Periodic

Definition · applies to functions, signals, and events

High school

Consider a function $f$ of time $t$, and a number $T > 0$. The function $f$ is periodic with period $T$ if

\[f(t + T) = f(t) \quad \text{for all } t.\]

Periodic events happen at regular intervals of length $T$.

Undergraduate (dynamical systems)

Consider a solution $x(t)$ of a dynamical system. The solution is periodic if, for some $T > 0$,

\[x(t + T) = x(t) \quad \text{for all } t.\]

Examples

A clock that chimes every hour Elementary
The chimes of a clock that chimes every hour are periodic, because they occur every hour.
The swinging pendulum of a grandfather clock High school

Things that may be mistaken for examples.

A point under rotation of a circle by the angle $2\pi\sqrt{2}$ Graduate
The point’s orbit is not periodic, because the angle is an irrational multiple of a full turn, so the orbit does not return exactly.

If $T$ is a period, then so is every positive integer multiple of $T$.
The trajectory of a nonconstant periodic solution is a closed curve, called a periodic orbit.