1 Programs for phase portrait plotting

Transcription

1 . 1 Programs for phase portrait plotting We are here looking at how to use our octave programs to make phase portraits of two dimensional systems of ODE, adding automatically or halfautomatically arrows on trajectories. We have the possibilities to give the initial values of a solution curve, by giving the coordinates of the initial point, or by clicking on the window for the plot where the curve should start. We also have programs for plotting only the arrows for the direction of the solution, without any curve, at points given by clicking with mouse. We start by description of the programs for linear systems. After that we give information how to do minimal modifications to the programs in order to get phase portraits for any system. We also have a collection of programs giving phase portraits for some different known systems. At the end we examine the programs in more detail and give some examples and exercises recommended to go through for those not very familiar with octave. 1.1 Presentation of programs The programs can be grouped into three groups, the main four programs with examples using them, the program for using the mouse for giving initial points of solution curves, and the programs for plotting only arrows for the direction. We start with presenting the four main programs, one for giving system parameters and initial conditions and values to other important program parameters used, one where we define the equations for the system, one for integration and plotting the solution curve and the last for plotting an arrow on the solution curve. These are used in example programs making whole phase portraits. The most important inputs are the parameters of the system, the initial condition, the time length of integration. As an example we use programs for phase portraits for a linear system x = ax + by y = cx + dy (1) We start with a shorter description and list of important program variables and then describe them in more details. In next section we tell you how to modify these programs when other systems are used. The four main programs for the linear system are: LinFunInit.m, LinFun.m, onecurve.m and arrow.m with the examples linex1.m and linex2.m The program for making solution curves by clicking with mouse at initial condition is: mousecurves.m 1

2 The programs for making only arrows for directions are: arrowstart.m and arrowmake.m LinFunInit.m makes default initial values, in function: for function parameters a, b, c and d for funkis, the name of the file where the function is defined (here LinFun ) in integration: for p, deciding whether we integrate forwards (p=1) or backwards (p=-1) for x0, the initial point of the integal curve for tfinal, the final time of integration for points, the number of point on the integral curve in plot: for xmin, xmax, the mininal and maximal x-value of the plot for ymin, ymax, the mininal and maximal y-value of the plot in making the arrows: for jpauto, telling whether the arrow should be placed automatically in the middle of the curve (jpauto=1) or if you wish yourself to give the value of jp the index of a point on the curve where the arrow should be (jpauto=0) for na, the ratio of the length of the arrow compared with plotting window, the smaller the bigger length. for vv, the wideness of arrow for jab, parameter for choosing two types of finding automatically the place where to put arrow LinFun.m gives the right hand side of the system. onecurve.m the program for making the integral curve and plotting it arrow.m the program for plotting the arrow on the curve needs variables na, vv, jpauto and jab. linex1.m is one example of full phase portrait for the case when a = 1, b = c = 0 and d = 1. mousecurves.m is a program for plotting solution curves by clicking at initial points. arrowstart.m and arrowmake.m are the programs for plotting arrows for directions and they need the same parameters as arrow.m for working, except for that also parameter le is needed for making the proportion between wing length and central line length for the arrow. The parameter arrownumber is used to give the number of arrows to be plotted for one call of arrowmake.m As you see from the example the most important is to give good initial conditions x0 (it is possible to use loops to create more curves) and determine 2

3 the direction of integration p. Other parameters which are important are tfinal determining how long the curve will be and the value of the integration parameter points which should be big enough to get a smooth curve. Some explanations how programs work The right hand sides of the system of differential equations are given in the function file LinFun.m by the vector xdot. The parameters a b c d are global which means that they have the same value in other programs also and can be given new value outside this program. The value of p is determining how fast forward or backward the integration is going to be done. Positive values (normally +1) means we go forward along the integral curve and negative values (normally -1) means that we are going backwards. The function is called by LinFun(x,t) where x is a two dimensional vector and t is the one dimensional time. Here the systems are autonomous and function is not depending on time, it is a dummy variable. In LinFunInit.m important default values are given for integration and plotting. The values of the system parameters a, b, c, d are needed in the function file for the right hand side of the system. The value of p is determining whether we integrate backwards or forwards. The integral curve starts at x0 for time equal zero and goes until time is the value given by tfinal. The numbers of points given on the curve is determined by the value of the variable points. The output of the integration in onecurve.m is a matrix x of size points 2. The vector x(i,1) gives the x-values on the curve at time t(i) which is tfinal*(i-1)/(points-1) as the time points are equally distributed in the interval [0,tfinal] and the number of them is given by the value of variable points. Analogously x(i,2) gives the y-values on the curve. The time interval is divided into points-1 intervals and the integral curve is plotted as straight lines between points on the real integral curve given at the endpoints of these time intervals. The value of parameter points must be big enough to see it is a smooth curve even if it consists of straight line edges. The whole phase portrait is plotted into a window where the x-coordinate is between xmin and xmax and the y-coordinate is between ymin and ymax. The values of na, vv and jpauto are used in the program arrow.m plotting the arrows on the curves to see the time direction. The value of na determines the length of the arrow and is smaller with bigger values of na. The vaule of vv gives the angle of the arrow, small value is giving a narrow arrow. The value of jpauto is determining whether arrow should automatically be put approximately in the middle of the curve. If it is something else than 1 you must manually give jp meaning that the arrow is at point x(jp,:) corresponding to time t(jp). In onecurve.m the integration is carried out and linspace gives the time points (equally distributed between 0 and tfinal) at which the solution is calculated as output matrix x. The solution is also plotted. The program arrow.m produces the arrow plot on the curve for the time 3

