Animations for

"Non-autonomous linear differential equations"

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

 Animation 1: The function r(x) = 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.)

 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.)

 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.)

 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.)