The existence of periodic solutions for evolution equations is of certain interest for both pure and applied mathematicians. Even for bidimensional systems of differential equations the detection of limit cycles by theoretical means is difficult. The bifurcation theory offers a strong tool for finding limit cycles, namely the theory concerning the Hopf bifurcation (when there is a varying param...

This paper is concerned with a delayed SIRS epidemic model with a nonlinear incidence rate. The main results are given in terms of local stability and Hopf bifurcation. Sufficient conditions for the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained by regarding the time delay as the bifurcation parameter. Further, the properties of Hopf bifurcation such ...

The bifurcation of limit cycles in single-input single-output control systems with saturation is considered. Under some non-degeneracy conditions, a theorem characterizing such bifurcation is stated for the cases of dimension two and three. In terms of the deviation from the critical value of the bifurcation parameter, expressions in form of power series for the period, amplitude and the charac...

A bstract |This paper reexamines the conventional currentmode control strategy as applied to dc/dc converters in the ligh t of \avoiding bifurcation". This alternativ eviewpoint permits convenien t selection of parameter values to guarantee stable operation. Slope compensation is viewed as a means to k eep the system suÆciently remote from the rst bifurcation point. It is shown that excessiv e ...

We consider iterated maps with a reflectional symmetry. Possible bifurcations in such systems include period-doubling bifurcations (within the symmetric subspace) and symmetry-breaking bifurcations. By using a second parameter, these bifurcations can be made to coincide at a mode interaction. By reformulating the period-doubling bifurcation as a symmetry-breaking bifurcation, two bifurcation eq...

The unfolding due to imperfections of a gluing bifurcation occurring in a periodically forced Taylor–Couette system is numerically analyzed. In the absence of imperfections, a temporal glide-reflection Z2 symmetry exists, and two global bifurcations occur within a small parameter region: a heteroclinic bifurcation between two saddle two-tori and a gluing bifurcation of three-tori. Due to the pr...

Quantitative aspects of models describing the dynamics of biological phenomena have been mostly restricted to results of numerical simulations, often by employing standard numerical methods. However, several studies have shown that these methods may fail to reproduce the actual dynamical behavior of the underlying continuous model when the integration time-step, model parameters, or initial con...

In this paper, by using the Hopf’s bifurcation theorem we will discuss the existence of small amplitude periodic solutions of the equation ẍ(t)+ f(x(t))ẋ(t) + g(x(t − r)) = 0, taking as bifurcation parameter c either d or r. We assume that r > 0, f ∈ C, f(0) = c > 0, g(0) = 0 and ġ(0) = d > 0.

We study a simple map as a minimal model of excitable cells. The map has two fast variables which mimic the behavior of class I neurons, undergoing a sub-critical Hopf bifurcation. Adding a third slow variable allows the system to present bursts and other interesting biological behaviors. Bifurcation lines which locate the excitability region are obtained for different planes in parameter space.

Period adding with the coexistence of successive at tract ing periodic orbit s is observed in the model of a chemical system when the bifurcation parameter is changed. This phenomenon is characterized by a family of one-dimensional return maps having a cusp shape with positive Schwarzian derivative that exhibits a saddle-node bifurcation.