Definition
If the cascade $ x(t) \to timedelay \to sys \to z(t) $ yields the same output as the cascade $ x(t) \to sys \to timedelay \to z(t) $ for any $ t_{0} $, then the system is called "time invariant".
Example of Time-Invariant System
System: $ x(t) \to y(t) = 3x(t) $
$ x(t) \to timedelay \to sys \to z(t)=3x(t-t_{0}) $
$ x(t) \to sys \to timedelay \to z(t)=3x(t-t_{0}) $
Since $ 3x(t-t_{0}) $ is equal to $ 3x(t-t_{0}) $, the system is time-invariant.
Example of Non-Time-Invariant System
System: $ x(t) \to y(t) = x(2t) $
$ x(t) \to timedelay \to sys \to z(t)=x(2(t-t_{0})) $
$ x(t) \to sys \to timedelay \to z(t)=x(2t-t_{0}) $
Since $ x(2(t-t_{0})) $ does not equal $ x(2t-t_{0}) $, the system is not time-invariant.
