Complex behaviors, signal propagation and amplification in chains of autonomous electromechanical systems with unidirectional coupling

Murielle Vanessa Tchakui (Supervisors: Paul Woafo and Pere Colet)
PhD Thesis (2019)

This thesis deals with the complex behaviors, the transmission of signal and its amplification in chains of unidirectionally coupled autonomous electro-mechanical systems (EMSs). Two models of EMS are used in this work: one with magnetic coupling and other with piezoelectric coupling.
The mathematical modelling of the networks of one way coupled EMSs as well as Hamiltonian formulation of the piezoelectric model are presented. Based on theoretical and numerical analysis involving the Smaller Alignment Index (SALI), we show for three coupled autonomous EMSs that when the unidirectional coupling coefficient reaches a threshold value, they get into motion through a Hopf bifurcation. Meanwhile a series of complex dynamical behaviors are observed ranging from steady state, periodic, quasiperiodic and chaotic states to hyperchaotic state. Taking into account time delay due to signal transmission, different bifurcation structures are also generated and we establish the range for time delay and coupling coefficient corresponding to a phase synchronization of oscillators. Regarding chains constituted of a large number of micro-elements, we demonstrate that weak sinusoidal signals are well propagated and exhibit two levels of amplification, each one followed by a saturation: the first amplification keeps the periodic form of the injected signal while the second amplification is an instability that leads to chaotic behavior in the chain. For stochastic input signals, the MEMS array acts as a bandpass filter converting noisy signals to sinusoidal oscillations and after just a few elements the signal has a narrow power spectra. In addition we explore the role of disorder in such a network. Namely, considering that the elements do not have identical parameters, in particular not having identical natural frequencies. We find out that in this 1D configuration, the network reduces the level of signal amplification and is also capable to avoid instabilities leading to chaotic behavior in a broaden frequency range.
This work contributes to the understanding of complex behaviors occurring in networks of coupled EMSs. Moreover it helps to bring a solution to the issue of energy supply for a chain of MaEMS, MEMS, micro machines or micro robots which is seen as a real challenge.

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