4 direction. Some midpoint of a curve part is chosen if jpauto=1. The variable jab can have the values 1 or 2 and usually 1 can be used but sometimes 2 is good in order to get the arrow in a better place. Be careful to give the initial conditions where the solution curve starts inside the window for plotting, otherwise the arrow-program could collapse. This program supposes that the window octave plots on the screen is quadratic, if it is far from quadratic the arrows may look a little bit strange and you should ask us for other versions. The program mousecurves.m first asks for whether we wish to add the curves to a previous plot or not. If we wish to start a complete new picture we should give 0 otherwise something else. After that it asks for the number of curves to be plotted (these can be added later by using the program once again saving the previous figure). And finally before plotting the program asks for whether we wish to change between integration forward and backward compared to the latest use and if we wish to change the length of the curves in time. After that it waits for clicking to get an input for initial condition and a curve with arrow should appear starting at the place clicked. The programs arrowstart.m and arrowmake.m you can use for easy plot of arrows for finding the directions for solutions of a system of ordinary differential equations. You just click on a point in the plot and an arrow showing you the direction of the solution exactly at that point should appear. To use the programs of course, you need a file for giving the right hand sides of the system (for example, LinFun.m for a linear system) and a file for giving values to important parameters (for example, LinFunInit.m for the linear system given by file mentioned above). Calling for arrowstart gives you the window where to plot and askes for parameter arrownumber which defines the number of arrows you will plot the first time. Then you call for arrowmake and just click the number of times you have ordered and you should see the arrows. If you need more later you can call again for arrowmake giving may be first a new value for arrownumber. The type of the arrows are defined by some parameters which are set in the programs for initial parameters (for example LinFunInit.m ) and in arrowstart.m The parameters are na: giving the length of the arrow as length of the window divided by na. Greater na will give smaller arrows. We give default as 30. le: gives the ratio between the length of the central line and the wings in the arrow. vv: gives the angle between a wing of the arrow and the central line. arrownumber: is the number of arrows you can plot calling once for arrowmake.m and this can be changed before next call for this program in order to plot more The program linex1.m gives a full phase portrait for the system for the parameters given above. One more linear example with other coefficients is given in linex2.m and examples for non-linear functions are given in answers to exercises below. 4

5 1.2 How to use for own systems You can define your own function in a file you name yourfun.m (or give some other convenient name) and change the file LinFunInit to yourfuninit.m and change the text to actual parameters for your system in yourfuninit.m and write funkis= yourfun instead of LinFun. You need not to be worried about checking all parameters you can just copy them from other files giving initial values. But the parameters for the function and the value of funkis you have to change to your own. Also, of course, the initial points will be different. Often you also need to change the integration between backwards and forwards by variable p and the length of the curve by variable tfinal. And finally you probably have other values for the boundaries of the plot xmin, xmax, ymin and ymax. This will give you possibility to modify the examples to phase portraits for your system. 1.3 More detailed explanations with exercises The function file LinFun.m calculates the values for the funtion given by F (x, y, t) = (ax + by, cx + dy) and it is a function from R 3 to R 2. The values are given by typing command LinFun([x,y],t) in the case p = 1. Example 1. If a = c = d = p = 1, b = 2 the command LinFun([1,2],1) will give the values 5 and 3 coming from F (1, 2, 1) = (a + 2b, c + 2d). Execise 1. If a = 1, b = 0, c = 1, d = 1, p = 1 which is then the expression for the function? What will the command LinFun([2,1],4) give? What will the command LinFun([2,1],5) give? What will the command LinFun([-2,1],5) give? Which command should you type if you wish to find the value at the point (0,2)? In the function file x(1) means the first component of the vector x, the argument of the function. In LinFun.m this corresponds to the x-coordinate. The second component (corresponding to the y-coordinate in LinFun.m) is given by x(2). Exercise 2. Write a function file for the function given by F (x, y) = ( x + 2x 2 + y 2, y + y 2 ) and check the value octave is giving with the expected for some arguments. Exercise 3. Write a function file for the function given by F (x, y) = (x(y 1), y(x 1)) and check the value octave is giving with the expected ones for some arguments. Exercise 4. Write a file for initializing parameter values for the parameters and a function file for the function given by F (x, y) = (x(k 1 ax by), y(k 2 cx dy)) and check the value octave is giving with the expected for some arguments. Examine the case when F (x, y) = (x(4 x 2y), y(6 2x y)) 5

