My Definition
A system is time invariant if the system and a time delay are commutative.
system + time delay = time delay + system.
Time Invariant Example
$ y(t)=5x(t) $
then
$ x(t)=t+4 $ yields $ y(t)=5(t+4)=5t+20 $ through a time delay of $ t_0 $ yields $ Z(t)=y(t-t_0)=5(t-t_0)+20 $
now switching the order,
$ x(t)=t+4 $ through a time delay of $ t_0 $ yields $ x(t-t_0)=(t-t_0)+4 $ now this input signal through the system yields $ y(t)=5((t-t_0)+4)=5(t-t_0)+20 $
the results of the two ways are the same. i.e. Time invariant.
non-time invariant
$ y(t)=t*cos(x(t)) $
with input $ x(t)=t $
1) System then delay
$ x(t)=t $ yields $ y(t)=tcos(t) $ through a time delay y(t) becomes $ y(t)=(t-t_0)*cos(t-t_0) $
2) delay then system
$ x(t)=t $ through a time delay becomes $ x(t)=t-t_0 $ now through the system, $ y(t)=tcos(t-t_0) $
$ (t-t_0)cos(t-t_0)\not=tcos(t-t_0) $, so not time invariant.
