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Oscillation analysis of linearly coupled piecewise affine systems

A lot of oscillatory phenomena exist in the natural world. In recent years, many of them have been found to play a crucial role in living organisms such as the circadian rhythms, neural networks, to list a few. This fact has prompted enormous theoretical research works on modeling/analysis of oscillatory phenomena. Among them, large scale arrays consisting of simple subsystems have drawn an intensive attention due to academic interest and also the similarity to actual cell models. In our work, we concentrate on the linearly coupled networks that have interesting applications such as Josephson junction networks. In general, the nonlinearity of the dynamics is indispensable for the occurrence of such phenomena. In this paper, we formulate the nonlinear individual subsystems within the framework of piecewise affine (PWA) systems, for which several practical analysis tools have been proposed. In summary, the overall dynamics is given as linearly coupled (a large number of) PWA systems. In this paper, we derive a sufficient condition under which the dynamics is Y-oscillatory. The Y-oscillation, originally introduced by Yakubovich, is a general notion of oscillatory phenomena that covers both periodic and aperiodic orbits. However, it is known that the analysis of PWA systems become more difficult to analyze as the number of modes increases, similarly to other switching systems. The main result is achieved by proving the well-posedness and ultimate boundedness. An important feature of the result is that, under the assumption that every subsystem has a property in common, the criteria can be rewritten in terms of connection topology and its complexity is considerably reduced so that it is applicable to large scale networks. For illustrative purpose, we analyze Fitzhugh-Nagumo equation that is a model for neural oscillator with the excitation property in mathematical physiology.