Astounding properties of biological sensors can often be mapped onto a dynamical system in the vicinity a bifurcation. For mammalian hearing, a Hopf bifurcation description has been shown to work across a whole range of scales, from individual hair bundles to whole regions of the cochlea. We reveal here the origin of this scale-invariance, from a general level, applicable to all neuronal dynami...

In this paper, we consider controlling and anti-controlling Hopf bifurcations in discrete maps using feedback controller of polynomial functions. It is shown that such a polynomial feedback controller is easy to be implemented, which not only preserves the system s equilibrium solutions, but also keeps the dimension of the system unchanged. Examples are used to show, with this type of controlle...

An explicit general formula is proposed for controlling Hopf bifurcation using state feedback. This method can be used to either delay (or even eliminate) an existing Hopf bifurcation or change a subcritical Hopf bifurcation to supercritical. The Lorenz system is used to illustrate the application of the formula.

In this paper, a delayed physiological control system is investigated. The sufficient conditions for stability of positive equilibrium and existence of local Hopf bifurcation are derived. Furthermore, global existence of periodic solutions is established by using the global Hopf bifurcation theory. Finally, numerical examples are given to support the theoretical analysis. Keywords—Physiological...

In this paper, a modified delayed mathematical model for the dynamics of HIV with cure rate is considered. By regarding the time delay as bifurcation parameter, stability and existence of local Hopf bifurcation are studied by analyzing the transcendental characteristic equation. Then the global existence of bifurcating periodic solutions is established with the assistance of global Hopf bifurca...

Complex dynamics in the reduced swing equations of a power system are investigated. As the change of bifurcation parameter μ, the system exhibits complex bifurcations, such as saddle-node bifurcation, Hopf bifurcation, cyclic fold bifurcation and torus bifurcations and complex dynamics, such as periodic orbits, quasiperiodic orbits, period-doubling orbits and chaotic attractors which link with ...

Hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system. The problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks. Most of these instabilities are starting from a single Hopf bifurcation. Because of the large delay, a delayed amplitude appears in the slow time bif...

A SI-type ecoepidemiological model that incorporates reproduction delay of predator is studied. Considering delay as parameter, we investigate the effect of delay on the stability of the coexisting equilibrium. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theore...

To study the nonlinear dynamics, such as Hopf bifurcation, of partial differential equations with delay, one needs to consider the characteristic equation associated to the linearized equation and to determine the distribution of the eigenvalues; that is, to study the spectrum of the linear operator. In this paper we study the projectors on the generalized eigenspaces associated to some eigenva...