Why do we go to sleep late and struggle to wake up on time? Historically, light-dark cycles were dictated by the solar day, but now humans can extend light exposure by switching on artificial lights. We use a mathematical model incorporating effects of light, circadian rhythmicity and sleep homeostasis to provide a quantitative theoretical framework to understand effects of modern patterns of light consumption on the human circadian system. The model shows that without artificial light humans wakeup at dawn. Artificial light delays circadian rhythmicity and preferred sleep timing and compromises synchronisation to the solar day when wake-times are not enforced. When wake-times are enforced by social constraints, such as work or school, artificial light induces a mismatch between sleep timing and circadian rhythmicity (‘social jet-lag’). The model implies that developmental changes in sleep homeostasis and circadian amplitude make adolescents particularly sensitive to effects of light consumption. The model predicts that ameliorating social jet-lag is more effectively achieved by reducing evening light consumption than by delaying social constraints, particularly in individuals with slow circadian clocks or when imposed wake-times occur after sunrise. These theory-informed predictions may aid design of interventions to prevent and treat circadian rhythm-sleep disorders and social jet-lag.

We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant partial di erential equations (PDEs). We restrict the space of solutions to those that are doubly-periodic with respect to a square or hexagonal lattice, and consider the bifurcation problem restricted to a nite-dimensional center manifold. For the square lattice we assume that the kernel of the linear operator, at the bifurcation point, consists of four complex Fourier modes, with wave vectors K 1 = ( ; ); K 2 = ( ; ), K 3 = ( ; ), and K 4 = ( ; ), where > > 0 are integers. For the hexagonal lattice, we assume that the kernel of the linear operator at the bifurcation point consists of six complex Fourier modes, also parameterized by an integer pair ( ; ). We derive normal forms for the bifurcation problems, which we use to compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. These solutions consist of rolls, squares, hexagons, and a countable set of rhombs, and also a countable set of planforms that are superpositions of all of the Fourier modes in the kernel. Since rolls and squares (hexagons) are common to all of the bifurcation problems posed on square (hexagonal) lattices, this framework can be used to determine their stability relative to a countable set of perturbations by varying and . For the square lattice an O(2( + ) 1) truncation of the normal form is required to completely determine the stability of the planforms, although many of the stability results are established at cubic order. For the hexagonal lattice, all of the solution branches guaranteed by the equivariant branching lemma are, generically, unstable due to the presence of a quadratic term in the normal form. We analyze the degenerate bifurcation problem that is obtained by setting the coe cient of the quadratic term to zero. We must retain terms through O(2 1) in the normal form of the bifurcation problem. The unfolding of the degenerate bifurcation problem reveals a new class of secondary bifurcations on the hexagons and rhombs solution branches.We also analyze the bifurcation problems for E(2) + Z 2 -equivariant PDEs, which leads to new results in the hexagonal lattice case, only.

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Sleep changes across the lifespan, with a delay in sleep timing and a reduction in slow wave sleep seen in adolescence, followed by further reductions in slow wave sleep but a gradual drift to earlier timing during healthy ageing. The mechanisms underlying changes in sleep timing are unclear: are they primarily related to changes in circadian processes, or to a reduction in the neural activity dependent build up of homeostatic sleep pressure during wake, or both? We review existing studies of age-related changes to sleep and explore how mathematical models can explain observed changes. Model simulations show that typical changes in sleep timing and duration, from adolesence to old age, can be understood in two ways: either as a consequence of a simultaneous reduction in the amplitude of the circadian wake-propensity rhythm and the neural activity dependent build-up of homeostatic sleep pressure during wake; or as a consequence of reduced homeostatic sleep pressure alone. A reduction in the homeostatic pressure also explains greater vulnerability of sleep to disruption and reduced daytime sleep-propensity in healthy ageing. This review highlights the important role of sleep homeostasis in sleep timing. It shows that the same phenotypic response may have multiple underlying causes, and identifies aspects of sleep to target to correct delayed sleep in adolescents and advanced sleep in later life.

Recent experiments [1] on two-frequency parametrically excited surface waves produce an intriguing "superlattice" wave pattern near a codimension-two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wavenumbers. The superlattice pattern is synchronous with the forcing, spatially periodic on a large hexagonal lattice, and exhibits small-scale triangular structure. Similar patterns have been shown to exist as primary solution branches of a generic 12-dimensional D6+T 2 -equivariant bifurcation problem, and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities [2]. Here we use the spatial and temporal symmetries of the problem to argue that weakly damped harmonic waves may be critical to understanding the stabilization of this pattern in the Faraday system. We illustrate this mechanism by considering the equations developed by Zhang and Viñals [3] for small amplitude, weakly damped surface waves on a semi-infinite fluid layer. We compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 2/3 and 6/7. For the 2/3 case, we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the codimension-two point. Also, we find that the 6/7 case is significantly different from the 2/3 case due to the presence of additional weakly damped harmonic modes. These additional harmonic modes can result in a stabilization of the superpatterns. * To appear in a special issue of Physica D dedicated to the memory of John David Crawford.