Animations for

"Non-autonomous linear differential equations"

 

Click on thumbnail for animation and .nb link for the Mathematica code to produce the animation.


Mathematica Code: bubbles_real.nb
 

Mathematica Code: vfield_animation.nb

Animation 1:
The function r(x) = <x, Bx> applied to the unit circle as the angle between the eigenvectors of B grows.  The eigenvectors of B are plotted in blue.  In the shaded regions, r is positive. (See Section 2.1.)

Animation 2:
A solution to Markus and Yamabe's example and its corresponding vector field in blue.  Here A(t) has complex eigenvalues with negative real part, but solutions are unstable. (See Section 3.)

 


Mathematica Code: vinograd_animationX.nb
 

Mathematica Code: vinograd_animationY.nb

Animation 3:
A solution to Vinograd's equation.  The rotated eigenvectors of B-G(-6) are plotted in blue and solutions tend away from the origin in the shaded regions.  (See Section 3.1.)

Animation 4:
The previous animation in the rotated coordinate system Y.  (See Section 3.1.)

 


Mathematica Code: 4_4H.nb
 

Animation 5:
An unstable solution to x'=A(t)x where A(t) = exp(H(-6,3)) B exp(-H(-6,3)) (see Section 4.4) and B is from Vinograd's example.  The eigenvectors of A(t) are plotted in blue and solutions tend away from the origin in the shaded regions.  (See Section 4.4.)

 


Mathematica Code: 4_4Fa.nb
 

Mathematica Code: 4_4Fb.nb

Animation 6:
An unstable solution to x'=A(t)x where A(t) = exp(F(3,1)) B exp(-H(3,1)) (see Section 4.4) and B is from Vinograd's example.  The eigenvectors of A(t) are plotted in blue.  (See Section 4.4.)

Animation 7:
An unstable solution to x'=A(t)x where A(t) = exp(F(1.5,1)) B exp(-H(1.5,1)) (see Section 4.4) and B is from Vinograd's example.  (See Section 4.4.)