MaplePrimes - answers and comments on Question, A question About the odeplot?

graph

jakubi — Sat, 20 Jun 2009 04:02:11 Z

E.g.:

sys:={diff(x(t),t)=x(t)^2+y(t)+z(t)+w(t), 
diff(y(t),t)=x(t)+y(t)^2+z(t)+w(t),
diff(z(t),t)=x(t)+y(t)+z(t)^2+w(t), 
diff(w(t),t)=x(t)+y(t)+z(t)+w(t)^2}:
IC:={x(0)=0,y(0)=0,z(0)=1,w(0)=0}:
sol:=dsolve(sys union IC,numeric):
plots:-odeplot(sol,[x(t),y(t)],t=0..0.5);

HOW ABOUT THE PHASEPLAN?

SPRING — Sat, 20 Jun 2009 18:13:32 Z

Hi！ everyone！ Day day study Day day up！

Thank you very much! I already know this. But can you tell me how to get the phaseplan of (x(t),y(t))?I do this withscene(x(t),y(t)),but the graph i get do not has arrows that i really want! I will greatly appreciate your help!

No method to get the phaseplane of (x(t),y(t))?

SPRING — Sun, 21 Jun 2009 10:32:04 Z

Hi！ everyone！ Day day study Day day up！

Thank you! Here what i mostly want is the phaseplan of (x(t),y(t)) of this ODE system. Is ther any method? Thank you again!

DEtools[DEplot]

PatrickT — Thu, 25 Jun 2009 14:54:15 Z

Hi Spring,

You may find useful hints under DEtools[DEplot]

The difficulty with your problem is that you have 4D system but you would like a 2D phase plot. There exist built-in routines to create 2D phase plots for 2D systems, namely dfieldplot and phaseportrait.

A fun example of 2D phase plot is given in the Help menu:

vdP := [diff(x(t),t)=10*(y(t)-x(t)^3/3+x(t)),diff(y(t),t)=-1/10*x(t)];
DEtools[DEplot](vdP,[x(t),y(t)],t=0..20,x=-3..3,y=-2..2,[[x(0)=0.2,y(0)=0.2]],
  title=`van der Pol oscillator`,dirfield=400,arrows=comet,
  size=magnitude,numpoints=505,linecolor=blue,animatecurves=true);

There is something for 3D systems and 3D phase plots, namely fieldplot3d, but I have found it difficult to make nice plots with it. You may have better luck.

At any rate you have a 4D system so none of these methods is directly applicable.

One second-best approach is to look for a 2D approximation of your system near some point of interest.

Another second-best is to use a color scheme to show the direction of motion. It doesn't have arrows though.

DEtools[phaseportrait]([D(x)(t)=y(t)-z(t),D(y)(t)=z(t)-x(t),D(z)(t)=x(t)-y(t)*2], 
[x(t),y(t),z(t)],t=-2..2,[[x(0)=1,y(0)=0,z(0)=2]],stepsize=.05, 
scene=[z(t),x(t)],linecolour=sin(t*Pi/2),method=classical[foreuler]);

As jakubi has suggested you may have to hard-code your own arrows based on the direction field in the plane of interest. If you manage to do something, I'd love to hear about it, so please do post your solution here once you have it.

Quite often, the more code you post, the more useful feedback you will get.

arrows

jakubi — Sun, 21 Jun 2009 01:31:32 Z

Phaseplan is one thing. Arrows is another price...

Nothing very suitable seems to be available for this case. But in principle, you could get some arrows tangent to a solution curve using e.g. VectorCalculus:-PlotPositionVector.

What phaseplane?

Robert Israel — Thu, 25 Jun 2009 22:14:38 Z

What do you mean by the phaseplane of two variables in a 4-variable system? The "arrows" would have to depend on the other two variables.