# 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 $\dot{x}=\mathrm{sin}x$ is time-invariant, while $\dot{x}=\frac{\mathrm{sin}x}{1+t}$ 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 |
---|---|

Canonical name | TimeInvariant |

Date of creation | 2013-03-22 15:02:14 |

Last modified on | 2013-03-22 15:02:14 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 5 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 00A05 |

Related topic | AutonomousSystem |

Defines | time-invariant |

Defines | shift-invariant |