Bifurcations and Hyperchaos in Mathematical Models of Sleep.

Sleep regulation is generated by interacting nonlinear feedback loops involving homeostatic pressure, circadian forcing, mutual inhibition, and neuromodulatory mechanisms. This work presents a nonlinear-dynamical interpretation of sleep with emphasis on bifurcation structure, Lyapunov stability, deterministic intermittency, and hyperchaotic fragmentation. After summarizing representative sleep models and the literature on sleep-stage dynamics, we combine bifurcation and Lyapunov exponent analyses of representative sleep-population models with a four-dimensional model in which cortical activation, sleep-promoting activity, homeostatic sleep drive, and the circadian pacemaker form a slow-fast feedback system. The model maps coordinate-dependent feedback, separated homeostatic accumulation and clearance, external sleep-debt forcing, refractory dynamics, and on-off intermittency onto physiological sleep-wake processes. The baseline diagrams identify routes from regular oscillations to period-doubling, chaotic bands, periodic windows, and positive Lyapunov regions, while the new Lyapunov maps of the four-dimensional model identify parameter regions where the largest and second-largest Lyapunov exponents are positive, supporting the hyperchaotic classification. The formulation also shows how chronic sleep debt and disease-associated or stress-related insomnia can be represented as shifts in homeostatic drive, inhibitory gain, and circadian coupling. Bifurcation theory therefore provides a useful framework for interpreting irregular sleep transitions and for inspiring future sleep-technology and digital twin applications.