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Entrainment of circadian rhythm synchronizes a circadian clock to a repeating cue, a zeitgeber. One-to-one entrainment of circadian rhythm gives one cycle per environmental cycle with a stable relative circadian phase.^[ref] The stable relation need not put an activity onset exactly at lights-off: a constant offset also expresses phase locking.

For an original local model, let $x_n$ be the activity-onset offset from lights-off on environmental cycle $n$, with a negative value meaning an earlier onset. Choose an intrinsic period $\tau =23.5\mathrm{h}$ and environmental period $T=24\mathrm{h}$. Let $D(x)$ be a stipulated daily delay, measured in hours; positive $D$ moves the next onset later. The recurrence relation isA fixed phase relationship $x_{\ast }$ requires $D(x_{\ast })=T-\tau$: here a half-hour delay balances the half-hour intrinsic advance each cycle. This equality alone describes balance; restoration after a small disturbance also requires fixed point stability for an iteration.

Choose the illustrative local law $D(x)=0.5\mathrm{h}-0.4x$. Thenfor the chosen starting offset $x_0=-3\mathrm{h}$. The error shrinks by a factor of $0.6$ each cycle, proving fixed point stability for an iteration in this model. After five cycles $x_5=-0.23328\mathrm{h}$, about fourteen minutes early. At the fixed point, the imposed delay is $0.5\mathrm{h}$ and the corrected onset interval is $\tau +D(x_{\ast })=24\mathrm{h}$, although the uncoupled period remains $23.5\mathrm{h}$.

The uncorrected comparison has $D=0$ and $x_n=-3\mathrm{h}+n(\tau -T)$, so after five cycles it is $5.5$ hours early. The two traces separate ongoing timekeeping from adjustment to a zeitgeber. The chosen linear delay law is only a local teaching construction: it is not a measured response to light, does not account for wrapping of circadian phase, and does not predict an organism's entrainment range. Nor does it include masking of circadian rhythm. In a real study, tracking several cycles after removal of the cue or after a timed perturbation supplies information that a single activity value under illumination cannot provide.

A zeitgeber is an external timing cue capable of entraining a circadian rhythm.