Dynamics of Delay Equations, Theory and Applications - Abstract
Klinshov, Vladimir
Pulse signals mediate interactions in numerous networks of various natures, including laser ensembles and optoelectronic systems, cardiac tissues and neuronal populations. We study the dynamics of simple oscillatory networks with pulse delayed interactions: a single oscillator with pulse delayed feedback and a unidirectional ring with pulse delayed connections. The oscillators are described in the phase approximation, and the impact of pulses is described by the phase reset curve (PRC). The basic regime of the systems under the consideration is periodical generation of pulses, or regular spiking. We report an unexpected scenario of regular spiking destabilization. For a sufficiently steep slope of the PRC, the regular spiking solution bifurcates with several multipliers crossing the unit circle at once. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous ?jittering? regimes with nonequal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution with a period roughly proportional to the delay. The number of different ?jittering? solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. We show that the theoretical results accurately predict the behavior of experimentally implemented electronic circuits.