= Entrainment of circadian rhythm
<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>.<Overview of Circadian Rhythms by Martha Hotz Vitaterna and others>[pp. 86–87, synchronization to external cues; p. 92, Entrainment and Zeitgeber]{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> is
$$
x_{n+1}=x_n+(\tau-T)+D(x_n).
$$
A fixed phase relationship $x_*$ requires $D(x_*)=T-\tau$: 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 $D(x)=0.5\,\mathrm h-0.4x$. Then
$$
x_{n+1}=0.6x_n,\qquad x_* =0,\qquad x_n=-3(0.6)^n\,\mathrm h
$$
for 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 $\tau+D(x_*)=24\,\mathrm h$, although the uncoupled <period> remains $23.5\,\mathrm h$.
\Image[/circadian-entrainment-phase-correction.png]
{title=Daily phase correction restores a stable relation in an idealized <entrainment of circadian rhythm> model}
{id=image-circadian-entrainment-phase-correction}
{description=Original conceptual <circadian phase> traces from the stated <recurrence relation>. With the stipulated delay law, the offset approaches zero and the delay approaches half an hour. With no correction, the offset drifts earlier. The coefficient $0.4$ and starting offset are selected for explanation, not measured biological parameters. Lines join discrete cycle values.}
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.
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