Vector Fields and Solutions to Ordinary Differential Equations using Andreas Stahel 6th December 29 Contents Vector fields. Vector field for the logistic equation...............................2 Solutions of ordinary differential equations with lsode.................. 2.3 Vector field for the equation of a damped pendulum...................... 3.4 Solution to the pendulum equation............................... 5 2 The ODE Package on -Forge 6 2. Examples for ode23, ode45 and ode78...................... 7 3 Codes from Lecture Notes 8 3. Using RK45, Runge-Kutta adaptiv............................. 9 3.2 Using ode Runge, Runge-Kutta with fixed step...................... 9 3.3 Using lsode........................................ 9 In these notes you find the methods to generate vector fields with the help of. In addition one of the methods to solve ordinary differential equations is illustrated. The notes assume that you have a recent version of installed. Vector fields A vector field in the plane R 2 is determined by a vector function F x, x 2 F x = F 2 x, x 2 At each point x R 2 the vector F x is attached. One of the possible applications of vectors fields is the visualization of solution of ordinary differential equations.. Vector field for the logistic equation The vector field for the logistic differential equation d dt xt = xt x2 t
VECTOR FIELDS 2 is given by Ft, x = x x 2 To visualize this field we have to choose the domain to be used define the functions for the vector field choose how many vectors to draw compute the vectors at the chosen points rescale the length of the vectors generate the graphic on screen save the result in a file The above tasks can be performed with, as shown in the code below with the resulting Figure. t = :. 5 : 8 ; %% choose the time values x = :. 2 : 2 ; %% choose the x values %% define the function function dx = l o g i s t i c x, t dx = x x ˆ 2 ; %% c r e a t e the zero vector f i e l d nt = length t ; nx = length x ; V = zeros nt, nx ; V2 = zeros nt, nx ; %% compute the vector f i e l d for i = : nt for j = : nx V i, j = ; V2 i, j = l o g i s t i c x j, t i ; endfor endfor %% choose the s c a l e f a c t o r s c a l e f a c t o r =. 2 ; f i g u r e ; quiver t, x,v,v2, s c a l e f a c t o r axis [, 8,, 2 ] ; xlabel time t ; ylabel population x grid on %% p r i n t Logistic. png %% p r i n t Logistic. eps.2 Solutions of ordinary differential equations with lsode provides a selection of commands to solve ordinary differential equations, including systems. Here we only use lsode to generate solutions for the differential equations arising from the above vector fields.
VECTOR FIELDS 3 2.5 population x.5 2 3 4 5 6 7 8 time t Figure : Vector field for the logistic equation To solve the initial value problem we proceed as follows. d dt xt = xt x2 t with x =.2 Choose the values of time t for which the solution is to be computed. Compute the solution with the command lsode. We may compute more solution by using different initial conditions, e.g. x =.4 and x = 2.. Plot the solutions. The solutions and the vector field can be shown in one graphic. Find the result in Figure 2. T = :. : 8 ; X = lsode l o g i s t i c,. 2,T ; X2 = lsode l o g i s t i c,. 4,T ; X3 = lsode l o g i s t i c, 2.,T ; f i g u r e 3 ; hold off p l o t T, [ X,X2,X3 ] ; grid on ; axis [, 8,, 2 ] ; hold on quiver t, x,v,v2,. 4.3 Vector field for the equation of a damped pendulum For atonomous differential equations the first component of the vector field equals. For general problems this is not the case. As an example we convert the differential equation for a damped pemdulum ÿt = k yt α ẏt
VECTOR FIELDS 4 2.5.5 2 3 4 5 6 7 8 Figure 2: Solutions and vector field for the logistic equation into a system of diffeential equations of order one d dt yt vt = vt k yt α vt and examine the resulting vector field. With the numerical values k = and α =. we have to examine the vector field Fy, v = y. v The code to be used is very similar to the logistic equations and thus shown without any further explanation. y = :.:; %% choose the y values v = :.:; %% choose the v values %% define the function function dy = Fy, v dy = v ; function dv = F2y, v k = ; alpha =. ; dv = k y alpha v ; %% c r e a t e the zero vector f i e l d ny = length y ; nv = length v ; V = zeros ny, nv ; V2 = zeros ny, nv ; %% compute the vector f i e l d for i = : ny for j = : nv V i, j = Fy i, v j ; V2 i, j = F2y i, v j ; endfor endfor
