3.7 The Trace-Determinant Plane

a movie of a one-parameter family

October 30, 1998

Presented below are two animations of the phase portrait of the planar system with coefficient matrix as varies from -2 to 2.  Click on the picture to get the animation controls (play, stop, forward, backward, etc.).  Take care not to reevaluate anything -- the commands can take 12 - 25 minutes (or more) to complete (and scads of memory - ~20M).  However, first, lets present a picture in the Trace-Determinant Plane of this system --  the red curve is the path followed as goes from -2 to 2.

>	plot([[5,6+a^2,a=-2..2],[T,T^2/4,T=-6..6],     [T,0,T=-6..6],[0,D,D=0..10]],thickness=2,     labels=["T","D"]);

>	with(plots):

>	with(DEtools):

>

for n from -20 to 20 do     a:=n/10;     myplot[n] := phaseportrait(          [D(x)(t)=2*x(t)+a*y(t),D(y)(t)=-a*x(t)+3*y(t)],          [x(t),y(t)],          t=-10..0.1,          [[x(0)=a,y(0)=0.5+0.5*sqrt(abs(1-4*a^2))],           [x(0)=-a,y(0)=-0.5-0.5*sqrt(abs(1-4*a^2))],           [x(0)=a,y(0)=0.5-0.5*sqrt(abs(1-4*a^2))],           [x(0)=-a,y(0)=-0.5+0.5*sqrt(abs(1-4*a^2))],           [x(0)=a,y(0)=0.5],           [x(0)=-a,y(0)=-0.5],           [x(0)=0,y(0)=1],           [x(0)=0,y(0)=-1]],          x=-1.5..1.5,          y=-1.5..1.5,          stepsize=0.1): od: display(seq(myplot[n],n=-20..20),insequence=true);

>

for n from -20 to 20 do     a:=n/10;     myplot[n] := phaseportrait(          [D(x)(t)=2*x(t)+a*y(t),D(y)(t)=-a*x(t)+3*y(t)],          [x(t),y(t)],          t=-10..0.1,          [[x(0)=0,y(0)=1],           [x(0)=-1,y(0)=1],           [x(0)=-1,y(0)=0],           [x(0)=-1,y(0)=-1],           [x(0)=0,y(0)=-1],           [x(0)=1,y(0)=-1],           [x(0)=1,y(0)=0],           [x(0)=1,y(0)=1]],          x=-1.5..1.5,          y=-1.5..1.5,          stepsize=0.1): od: display(seq(myplot[n],n=-20..20),insequence=true);

>