Controlling the collective dynamics in systems of active oscillators through geometry and hydrodynamic entrainment
Synchronisation is broadly defined as the coordinated action of two or more individual elements that exhibit some time periodic behaviour. It is widely observed across different systems, where it is often a help or hindrance. Here the focus is on synchronisation facilitated by hydrodynamic coupling, with the viscous forces dominating. This style of coupling is spatially dependent, and so the steady state dynamics of the oscillators can be controlled using their positions. I study 'rower' oscillators, a highly simplified model for motile cilia that approximates each cilium by a rigid sphere that is driven by a geometrically updated force. The simplicity of the model lends itself to generic results that could be observed in many systems with hydrodynamic coupling. This thesis is broken into two main parts. The first focuses on developing an analytical framework to further understand the synchronisation between two oscillators coupled through hydrodynamic forces. To achieve this a phase reduction is applied to the geometric oscillators. To apply a phase reduction first the transformation the natural phase is determined; the natural phase is characterised by constant phase velocity and involves a moving reference frame. Following the transformation, the interaction is subjected to an averaging process. The result is a continuous interaction function characterised by the phase difference of two oscillators. This dramatically simplifies the system and allows standard dynamical system techniques to be applied. The new interaction is verified through a comparison of relaxation time, before it is used to predict the steady state through the examination of fixed points. This framework is then used to demonstrate the ability to entrain a rower is not sensitive to changes in the characteristics of the rower motion, while the synchronisation between rowers is susceptible. Estimates of the relative synchronisation strength vis-a-vis the entrainment were calculated for two single cells flagellates and two types ciliated epithelium. Early results indicate different susceptibility between the species, and conclusions regarding interactions between oscillators should be drawn carefully from their behaviour under external flow. Applying phase reduction to active, `rower' oscillators reduces their dynamics to an interaction function depending exclusively on the phase difference. This ties them into the larger context of Kuramoto oscillators, one of the simplest and most widely studied types of phase oscillators. The phase reduction also allows fixed point analysis and other standard nonlinear techniques to be applied. Facilitating direct comparison between different oscillator motions and the prediction of the steady state dynamics. The second part of this thesis departs from the earlier framework and instead phenomenologically explores how to control subset formation in rower arrays. This is achieved either through the array configuration or through characteristics of the oscillators. Often the arrays in this section are too large to be easily understood through the phase reduction approach, leading to investigations carried out predominantly by simulation. The hydrodynamic nature of the coupling allows the strength and coupling range to be altered using the geometry of the array. This can then influence the preferred geometry of any sub-populations that form in the array. Inspiration for specific controlling geometries is drawn from biological systems and abstract work focused on generic oscillators. Irregularities in the oscillators are also introduced in a regular way as an alternative control option that could be relevant for biological organisms. The geometric controls investigated showed the most consistent control of subset formation, but previously transient states were stabilised with each control mechanism. This section demonstrates the rich and complex collection of behaviours that can occur in systems with hydrodynamic coupling. This has applications in actualising states that are usually investigated more abstractly, and in biological systems where hydrodynamic forces are suspected to play a defining role.
rower model, hydrodynamic forces, synchronisation, cilia
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