Stability
Example
The solutiony=0 of the system
yt+ 2 +ayt+ 1 +byt= 0
is asymptotically stable if the absolute value of the characteristic roots is
smaller than one. If the absolute value of the characteristic roots are not
bigger than one and the characteristic roots are unique then the system is
stable.
Whenλ=1 is a double root, thenyt+ 2 2 yt+ 1 +yt= 0 .In this case
the general solution isyt=( 1 )t(c 1 +tc 2 )which is not stable. The
general solution
yt=c 1 cosθt+c 2 sinθt
is stable but not asymptotically stable.
