Chattering and singular perturbation in discontinuous dynamics

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2017
Çağ, Sabahattin
The main purpose of this dissertation is to address the chattering and singularity phenomena in discontinuous dynamical systems. The study describes models of singular impulsive differential equations such that in the system, not only the differential equation is singularly perturbed, but also the impulsive function is singular. Tikhonov Theorem is extended for the impulsive differential equations. Interestingly, in some models described here, a solution of the problem approaches more than one root of the differential equation as the parameter decreases to zero. Wilson-Cowan neuron model is studied with impulse function in which the membrane time constant is considered as both the singularity and bifurcation parameter. A new technique of analysis of the phenomenon is suggested. This allows to consider the existence of solutions of the model and bifurcation in ultimate neural behavior is observed through numerical simulations. The bifurcations are reasoned by impulses and singularity in the model and they concern the structure of attractors, which consist of newly introduced sets in the phase space such that medusas and rings. Moreover, the singular impact moments are introduced and they are utilized for the problems with chattering solutions. The singular impulse moments gives the advantages that the chattering arising in models, e.g., a bouncing ball, an inverted pendulum and a hydraulic relief valve, can be analyzed through the singularity point of view. The presence of chattering is shown exclusively by examination of the right hand side of impact models. Criteria for the sets of initial data which always lead to chattering are established.