# time invariant

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 $\dot{x}=f(x,t)$ 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 $x[n]=2x[n-1]$ is time-invariant, while $x[n]=2nx[n-1]$ is time-varying.

Title time invariant TimeInvariant 2013-03-22 15:02:14 2013-03-22 15:02:14 Mathprof (13753) Mathprof (13753) 5 Mathprof (13753) Definition msc 00A05 AutonomousSystem time-invariant shift-invariant