6 The text in onecurve.m is t=linspace(0,tfinal,points); x=lsode("linfun",x0,t); plot(x(:,1),x(:,2)) axis([xmin,xmax,ymin,ymax]); It first gives the time points 0, dt, 2dt,..., tfinal-dt, tfinal, where dt=tfinal/(points-1). The output x of integration is a matrix where each row is a point on the solution curve. The vector x(1,:) is the first point which is the initial condition x0. The vector x(points,:) is the last point on the solution curve. The point x(j,:) is the point on the solution curve for time t(j). The axis command gives coordinates for corners of the window where the solutions are plotted. Example 2. If we have a = 1, b = c = 0, d = 2 integrating the system in LinFun.m and x0=[1,0] and tfinal=10 and points=11, then x(5,:)=[e 4, 0] and if x0=[2,3] then x(7,:)=[2e 6, 3e 12 ]. Exercise 5. If we have a = 1, b = c = 0, d = 3 and we integrate the system in LinFun.m and tfinal=20 and points=41, what is then the value of t(20) and x(20,:) if x0=[1,-1]? Exercise 6. If we have a = 1, b = 1, c = 1, d = 1 and we integrate the system in LinFun.m and tfinal=10 and points=11, what is then the value of t(9) and x(9,:) if x0=[1,0]? In the following exercises it is useful to do some analysis of the system, calculating zero-isoclines, equilibria and carrying out sign analysis for the right hand sides of the equations, before starting to plot solution curves. Exercise 7. Write a program making a phase portrait for the system in exercise 2. Exercise 8. Write a program making a phase portrait for the system in exercise 3. Exercise 9. Write a program making a phase portrait for the system in exercise 4. Except system given by F (x, y) = (x(4 x 2y), y(6 2x y)) examine also F (x, y) = (x(1 x y), y(6 2x y)). Exercise 10. Write a program making a phase portrait for the system S = βis +ω(1 I S), I = βis γi in the case when β = 6, ω = 1, γ = 4 and in the case when β = 3, ω = 1, γ = 4. All variables and parameters are assumed non-negative. Exercise 11. Write a program making a phase portrait for the Ross system x = r 1 T 1 (1 x)y r 1 x, y = r 2 T 2 (1 y)x r 2 y in the case when r 1 = 0.01, r 2 = 0.1, T 1 = 6, T 2 = 3. Variables x and y are assumed to be in the closed interval from 0 to 1. 6

7 Exercise 12. Write a program making a phase portrait for the system x = ax ( ) 1 x K bxy 1+Ax y = cy + dxy 1+Ax in the cases a = 1, b = 1, c = 1, d = 1, K = 10, A = 0.2 and a = 1, b = 1, c = 1, d = 1, K = 4, A = 0.2 and a = 1, b = 1, c = 1, d = 1, K = 1.1, A = 0.2 and a = 1, b = 1, c = 3, d = 1, K = 1.5, A = 0.5. Variables are assumed non-negative. (2) 7

8 Answers Exercise 1. The function is given by F (x, y) = ( x, x y) The value of LinFun([2,1],4) and LinFun([2,1],5) will both be (-2,1). The function is not time depent as t does not occur in the function expressions. The command LinFun([-2,1],5) will give (2,-3). The command to get the value for (0,2) could be LinFun([0,2],3.5) or LinFun([0,2],-100) or LinFun([0,2],t0) where t0 is any number. Exercise 2. If the file is called BBFun.m a possible text could be function xdot=bbfun(x,t) global p xdot(1)=p*(-x(1)+2*(x(1)^2)+(x(2))^2); xdot(2)=p*(-x(2)+x(2)^2); endfunction Exercise 3. If the function file is called APfun.m a possible text could be function xdot=apfun(x,t) global p xdot(1) = p*(-(1 - x(2))*x(1)); xdot(2) = p*(-(1 - x(1))*x(2)); endfunction Exercise 4. A possible file for initial conditions could be called ecolinit.m and contain the text below. (Here we also give initial values for parametes used for integration and plotting arrows, only the first row is necessary for giving the actual parameters). global a b c d k1 k2 p a=1;b=2;k1=4;k2=6;c=2;d=1;p=1; points=1000;pmid=500; tfinal=200; xmin=0;xmax=1.5;ymin=0;ymax=1; na=30;vv=0.5;xratio=2;jauto=1; jpauto=1; 8

9 The function file could be called ecol.m and contain the text. function xdot=ecol(x,t) global p a b c d k1 k2 xdot(1)=p*x(1)*(k1-a*x(1)-b*x(2)); xdot(2)=p*x(2)*(k2-c*x(1)-d*x(2)); endfunction Exercise 5. The point is x(20,:)=[e 9.5, e 28.5 ] for time t(20)=9.5. Exercise 6. The point is x(9,:)=[e 8 cos 8, e 8 sin 8] for time t(9)=8. The solution has the form (x(t), y(t)) = (e t cos t, e t sin t). Exercise 7. Example programs are in folder BBpp Exercise 8. Example programs are in folder APpp Exercise 9. Example programs are in folder predprey Exercise 10. Example programs are in folder sir Exercise 11. Example programs are in folder ross Exercise 12. Example programs are in folder oros 9