A tri-phasic mode is stable when three non-linear oscillators interact with each other

Oregonator generalization as a minimal model of quorum sensing in Belousov–Zhabotinsky reaction with catalyst confinement in large populations of particles

The Oregonator demonstrates that quorum sensing in populations of Belousov–Zhabotinsky oscillators arises from modification of the stoichiometry by catalyst confinement.

Modeling of a density oscillator

A density oscillator is a well-known system, which exhibits relaxation oscillation. It alternately exhibits up and down flows through a pipe that connects two containers filled with fluids that have different densities. Although the up-flow, down-flow, and flow-reversal processes have been studied separately, the entire oscillatory dynamics has not been modeled quantitatively. In this study, we derive a model of a density oscillator by considering all the above mentioned processes. The model thus obtained describes the oscillatory behavior in a unified manner, and its viscosity and pipe-length dependence is well described. Moreover, for the demonstration of this model, we have extended it to describe the dynamical behaviors observed in coupled density oscillators. Thus, this model provides a general expression for density oscillators.

Phase resetting, phase locking, and bistability in the periodically driven saline oscillator: experiment and model

The saline oscillator consists of an inner vessel containing salt water partially immersed in an outer vessel of fresh water, with a small orifice in the center of the bottom of the inner vessel. There is a cyclic alternation between salt water flowing downwards out of the inner vessel into the outer vessel through the orifice and fresh water flowing upwards into the inner vessel from the outer vessel through that same orifice. We develop a very stable (i.e., stationary) version of this saline oscillator. We first investigate the response of the oscillator to periodic forcing with a train of stimuli (period=Tp) of large amplitude. Each stimulus is the quick injection of a fixed volume of fresh water into the outer vessel followed immediately by withdrawal of that very same volume. For Tp sufficiently close to the intrinsic period of the oscillator (T0) , there is 1:1 synchronization or phase locking between the stimulus train and the oscillator. As Tp is decreased below T0 , one finds the succession of phase-locking rhythms: 1:1, 2:2, 2:1, 2:2, and 1:1. As Tp is increased beyond T0 , one encounters successively 1:1, 1:2, 2:4, 2:3, 2:4, and 1:2 phase-locking rhythms. We next investigate the phase-resetting response, in which injection of a single stimulus transiently changes the period of the oscillation. By systematically changing the phase of the cycle at which the stimulus is delivered (the old phase), we construct the new-phase--old-phase curve (the phase transition curve), from which we then develop a one-dimensional finite-difference equation ("map") that predicts the response to periodic stimulation. These predicted phase-locking rhythms are close to the experimental findings. In addition, iteration of the map predicts the existence of bistability between two different 1:1 rhythms, which was then searched for and found experimentally. Bistability between 1:1 and 2:2 rhythms is also encountered. Finally, with one exception, numerical modeling with a phenomenologically derived Rayleigh oscillator reproduces all of the experimental behavior.

Viscosity-dependent flow reversal in a density oscillator

The density oscillator is a simple system that exhibits self-sustained oscillation. It alternately exhibits up and down flow through a pipe which connects two containers filled with fluids of different densities. However, the mechanism of the flow reversal has not yet been fully understood. From the detailed measurements, we have found that flow reversal begins with an intrusion of fluid, which is followed by rapid growth. This process is definitely sensitive to the viscosities of the fluids, and as a consequence, the critical heights leading to flow reversal are clearly viscosity dependent. These experimental results are explained by a simple model, derived by considering forces acting on a unit volume element located at the tip of the intrusion. Using this model, we can successfully explain the mechanism of flow reversal, which is the most essential process in a density oscillator.

Chaotic patterns in a coupled oscillator-excitator biochemical cell system

In this paper we examine dynamical modes resulting from diffusion-like interaction of two model biochemical cells. Kinetics in each of the cells is given by the ICC model of calcium ions in the cytosol. Constraints for one of the cells are set so that it is excitable. One of the constraints in the other cell - a fraction of activated cell surface receptors-is varied so that the dynamics in the cell is either excitable or oscillatory or a stable focus. The cells are interacting via mass transfer and dynamics of the coupled system are studied as two parameters are varied-the fraction of activated receptors and the coupling strength. We find that (i) the excitator-excitator interaction does not lead to oscillatory patterns, (ii) the oscillator-excitator interaction leads to alternating phase-locked periodic and quasiperiodic regimes, well known from oscillator-oscillator interactions; torus breaking bifurcation generates chaos when the coupling strength is in an intermediate range, (iii) the focus-excitator interaction generates compound oscillations arranged as period adding sequences alternating with chaotic windows; the transition to chaos is accompanied by period doublings and folding of branches of periodic orbits and is associated with a Shilnikov homoclinic orbit. The nature of spontaneous self-organized oscillations in the focus-excitator range is discussed. (c) 1999 American Institute of Physics.

Spatial phase pattern in one-dimensional arrays of limit cycle oscillators with discrete coupling

We present a model of limit cycle oscillators for collective oscillations in intracellular calcium concentration in cell communities. A phase-dependent discrete coupling between nearest neighbors is introduced into the model on the basis of the experimental observation that intercellular transmission of calcium or calcium mobilizing messenger is effected by gap junction and gap junctional permeability is affected by intracellular calcium concentration. The spatial phase pattern of several clusters in which oscillations are in phase is found with the phase-dependent discrete coupling.