00A05-TimeInvariant.tex

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A dynamical system is {\bf 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 $\dot{x}=f(x,t)$ is time-invariant if and only if $f(x,t_1)\equiv f(x,t_2)$ for all valid states $x$ and times $t_1$ and $t_2$.  Thus $\dot{x}=\sin x$ is time-invariant, while $\dot{x}=\frac{\sin x}{1+t}$ is time-varying.

Likewise, the discrete-time system $x[n]=f[x,n]$ is time-invariant (also called shift-invariant) if and only if $f[x,n_1]\equiv f[x,n_2]$ for all valid states $x$ and time indices $n_1$ and $n_2$.  Thus $x[n]=2 x[n-1]$ is time-invariant, while $x[n]=2 n x[n-1]$ is time-varying.
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