Considering two identical, coupled Rössler systems, the paper first examines the bifurcations through which low-periodic orbits embedded into the synchronized chaotic state lose their transverse stability and produce the characteristic picture of riddled basins of attraction. The paper hereafter addresses the issue of the robustness of the synchronized chaotic state to a mismatch of the parameter values between the two subsystems. It is shown that the synchronized state is shifted away from the symmetric manifold, and the magnitude of this shift is expressed in terms of the coupling strength and the mismatch parameter. Finally, the paper illustrates how similar phenomena can be observed in a system of two coupled chaotic oscillators describing the spiking behavior of biological cells. Int. J. Bifurcation Chaos 2000.10:2629-2648. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 02/04/15. For personal use only.
The paper first describes the main bifurcation structure for a typical model of an insulin producing pancreatic β-cell. Considering a system of two coupled identical and chaotically spiking β-cells, the paper continues to examine the bifurcations through which low periodic orbits embedded in the synchronized chaotic state lose their transverse stability and produce the characteristic picture of locally and globally riddled basins of attraction. We discuss the different types of riddled basins with the associated phenomena of attractor bubbling and on-off intermittency. §1. IntroductionThere is experimental evidence to show that the insulin producing β-cells of the pancreas can synchronize their behavior such that not only the transitions between the silent and the active phases (the bursts), but also the individual spikes (during the bursting phase) occur simultaneously. 1), 2) Interaction between the cells arises through the diffusive exchange of ions via gap junctions, and a number of workers have investigated how this mechanism can produce synchronization in a system of periodically bursting cells. 3), 4) The β-cells can also exhibit chaotic oscillations. 5), 6) This will typically occur in the transitions between the various bursting states and between bursting and continuous spiking, and it is clearly of biological interest to understand the possible phenomena that can arise from shifts between different types of synchronous and asynchronous chaotic behavior.Systems of coupled identical chaotic oscillators may attain a state of complete synchronization in which both the phases and the amplitudes develop in precisely the same manner. 7), 8) Partial synchronization 9) (or clustering 10) ), where some of the oscillators synchronize and others do not, may be observed for systems of three or more interacting chaotic oscillators. An interesting problem relates to what happens when the synchronization breaks down. Another important question concerns the stability of the synchronized state to desynchronizing perturbations in the form of noise or of a small parameter mismatch between the interacting oscillators. Recent studies of these problems have led to the observation of a variety of new phenomena, including riddled basins of attraction, 11), 12) attractor bubbling, 13) and on-off intermittency. 14), 15) Riddled basins of attraction are observed in regions of parameter space where the synchronized chaotic state is attracting on the average (the largest transverse Lyapunov exponent is negative), while at the same time particular orbits embedded in the chaotic set are transversely unstable. 11), 12) The basin of attraction for
Abstract-An efficient method is presented for the estimation of the bit-error rate (BER) of a system employing all-optical regenerators influenced by pattern effects. We theoretically study noise accumulation and noise redistribution in long distance transmission systems employing a delayed interference signal wavelength converter for all-optical regeneration. The BER is studied for return-to-zero signals at bit rates of 2.5 Gb/s (no patterning) up to 40 Gb/s (strong patterning). The calculation of the BER is based on pattern dependent transfer functions, which may be obtained numerically or measured.
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