Bifurcation control and the existence of chaos in a class of linear impulsive systems are discussed by means of both theoretical and numerical ways. Chaotic behaviour in the sense of Marotto＇s definition is rigorously proven. A linear impulsive controller, which does not result in any change in one period-1 solution of the original system, is proposed to control and anti-control chaos. The numerical results for chaotic attractor, route leading to chaos, chaos control, and chaos anti-control, which are illustrated with two examples, are in good agreement with the theoretical analysis.
In this paper, we develop a new mathematical model for the mammalian circadian clock, which incorporates both transcriptional/translational feedback loops （TTFLs） and a cAMP-mediated feedback loop. The model shows that TTFLs and cAMP signalling cooperatively drive the circadian rhythms. It reproduces typical experimental observations with qualitative similarities, e.g. circadian oscillations in constant darkness and entrainment to light dark cycles. In addition, it can explain the phenotypes of cAMP-mutant and Rev-erba^-/^- -mutant mice, and help us make an experimentally-testable prediction： oscillations may be rescued when arrhythmic mice with constitutively low concentrations of cAMP are crossed with Rev-erba^-/- mutant mice. The model enhances our understanding of the mammalian circadian clockwork from the viewpoint of the entire cell.
