Considering systems of diffusively coupled identical chaotic oscillators, an effective method to determine the possible states of cluster synchronization and ensure their stability is presented. The method, which may find applications in communication engineering and other fields of science and technology, is illustrated through concrete examples of coupled biological cell models.

Oscillations in human insulin secretion have been observed in two distinct period ranges, 10-15 min (i.e. rapid) and 100-150 min (i.e., ultradian). The cause of the ultradian oscillations remains to be elucidated. To determine whether the oscillations could result from the feedback loops between insulin and glucose, a parsimonious mathematical model including the major mechanisms involved in glucose regulation was developed. This model comprises two major negative feedback loops describing the effects of insulin on glucose utilization and glucose production, respectively, and both loops include the stimulatory effect of glucose on insulin secretion. Model formulations and parameters are representative of results from published clinical investigations. The occurrence of sustained insulin and glucose oscillations was found to be dependent on two essential features: 1) a time delay of 30-45 min for the effect of insulin on glucose production and 2) a sluggish effect of insulin on glucose utilization, because insulin acts from a compartment remote from plasma. When these characteristics were incorporated in the model, numerical simulations mimicked all experimental findings so far observed for these ultradian oscillations, including 1) self-sustained oscillations during constant glucose infusion at various rates; 2) damped oscillations after meal or oral glucose ingestion; 3) increased amplitude of oscillation after increased stimulation of insulin secretion, without change in frequency; and 4) slight advance of the glucose oscillation compared with the insulin oscillation.(ABSTRACT TRUNCATED AT 250 WORDS)

Riddled basins denote a characteristic type of fractal domain of attraction that can arise when a chaotic motion is restricted to an invariant subspace of total phase space. An example is the synchronized motion of two identical chaotic oscillators. The paper examines the conditions for the appearance of such basins for a system of two symmetrically coupled logistic maps. We determine the regions in parameter plane where the transverse Lyapunov exponent is negative. The bifurcation curves for the transverse destabilization of low-periodic orbits embedded in the chaotic attractor are obtained, and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations. It is shown that the appearance of transversely unstable orbits does not necessarily lead to an observable basin riddling, and that the loss of weak stability (when the transverse Lyapunov exponent becomes positive) does not necessarily destroy the basin of attraction. Instead, the symmetry of the synchronized state may break, and the attractor may spread into two-dimensional phase space.