portrait in the x-y graph window, popup the graph menu (right-click, or click the Menu icon), choose Edit, and then the Dir Field tab. Put a check in the Show box, and choose Arrows for Vector Heads. Change the intervals from 15 to 20 for both X and Y. Click "Ok" to draw the vector fields.

Use the Scales graph menu item to change the X and Y scales to a range of 0 to 1.5 for each. (Make sure to turn off Auto Scale in both the X-Scale and Y-Scale tabs so that you can change the Minimum and Maximum for each.) Again, click "Ok."

One way to draw several solutions is to choose the Solve icon (third from top, calculator) at the right of the graph window, and then click anywhere in the graph. A curve will be drawn, with t values running from 0 to 10 (by default). To extend the curve for higher values of t, click the Extend button in the solver region. Do this a few times until you have an idea of the limiting behavior of the run. You should also probably do this for the backwards direction (click the left pointing arrow in the solver region, and Extend the run). Repeat this for a few other initial conditions (click on different points in the graph).

What happens to the solutions as t ® ¥ ? What happens as t ® - ¥ ?

Another way to get several solutions at once is to do a sweep on initial conditions. First, with the IC tab chosen in the solver region, Clear All Runs, make the t Initial Condition 0, set Interval to 100, and set # Points to 400. (This will enable us to avoid using Extend, other than to change direction. Interval gives us the length of the t-interval to use, starting at the t initial condition, and # Points is the number of steps to use in the solver.)

Next switch to the Sweep tab. Since you want to vary both the x and y values, you need to do a Dual sweep. Choose Dual (Matrix) to create a rectangular array of points to use as initial conditions. Make Sweep 1 a sweep on x and Sweep 2 a sweep on y. For each, use Start 0.2, Stop 0.8, and # Points 3. (This will create 9=3*3 data sets.) Make sure the forward (right) solution arrow has been chosen. Press the Sweep button (three calculators), and wait for the calculations to finish. In order to go backward in time, switch to the left pointing arrow, and Extend the sweep. Unfortunately, you will get an error message during the sixth run, and the solver will crash, but you should still get a good idea as to what is happening on the graph.

For the next portrait, choose Dual (Linear) for the sweep, but keep everything else the same. In this case, the Start and Stop values for x and y determine a line, and # Points picks evenly spaced points on the line to use as initial conditions for x and y. Go to the IC tab and Clear All Runs. Change back to the forward (right) solution arrow, and press the Sweep button. This time you will only get three runs of data. Extend them in the backwards direction, first setting Interval to 50 and # Points to 200. Although you still get the error message in the third run, the graph will look complete (as the missing data corresponds to points off the screen).

This final portrait should be reasonably easy to analyze. In the x-y graph window, choose the Select icon (second from top, graph with a pointer). Examine the three solution curves. You should notice that two of them spiral out from a single point to a limit cycle, while the third spirals in to that same cycle.

Animation

This is a situation where animation might be nice. Choose one of the solution curves, and press the Animate Dialog button (pendulum with a list), or equivalently the F9 key. Move the rate slider to 100% and press OK. You will see the population trajectory for your solution sketched out. Switch to each of the other solutions in turn, and press the Animate button (pendulum) to trace the curve.

One warning - you need to keep the ratio between # Point and Interval constant anytime you Extend if you want to be able to analyze the speed of the trace. We did this by cutting both of these in half when we did the reverse solution.

Other 2D graphs

You might wish to see the graphs of the individual populations against time. To see the graph of the predator population against time, choose the t-y tab in the bottom graph window. If you also click the Selector icon for this window, you can simultaneously see the t-y graph and the x-y graph for a single chosen data run by clicking on one of the curves in either window. (The corresponding curve will be chosen in the other window.

The same is true for the prey, i.e. the t-x graph, but what you get with Auto Scales is hardly informative. You will need to change the Y-scale (actually for the variable x in this case!) to something like 0 to 1.2 in order to get a good picture.

Equilibrium

For most systems of equations, you probably will want to find any equilibrium points. This is easy to do here. In fact, you can probably pick out one equilibrium point for this system from the phase portraits you have already drawn. To get the ODE Architect Tool to do it for you, click the Equilibrium icon (fourth from top, graph with EQ) from the icons at the right of the x-y graph window. Click anywhere in the graph window near where you think an equilibrium point is located. The tool will search for the nearest equilibrium, and put a symbol on the graph indicating its location and classification. To see a report of what has been found, click on the Equil tab in the lower graph window.

