The main purpose of this article is to demonstrate that bursting oscillations can be observed not only in the slow–fast autonomous dynamical systems with multiple scales associated with time domain, but also in the non-autonomous dynamical systems with periodic excitations when an order gap exists between the exciting frequency and the natural frequency, implying multiple scales in frequency do...

Crank-Nicolson is a popular method for solving parabolic equations because it is unconditionally stable and second order accurate. One drawback of CN is that it responds to jump discontinuities in the initial conditions with oscillations which are weakly damped and therefore may persist for a long time. We present a selection of methods to reduce the amplitude of these oscillations.

Sustained oscillations are frequently observed in biological systems consisting of a negative feedback loop, but a mathematical model with two ordinary differential equations (ODE) that has a negative feedback loop structure fails to produce sustained oscillations. Only when a time delay is introduced into the system by expanding to a three-ODE model, transforming to a two-DDE model, or introdu...

We extend the study of a computational model recently proposed for the mammalian circadian clock (Proc. Natl Acad. Sci. USA 100 (2003) 7051). The model, based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, and Clock genes, can give rise to sustained circadian oscillations in conditions of continuous darkness. These limit cycle oscillations correspond to...

A deterministic system that operates in the vicinity of a Hopf bifurcation can be described by a single equation of a complex variable, called the normal form. Proximity to the bifurcation ensures that on the stable side of the bifurcation (i.e. on the side where a stable fixed point exists), the linear-response function of the system is peaked at the frequency that is characteristic of the osc...

Reaction temperature plays a major role in product selectivity in the oscillatory mode of the palladium-catalyzed phenylacetylene oxidative carbonylation reaction. At 40 °C, dimethyl (2Z)-2-phenyl-2-butenedioate is the major product whereas at 0 °C the major product is 5,5-dimethoxy-3-phenyl-2(5H)-furanone. The occurrence of oscillations in pH coincides with an increase in the rate of phenylace...

p53 is activated in response to events compromising the genetic integrity of a cell. Recent data show that p53 activity does not increase steadily with genetic damage but rather fluctuates in an oscillatory fashion. Theoretical studies suggest that oscillations can arise from a combination of positive and negative feedbacks or from a long negative feedback loop alone. Both negative and positive...

We show how an island (isola) evolves out of the usual S-curve of steady states of diffusion flames when radiation losses are accounted for and how it eventually disappears when radiation increases further. At small activation temperatures there are never any islands. We show that stable oscillations evolve first out of perturbations of steady states on the S-curve at large Damköhler numbers. O...

Energy dissipation is an unavoidable phenomenon of physical systems that are directly coupled to an external environmental bath. In an oscillatory system, it leads to the decay of the oscillation amplitude. In situations where stable oscillations are required, the energy dissipated by the vibrations is usually compensated by replenishment from external energy sources. Consequently, if the exter...

Activation of hormone receptors was recently found to evoke oscillations of the cAMP concentration ([cAMP]) beneath the plasma membrane of insulin-secreting cells. Here we investigate how different time courses of cAMP signals influence the generation of cytoplasmic Ca(2+) signals and nuclear translocation of the PKA (protein kinase A) catalytic subunit in individual INS-1 beta-cells. [cAMP] wa...