VECTOR FIELDS 5 %% choose the s c a l e f a c t o r s c a l e f a c t o r =. 5 ; f i g u r e ; quiver y, v,v, V2, s c a l e f a c t o r grid on ; axis [ min y,max y, min v,max v ] xlabel p o s i t i o n x ; ylabel speed v The only additional feature is the explicit choice of the domain shown in the graphic by the command axis. As can be seen in the left of Figure 3 the vectors at the origin have a small length and thus it is difficult to read of the direction. This can be improved by normalizing all the vectors to have length and then rescale them to have a good length for visalisation. This is done in the code below witrh the result on the right in Figure 3. f i g u r e 2 ; Vlength = s q r t V.ˆ2+V2. ˆ 2 ; Vn = V. / Vlength ; V2n = V2. / Vlength ; quiver y, v, Vn,V2n, s c a l e f a c t o r grid on ; xlabel p o s i t i o n x ; ylabel speed v axis [ min y,max y, min v,max v ].5.5 speed v speed v -.5 -.5 - - -.5.5 position x - - -.5.5 position x Figure 3: Standard and normalized vector field for the pendulum equation.4 Solution to the pendulum equation To determine solutions of the system of differential equations d dt yy vt = vt yt. vt with y v =.9 we have to rewrite the function to be used by lsode. T = :. 5 : 2 5 ; function dy = ODEPendy, t
2 THE ODE PACKAGE ON OCTAVE-FORGE 6 k = ; alpha =. ; dy = [ y 2 ; k y alpha y 2 ] ; YV = lsode ODEPend, [ ;. 9 ],T ; f i g u r e 3 ; p l o t T,YV ; grid on xlabel time ; ylabel p o s i t i o n and velocity ; f i g u r e 4 hold off quiver y, v, Vn,V2n, s c a l e f a c t o r grid on axis [ min y,max y, min v,max v ] hold on p l o t YV :,,YV :, 2, r xlabel position ; ylabel velocity ; hold off position and velocity.5 -.5 velocity.5 -.5-5 5 2 25 time - - -.5.5 position Figure 4: One solution and the vector field for the pendulum equation 2 The ODE Package on -Forge The standard command in to solve ordinary differential equations is lsode, as used in the previous section. On Forge at http://octave.sourceforge.net/ you can find an package with additional commands to solve ODEs. Dowload the package odepkg in a local directory Launch and install the package with the command pkg install odepkg-.6.4.tar.gz Depending on your local configuration you might have to load the package with pkg load odepkg
2 THE ODE PACKAGE ON OCTAVE-FORGE 7 With this package many commands from MATLAB are now available in, and some additional commands. Table shows some of the commands. Solver lsode ode23 Description efficient, addaptive solver based on Hindmarsh s work [Hind93] addaptive, explicit,solver, based on Heun s method ode45 addaptive, explicit solver, based on Runge-Kutta method of order 4, resp. 5 ode54 Runge Kutta solver ode78 addaptive, explicit solver, based on Runge-Kutta method of order 7, resp. 8 ode2r ode5r solver for stiff problems, based on Hairer and Wanner s code solver for stiff problems, based on Hairer and Wanner s code Table : commands to solve ordinary differential equations The package provides many additional commands, some of them shown in Table 2. Find a description of the algorithms, their advantages and disadvantages in the file odepkg.pdf, to be found in the subdirectory with the package. Command odeexamples ode23d ode45d ode78d Description launch demos for ordinary differential equations solver for delay differential equations solver for delay differential equations solver for delay differential equations Table 2: Additional commands in the ODE package 2. Examples for ode23, ode45 and ode78 As an example we consider again the pendulum equation d yy vt = dt vt yt. vt with y v = Thus a function file with this function may be generated and then used by all examples below. function dy = ODEPend t, y k = ; alpha =. ; dy = [ y2; k y alpha y 2 ] ; ODEPend.m As a general rule we first choose the parameters for the solver. As a first example we choose a relative tolerance of 3 and an absolute tolerance of 3. We want a graph to be generated while the differential equation is solved. vopt = odeset RelTol, e 3, AbsTol, e 3, NormControl, on,\ OutputFcn, @odeplot ; Then we use ode78 to solve the system of equations..9
