The dynamics of a delayed mathematical model for Hes1 oscillatory expression are investigated. The linear stability of positive equilibrium and existence of local Hopf bifurcation are studied. Moreover, the global existence of large periodic solutions has been established due to the global bifurcation theorem. Keywords—Hes1, Hopf bifurcation, time delay, transcriptional repression loop

We report on transcritical bifurcations of periodic orbits in nonintegrable two-dimensional Hamiltonian systems. We discuss their existence criteria and some of their properties using a recent mathematical description of transcritical bifurcations in families of symplectic maps. We then present numerical examples of transcritical bifurcations in a class of generalized Hénon-Heiles Hamiltonians ...

School of Mathematical Sciences, Zhejiang University, Zhejiang 310027, China. Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China. Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China. School of Engineering Science, University of the Chinese Academy Sciences, Beijing 10049, C...

The slow passage through a Hopf bifurcation leads to the delayed appearance of large amplitude oscillations. We construct a smooth scalar feedback control which suppresses the delay and causes the system to follow a stable equilibrium branch. This feature can be used to detect in time the loss of stability of an ageing device. As a by-product, we obtain results on the slow passage through a bif...

This pdf document provides the textual background in the mini course on bifurcation analysis, by George van Voorn. It is the companion material by the powerpoint presentations posted on the site www.bio.vu.nl/thb. Goal of this manuscript is to explain some of the basics of bifurcation theory to PhD-students at the group. The emphasis is strongly on the biological interpretation of bifurcations;...

There is a vast body of literature devoted to the study of bifurcation phenomena in autonomous systems of differential equations. However, there is currently no well-developed theory that treats similar questions for the non-autonomous case. Inspired in part by the theory of pullback attractors, we discuss generalizations of various autonomous concepts of stability, instability, and invariance....

The slow drift (with speed ε) of a parameter through a pitchfork bifurcation point, known as the dynamic pitchfork bifurcation, is characterized by a significant delay of the transition from the unstable to the stable state. We describe the effect of an additive noise, of intensity σ, by giving precise estimates on the behaviour of the individual paths. We show that until time √ ε after the bif...

The aim of this paper is to study Hopf bifurcation with Dn-symmetry assuming Birkhoff normal form. We consider the standard action of Dn on C. This representation is absolutely irreducible and so the corresponding Hopf bifurcation occurs on C ⊕ C. Golubitsky and Stewart (Hopf bifurcation with dihedral group symmetry: Coupled nonlinear oscillators. In: Multiparameter Bifurcation Series (M. Golub...

We give conditions under which a general class of delay differential equations has a point of Bogdanov-Takens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to a two or three dimensional systems of ordinary differential equations. We put these equations in normal form and determine how the coefficients of the normal forms depen...

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in s...