Narrow Results

The methodology of open-plus-close-loop (OPCL), which is up till now mainly used for the case of ordinary dynamical systems is extended for the first time to the case of delay dynamical process, to achieve complete, lag synchronization. Both the one and two way couplings are examined in detail, with the help of delay logistics, Ikeda and Mackey–Glass system. Robustness of the process is exhibited when the same analysis is done in presence of Gaussian noise. Influence of the strength of such a noise is shown. Our procedure opens up a potentially new approach for the synchronization study of delay dynamical systems.

A higher-order oscillator, including a nonlinear unit and an 8th-order low-pass active Bessel filter is described. The Bessel unit plays the role of "three-in-one": a delay line, an amplifier and a filter. Results of hardware experiments and numerical simulation are presented. Depending on the parameters of the nonlinear unit the oscillator operates either in a one-scroll or two-scroll mode. Two positive Lyapunov exponents, found at larger values of the negative slopes of the nonlinear function, characterize the oscillations as hyperchaotic.

In most vertebrate species, the body axis is generated by the formation of repeated transient structures called somites. This spatial periodicity in somitogenesis has been related to the genetic network oscillations in certain mRNAs and their associated gene products in the cells forming the presomitic mesoderm. The current molecular view of the mechanism underlying these oscillations involves negative-feedback regulation at transcriptional and translational levels. The spatially periodic nature of somite formation suggests that the genetic network involved must display intracellular oscillations that interact with a longitudinal positional information gradient, called determination front, down the axis of vertebrate embryos to create this spatial patterning. Here, we consider a simple model for diploid cells based on this current biological picture considering gene regulation as a noisy process relevant in a real developmental situation and study its consequences for somitogenesis. Comparison is made with the known properties of somite formation in the zebrafish embryo.

A popular biomathematics model of the Goodwin oscillator has been previously generalized to a more biologically plausible construct by introducing three time delays to portray the transport phenomena arising due to the spatial distribution of the model states. The present paper addresses a similar conversion of an impulsive version of the Goodwin oscillator that has found application in mathematical modeling, e.g. in endocrine systems with pulsatile hormone secretion. While the cascade structure of the linear continuous part pertinent to the Goodwin oscillator is preserved in the impulsive Goodwin oscillator, the static nonlinear feedback of the former is substituted with a pulse modulation mechanism thus resulting in hybrid dynamics of the closed-loop system. To facilitate the analysis of the mathematical model under investigation, a discrete mapping propagating the continuous state variables through the firing times of the impulsive feedback is derived. Due to the presence of multiple time delays in the considered model, previously developed mapping derivation approaches are not applicable here and a novel technique is proposed and applied. The mapping captures the dynamics of the original hybrid system and is instrumental in studying complex nonlinear phenomena arising in the impulsive Goodwin oscillator. A simulation example is presented to demonstrate the utility of the proposed approach in bifurcation analysis.