3 CODES FROM LECTURE NOTES 8 ode78 @ODEPend, [ 25], [. 9 ], vopt ; The resulting animation and final solution will look rather ragged. Since the Runge Kutta algorithm of order 7 is very efficient, only very few points have to be computed. Thus the graphic does not look nice, but the numerical results are reliable. The command ode23 @ODEPend, [ 25], [. 9 ], vopt ; will generate a nice looking graph, but require more computation time. With all of the above codes the numerical values will not be returned and are thus not available for further computation. Use the code below if further computations have to be performed. You will find a plot of position and velocity as function of time and a phase plot. vopt = odeset RelTol, e, AbsTol, e, NormControl, on ; [ t, y ] = ode78 @ODEPend, [ 25], [. 9 ], vopt ; f i g u r e ; p l o t t, y ; xlabel time ; ylabel p o s i t i o n and velocity ; grid on f i g u r e 2 ; p l o t y :,, y :, 2 ; xlabel position ; ylabel velocity ; grid on 3 Codes from Lecture Notes To illustrate the codes presented in the lecture notes we use the example of a diode circuit examined in the notes. The behavior of the diode is given by a function { i = D u = for u u s R D u + u s for u < u s Based on Kirchhoff s law we find the differential equations u h = D u h u in + D u out u h C u out = u in D u out u h C 2 We define the initial conditions and the two function in a script file. u in t = cost. Tend = 3; u = [ ; ] ; We examine an input voltage of function curr = Diode u Rd = ; us =. 7 ; i f u>= us curr =; e l s e curr=rd u+us ; endif function y = c i r c u i t u, t C = ; C2 = ; y = [ /C Diode u sin t Diode u2 u ; cos t /C2 Diode u2 u ] ;
3 CODES FROM LECTURE NOTES 9 3. Using RK45, Runge-Kutta adaptiv We can use the adaptive Runge-Kutta algorithm with relative and absolute tolerance of 5 and generate the plot by t = cputime ; [ t, u ] = rk45 c i r c u i t,, Tend, u, e 5,e 5; % Runge Kutta adaptiv timer = cputime t f i g u r e ; p l o t t, u :, 2. grid on ; xlabel time ; ylabel tension ; The result in Figure 5 seems reasonable, but it took 8 seconds of CPU time to compute. 3.2 Using ode Runge, Runge-Kutta with fixed step We can compare the result with a Runge-Kutta calculation with steps, resulting in Figure 5. tfix = linspace, Tend, ; t = cputime ; [ tfix, ufix ] = ode Runge c i r c u i t, tfix, u, ; % Runge Kutta timer = cputime t p l o t tfix, ufix :, 2, t, u :, 2 grid on ; xlabel time ; ylabel tension ; legend u fix, u adapt It took only.5 seconds, but the solution is ragged at the turning points. This can be improved by using intermediate steps between the output times. Use [ tfix, ufix ] = ode Runge c i r c u i t, tfix, u, ; % Runge Kutta to find a competitive solution in.3 seconds. This is one of the rare occasion where a fixed size algorithm outperforms an adaptive algorithm. 3.3 Using lsode We can compare the above result with the performance of lsode. Unfortunately the arguments t and u have to be given in reverse order for lsode and the above codes and thus the header of the function circuit has to be modified slightly. We have to specify the tolerances. With a computation time of.48 seconds we find a good solution. This illustrates the quality of the algorithm in lsode.m. Tend = 3; u = [ ; ] ; DoubleTensionLSODE.m function curr = Diode u Rd = ; us =. 7 ; i f u>= us curr =; e l s e curr=rd u+us ; endif function y = c i r c u i t u, t C = ; C2 = ; y = [ /C Diode u sin t Diode u2 u ; cos t /C2 Diode u2 u ] ; t = linspace, Tend, ;
REFERENCES 3 25 u fix u adapt tension 2 5 5-5 5 5 2 25 3 time Figure 5: Solution for the diode circuit with Runge-Kutta, fixed step and adaptive lsode options absolute lsode options r e l a t i v e t o l e r a n c e, e 5; t o l e r a n c e, e 5; t = cputime ; u = lsode c i r c u i t, u, t ; timer = cputime t p l o t t, u :, 2 grid on ; xlabel time ; ylabel tension ; In the previous section the codes and data files in Table 3 were used. filename ode Euler.m ode Heun.m ode Runge.m rk45.m DoubleTension.m DoubleTensionLSODE.m function algorithm of Euler, fixed step size algorithm of Heun, fixed step size algorithm of Runge-Kutta, fixed step size adaptive Runge Kutta algorithm sample code for the diode circuit using lsode Table 3: Codes and data files References [Hind93] A. C. Hindmarsh and K. Radhakrishnan. Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations. NASA, 993.