In this paper we study the Hopf bifurcation for the tumor-immune system model with one delay. This model is governed by a system of two differential equations with one delay. We show that the system may have periodic solutions for small and large delays for some critical value of the delay parameter via Hopf bifurcation theorem bifurcating from the non trivial steady state.

In this paper we analyze a simple mathematical model which describes the interaction between the proteins p53 and Mdm2. For the stationary state we discuss the local stability and the existence of the Hopf bifurcation. Choosing the delay as a bifurcation parameter we study the direction and stability of the bifurcating periodic solutions. Some numerical examples and the conclusions are finally ...

Imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on Banach spaces. In particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined, and two-parameter unfoldings of singularities are rigorously established. Applications include semilinear elliptic equations, imperfect Euler buckling beam problem and pertu...

One of the most important tasks in nonlinear dynamics is the determination of bifurcation sets of various dynamical systems. If the dynamical equations are known, bifurcation points in parameter space can be detected by solving some specific fixed point equations. Using suitable continuation algorithms these bifurcation points can be traced in parameter space in order to compute bifurcation cur...

An optically injected semiconductor laser can produce excitable multipulses. Homoclinic bifurcation curves confine experimentally accessible regions in parameter space where the laser emits a certain number of pulses after being triggered from its steady state by a single perturbation. This phenomenon is organized by a generic codimension-two homoclinic bifurcation and should also be observable...

In this paper we study the eigenvalues associated with a positive eigenfunction of a quasilinear elliptic problem with an operator that is not necessarily bounded. For that, we use the bifurcation theory and obtain the existence of positive solutions for a range of values of the bifurcation parameter.

Abstract Analytical determination of the bifurcation thresholds is of important interest for practical electrical machine design and analysis control. This paper presents mathematical investigation of the qualitative behavior of indirect filed oriented control induction motor based on bifurcation theory. In this context, steady-state responses analysis of the motor model is discussed and an ana...

From the periodicity of regional climate change to sustained oscillations in living cells, the transition between stationary and oscillatory behavior is often through a Hopf bifurcation. When a parameter slowly passes or ramps through a Hopf bifurcation there is a delayed transition to sustained oscillations and an associated memory effect where onset is dependent on the initial state of the sy...

Novak and Tyson have proposed a realistic mathematical model of the biochemical mechanism that regulates M-phase promoting factor (MPF), the major enzymatic activity controlling mitotic cycles in frog eggs, early embryos, and cell-free egg extracts. We use bifurcation theory and numerical methods (AUTO) to characterize the codimension-one and -two bifurcation sets in this model. Our primary bif...

Bifurcation diagrams which allow one to visualise changes in the behaviour of low dimensional nonlinear maps as a parameter is altered are common. Visualisation in higher dimensional systems is more difficult. A straightforward method to visualise bifurcations in flows of high dimensional nonlinear dynamical systems is presented, using the Lorenz ’96 systems with dimension 8 and dimension 40 as...