Phase Portrait for a 3 Dim Linear System

restart: with(plots):

with(DEtools):

Define the differential equation using parameters:

LinDE := [diff(x(t),t) = a11*x(t)+a12*y(t)+a13*z(t), diff(y(t),t) = a21*x(t)+a22*y(t)+a23*z(t), diff(z(t),t) = a31*x(t)+a32*y(t)+a33*z(t)];

Input the values of the coefficients.

a11:= 0; a12:= 3; a13:= 1; a21:= 4; a22:= 1; a23:=-1; a31:= 2; a32:= 7; a33:=-5;

Input the inital conditions.

IC1:=[[x(0)=0.05,y(0)=0.05,z(0)=0.05], [x(0)=-0.05,y(0)=-0.05,z(0)=-0.05], [x(0)=1.0,y(0)=-1.0,z(0)=-5.0], [x(0)=-1,y(0)=1,z(0)=5], [x(0)=1.05,y(0)=-0.95,z(0)=-4.95], [x(0)=0.95,y(0)=-1.05,z(0)=-5.05], [x(0)=-1.01,y(0)=0.99,z(0)=4.99], [x(0)=-0.99,y(0)=1.01,z(0)=5.01] ];

Plot the phaseportrait using Runge-Kutta method; the initial conditions are input on the line above and can be changed.

DEplot3d(LinDE, [x(t),y(t),z(t)], t=0..10, IC1,linecolour=BLUE, x=-5..5, y=-5..5, z=-5..5, stepsize=0.01, arrows=NONE, method=classical[rk4]);

A plot with more initial conditions.

IC2:=[ [x(0)=0.05,y(0)=0.05,z(0)=0.05], [x(0)=-0.05,y(0)=-0.05,z(0)=-0.05], [x(0)=1.05,y(0)=-0.95,z(0)=-4.95], [x(0)=0.95,y(0)=-1.05,z(0)=-5.0], [x(0)=-1.05,y(0)=0.95,z(0)=4.95], [x(0)=-0.95,y(0)=1.05,z(0)=5.0], [x(0)=2.05,y(0)=-1.95,z(0)=0.05], [x(0)=1.95,y(0)=-2.05,z(0)=-0.05], [x(0)=-2.05,y(0)=1.95,z(0)=-0.05], [x(0)=-1.95,y(0)=2.05,z(0)=0.05]];

DEplot3d(LinDE, [x(t),y(t),z(t)], t=0..10, IC2,linecolour=BLUE, x=-5..5, y=-5..5, z=-5..5, stepsize=0.01, arrows=NONE, method=classical[rk4]);

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