Saddle-node Bifurcation Of Limit Cycles Research Articles

Bifurcations in a white-blood-cell production model

We study the dynamics of a model of white-blood-cell (WBC) production. The model consists of two compartmental differential equations with two discrete delays. We show that from normal to pathological parameter values, the system undergoes supercritical Hopf bifurcations and saddle-node bifurcations of limit cycles. We characterize the steady states of the system and perform a bifurcation analysis. Our results indicate that an increase in apoptosis rate of either hematopoietic stem cells or WBC precursors induces a Hopf bifurcation and an oscillatory regime takes place. These oscillations are seen in some hematological diseases. To cite this article: S. Bernard et al., C. R. Biologies 327 (2004).

Bifurcation of limit cycles and separatrix loops in singular Lienard systems

We give sufficient conditions for the existence of one or two limit cycles of singular Lienard systems through the construction of a Poincare–Bendixson domain. With the help of the theory of rotated vector fields,we develop a method to compute bifurcation value at Saddle-node bifurcation of limit cycles and homoclinic or symmetric heteroclinic bifurcations. We also present application examples and prove the existence of duck cycles.

Oscillations in cyclical neutropenia: new evidence based on mathematical modeling

We present a dynamical model of the production and regulation of circulating blood neutrophil number. This model is derived from physiologically relevant features of the hematopoietic system, and is analysed using both analytic and numerical methods. Supercritical Hopf bifurcations and saddle-node bifurcations of limit cycles are shown to exist. We make the estimation of kinetic parameters for dogs and then apply the model to cyclical neutropenia (CN) in the grey collie, a rare disorder in which oscillations in all blood cell counts are found. We conclude that the major cause of the oscillations in CN is an increased rate of apoptosis of neutrophil precursors which leads to a destabilization of the hematopoietic stem cell compartment.

Bifurcation Analysis of a Predator-Prey System with Nonmonotonic Functional Response

We consider a predator-prey system with nonmonotonic functional response: $p(x)=\frac{mx}{ax^2+bx+1}$. By allowing b to be negative ($b > -2\sqrt a$), p(x) is concave up for small values of x > 0 as it is for the sigmoidal functional response. We show that in this case there exists a Bogdanov--Takens bifurcation point of codimension 3, which acts as an organizing center for the system. We study the Hopf and homoclinic bifurcations and saddle-node bifurcation of limit cycles of the system. We also describe the bifurcation sequences in each subregion of parameter space as the death rate of the predator is varied. In contrast with the case $b\geq 0$, we prove that when $-2\sqrt a < b < 0$, a limit cycle can coexist with a homoclinic loop. The bifurcation sequences involving Hopf bifurcations, homoclinic bifurcations, as well as the saddle-node bifurcations of limit cycles are determined using information from the complete study of the Bogdanov--Takens bifurcation point of codimension 3 and the geometry of the system. Examples of the predicted bifurcation curves are also generated numerically using XPPAUT. Our work extends the results in [F. Rothe and D. S. Shafer, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), pp. 313--347] and [S.Ruan and D. Xiao, SIAM J. Appl. Math., 61 (2001), pp. 1445--1472].

Ghostbursting: a novel neuronal burst mechanism.

Pyramidal cells in the electrosensory lateral line lobe (ELL) of weakly electric fish have been observed to produce high-frequency burst discharge with constant depolarizing current (Turner et al., 1994). We present a two-compartment model of an ELL pyramidal cell that produces burst discharges similar to those seen in experiments. The burst mechanism involves a slowly changing interaction between the somatic and dendritic action potentials. Burst termination occurs when the trajectory of the system is reinjected in phase space near the "ghost" of a saddle-node bifurcation of fixed points. The burst trajectory reinjection is studied using quasi-static bifurcation theory, that shows a period doubling transition in the fast subsystem as the cause of burst termination. As the applied depolarization is increased, the model exhibits first resting, then tonic firing, and finally chaotic bursting behavior, in contrast with many other burst models. The transition between tonic firing and burst firing is due to a saddle-node bifurcation of limit cycles. Analysis of this bifurcation shows that the route to chaos in these neurons is type I intermittency, and we present experimental analysis of ELL pyramidal cell burst trains that support this model prediction. By varying parameters in a way that changes the positions of both saddle-node bifurcations in parameter space, we produce a wide gallery of burst patterns, which span a significant range of burst time scales.

Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations

This paper presents the dynamics and bifurcations of a semiconductor laser subject to instantaneous phase-conjugate feedback. Recently, the behavior of such a laser has been explored by means of bifurcation diagrams. However, the exact nature of the involved dynamics and bifurcations remained unclear. Here we present a detailed study of the changes of the dynamics as the feedback strength is varied. Most prominent are symmetry-breaking and -restoring bifurcations, tori and their bifurcations, and a sudden transition between chaos and a stable limit cycle due to a saddle-node bifurcation of limit cycles.

Coherence resonance in a Hodgkin-Huxley neuron

We study the nonlinear response of the Hodgkin-Huxley model without external periodic signal to the noisy synaptic current near the saddle-node bifurcation of limit cycles. The coherence of the system, estimated from the interspike interval histogram and from the power spectra of membrane potentials and spike trains, is maximal at a certain noise intensity, so that the coherence resonance occurs. The mechanism of this phenomenon is found to be different from previously studied models of coherence resonance and explained in terms of rigid excitations of periodic oscillations, and the combined effect of amplitude and phase fluctuations.