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A dynamical system is time-invariant if its generating formula is dependent on state only, and independent of time. A synonym for time-invariant is autonomous. The complement of time-invariant is time-varying (or nonautonomous).
For example, the continuous-time system
 is time-invariant if and only if
 for all valid states  and times  and  . Thus
 is time-invariant, while
 is time-varying.
Likewise, the discrete-time system
![$ x[n]=f[x,n]$](http://images.planetmath.org:8080/cache/objects/6748/l2h/img8.png "$ x[n]=f[x,n]$") is time-invariant (also called shift-invariant) if and only if
![$ f[x,n_1]\equiv f[x,n_2]$](http://images.planetmath.org:8080/cache/objects/6748/l2h/img9.png "$ f[x,n_1]\equiv f[x,n_2]$") for all valid states  and time indices  and  . Thus
![$ x[n]=2 x[n-1]$](http://images.planetmath.org:8080/cache/objects/6748/l2h/img13.png "$ x[n]=2 x[n-1]$") is time-invariant, while
![$ x[n]=2 n x[n-1]$](http://images.planetmath.org:8080/cache/objects/6748/l2h/img14.png "$ x[n]=2 n x[n-1]$") is time-varying.
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