Bifurcations of sleep patterns due to homeostatic and circadian variation in a sleep-wake flip-flop model

The paper numerically constructs sleep-onset circle maps and documents: (i) a Farey/period-adding organization with a devil’s-staircase-like rotation-number plot as k decreases, explicitly noting mediants between neighboring plateaus (Devil’s staircase-like; Farey sequence) ; (ii) the primary delimiting sequence SN → BC-U → BC-S for windows with ρ in [1/2,1], illustrated for ρ=2/3 and ρ=1/2 ; (iii) deformations to SN → BC-U → BC-U → SN and to SN → SN near the continuous-map regime ; and (iv) in the circadian hard-switch (CHS) limit α→0+, winnowing of windows with ρ∈(1/2,1) and ρ∈(1/4,1/3), with the dyadic windows ρ=1/2 and ρ=1/4 persisting and reversing the delimiting order to BC-S → BC-U → SN , . The candidate solution derives the same qualitative structures from rotation theory for monotone (possibly discontinuous) circle maps and standard border-collision theory, furnishing explicit hypotheses (monotonicity on branches, single-jump “Lorenz-like” regime, one-sided slopes) that explain the observed sequences and the CHS-limit winnowing. Small differences reflect scope: the paper reports additional compound sequences and multi-gap cases, whereas the model solution confines attention to a single-gap Lorenz-like regime. Overall the two accounts agree on substance; the paper’s support is numerical/constructive, while the model leans on known one-dimensional theory.