Mathematical modeling of spontaneous oscillations in flagellar and bio-inspired systems.

In this thesis, we analyze the structure and dynamics of a single cilium, aiming to understand its multi-scale dynamics. At the microscopic level, ciliary motion is generated by ensembles of dynein molecular motors, which convert chemical energy into mechanical force along adjacent microtubule filaments. To describe this process, we formulate the two-row model, a system of coupled nonlinear partial differential equations, representing the interaction of two opposing rows of motors with the filament backbone in an isolated environment. We establish well-posedness by proving the existence and the uniqueness of solutions, and we demonstrate the onset of oscillatory instabilities via a supercritical Hopf bifurcation. The theoretical analysis is complemented with numerical simulations that confirm the predicted bifurcation structure and nonlinear dynamics. At the mesoscale, the internal activity described by the two-row model is coupled to the elastic bending of the axonemal filaments and to the surrounding viscous fluid, modeled in the low Reynolds number regime. The resulting composite system, denoted mu-chemoEH, captures the multiscale interaction between motor activity, filament elasticity, and hydrodynamics. Numerical simulations of mu-chemoEH reveal the emergence of periodic beating patterns characteristic of ciliary motion. To gain further analytical insight, we introduce a model inspired by the mu-chemoEH one, coupling motor ODEs with filament PDEs. This system exhibits a self-organized mode selection mechanism in which a dominant spatial wavelength emerges beyond the Hopf bifurcation. Finally, we compare the ciliary dynamics with an artificial active system, namely a photo-deformable liquid crystal elastomer. This reduction allows for explicit analytical expressions of activation thresholds and oscillation frequencies, providing an analytical approach to both natural and artificial oscillatory systems. These findings offer new mathematical insights and modeling approaches that link molecular-scale activity with the coordinated, large-scale beating patterns observed in filamentous systems.