There are other equilibrium points for this system. First, change the scales on both axes to -0.5 to 1.5 on both axes. Then perform equilibrium searches until you have found three equilibria with x ³ 0 and y ³ 0.

You should notice that there is a color/symbol scheme differentiating the types of equilibrium. One of your equilibrium points should be marked by a (probably cyan) open circle, and the other two should be marked by (probably purple) squares. To understand these symbols, right-click in the x-y graph window and choose Edit. Click the Equilibrium tab. You will see a list of types of equilibria. If you click on one of the types, the Style and Color for the marker will be shown. These are user adjustable, if you wish to change them.

For more information on the equilibria, choose one of them in the Equil window. Then right-click, and pick Eigenvalues. A pop-up window will tell you the type of equilibrium point you have chosen, and give you the Jacobian/eigenvalue/eigenvector information the program used in determining the type. You should probably examine the Eigenvalues window for each of the three equilibrium points you found.

Varying a parameter - a bifurcation

An interesting thing happens in this model when the parameter k is varied. You might want to see what happens to the long-term behavior of the system as k is increased. To do this experiment, you should first clear everything from the graph. Start by changing the scale to run from 0 to 1.5 for both the X-axis and Y-axis. Clear the equilibrium points you have found by right-clicking in the Equil window, and choosing Clear All. Then choose the IC tab in the solver region, and Clear All Runs. Set 1 as the initial conditions for both x and y (keeping t as 0), set Interval back to 100, # Points back to 400, and choose the forward arrow. You are now ready to see what happens when k is changed.

Set k = .25 in the equations region, and press Enter to tell ODE Architect the new value for k. The direction field in the graph window will change to reflect this new parameter. Press Solve to draw a solution curve. You should get a solution curve that spirals in to a limit cycle. Find and classify the equilibrium point that is inside this limit cycle.

Repeat the preceding paragraph, but this time with k = .45. You should notice that the new solution behaves quite differently than the first one, especially in the long-term.

You should notice that the symbol for the second equilibrium point is different from that for the first. You can verify that this point is an attracting spiral in one of two ways. Either look up the symbol in the Equilibrium tab of the Edit pop-up menu (right-click in the graph), or click in the row for this point in the Equil window table, right-click, and then choose Eigenvalues from the resulting pop-up menu. You should also probably display the parameters in the Equil window (right-click to pop-up the menu, choose Display and then Parameters).

This change in solutions from having a limit cycle to having an attracting spiral equilibrium is called a bifurcation. It happens for a k-value somewhere between 0.25 and 0.45. You can do a sweep on k to try and estimate where this occurs.

Click on the IC tab in the solver region, and Clear All Runs. Keep the values in this region the same as before (0 for t, 1 for x, and 1 for y). Click on the Sweep tab, and choose a Single sweep. Make the Sweep 1 variable k, and set the Start as .25, the Stop as .45, and # Points as 3. You may need to Extend the runs at least once to see what might be happening at the middle sweep value of .35.

If it is still too difficult to see what is happening for k = .35, you will probably have to use the ODE Architect Tool to find the equilibrium point corresponding to this value. In the equations region, set k = .35 and click Enter. Find the equilibrium point. You should find that it is an attracting spiral. This indicates that the bifurcation occurs at a k value between .25 and .35. Repeat the preceding paragraph, but this time use .25 as Start and .35 as Stop. The midpoint of these two, with k value .3, yields another limit cycle.

To narrow our search a bit faster, Clear All Runs in the IC tab. In the Sweep tab, set the Start to .3, the Stop to .35, and # Points to 6. Press the Sweep button, and Extend the sweep one time.

The resulting graph is a bit of a mess. To be able to interpret the results, click the Data tab in the bottom graph window and choose the Select icon in the x-y graph window. You can then click on each data tab, one at a time, and see the corresponding solution curve. Data tabs 1, 2, and 3 seem to correspond to limit cycles (although the successive limit cycle seems to shrink). Data tab 6 corresponds to an attracting spiral. Data tabs 4 and 5 are similar to 1, 2, and 3, but have "cycles" small enough to warrant a closer look. Click the Extend button a few times, and continue to observe the curves corresponding to data tabs 4 and 5. You might guess that 4 corresponds to a limit cycle and 5 to an attracting spiral. Since 4 has k-value .33 and 5 has k-value .34, that would mean the bifurcation occurs between k = .33 and k = .34. You probably should find and classify the equilibria corresponding to these k-values.

To do this, enter k = .33 in the equation section, press Enter, choose the EQ icon in the graph window, and click to find the equilibrium point. Check its classification, and repeat this for k = .34. Do the results bear out the guess from the preceding paragraph?

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