Study of coupled self-sustained oscillators : circadian oscillations in cyanobacteria and powering of biologicals loads

Abstract

This thesis deals firstly with the analysis of the dynamics of coupled systems considered here as a set of self-sustained oscillators (Van der Pol, Grudzinski-Zebrowski, Hindmarsch-Rose) powering an electrical load (RLC, RL, RC or R). Secondly, to simulate the circadian clock in cyanobacteria made up of three proteins under its simplified structure (monomer) and its basic structure (hexamer). The harmonic balance method, the Kirchoff’s laws, RK4 method and the criteria of synchronization have been used to obtain the following results: In the case of an array of Van der Pol oscillators coupled to a load, it is found that after a threshold number of oscillators under which the power is equal to zero, increase versus number of oscillator. A high order nonlinearity in the damping of the Van der Pol oscillator increases the power in the load. In the case of an array of oscillator Grudzinski-Zebrowski oscillators coupled to a load, we design an equivalent electrical circuit whose equation is similar to the self-sustained oscillator model presented by Grudzinski-Zebrowski. It is then demonstrated that the power in electrical loads (RLC, RL, RC and R) coupled to an array of such oscillator’s increases with the number of oscillators till a constant values depending on the types of loads and values of the load parameters. The synchronization domain is seen to depend on the values of the direct coupling, the values of the indirect coupling and on the number of oscillators in the array. In the case of an array of Hindmarsch-Rose oscillators coupled to a load, it is shown that varying the coupling coefficient leads to the appearance of chaotic dynamics in the system. It is also found that the voltage amplitudes decrease when the size of the array of the HR oscillators increases. For the hexameric model of the cyanobacterial circadian clock, we propose an in vivo model of cyanobacterial circadian clock based on the in vitro model and it is shown that there is a large range of values where there are oscillations.