Mihaela Sterpu

CONTROLLABLE HOPF BIFURCATIONS OF CODIMENSIONS ONE AND TWO IN LINEAR CONTROL SYSTEMS

Given a linear time-invariant control system, the purpose of this work is to define a four-parameter family of static state feedback such that the corresponding closed-loop control system exhibits controllable Hopf bifurcation of codimensions one and two.

Bifurcation in coupled Hopf oscillators

Two identical dynamical systems, representing the normal form corresponding to the Hopf bifurcation, were coupled using two parameters. The 4D dynamical system obtained possesses additional equilibria. Our study concerns the bifurcations of this system around the origin. We found that Hopf bifurcation takes place in two cases and it is of the same type as the Hopf bifurcation of the single model. In the first case the center manifold is a 2‐plane and the limit cycle does not depend on the coupling parameters. In the second case, if the coupling parameters are equal, limit cycles with four regimes of behavior emerge, while if the coupling parameters are different, limit cycles with eight regimes of behavior are emphasized and different amplitudes of the oscillations occur in addition. For some values of the parameters, other bifurcations are present: degenerated fold bifurcation, degenerated double‐zero bifurcation and symmetric Hopf bifurcation.

Approximations of the Homoclinic Orbits Near a Double-Zero Bifurcation with Symmetry of Order Two

The two-dimensional system of differential equations corresponding to the normal form of the double-zero bifurcation with symmetry of order two is considered. This is a codimension two bifurcation. The associated dynamical system exhibits, among others, a homoclinic bifurcation. In this paper, we obtain second order approximations both for the curve of parametric values of homoclinic bifurcation and for the homoclinic orbits. To perform this task, we reduce first the normal form to a perturbed Hamiltonian system, using a blow-up technique. Then, by means of a perturbation method, we determine explicit first and second order approximations of the homoclinic orbits. The solutions obtained theoretically are compared with those obtained numerically for several cases. Finally, an application of the obtained results is presented.

Stability and Hopf Bifurcation for Two Advertising Systems, Coupled with Delay

Two advertising systems were linearly coupled via the first variable, with time delay. The stability and the Hopf bifurcation corresponding to the symmetric equilibrium point (the origin) in the 4D system are analyzed. Different types of oscillations corresponding to the limit cycles are compared.

Local Bifurcation for the Fitzhugh-Nagumo System

The 2-D FitzHugh-Nagumo (F-N) system depending on three real parameters a, b, and c is considered. It models the electrical potential of the nodal system in the heart. All local bifurcations of equilibria are emphasized in three qualitatively distinct situations concerning the parameter c(0 1). We found codimension-one bifurcations (saddle-node, Hopf), codimension two bifurcations (Bogdanov-Takens, Bautin, cusp, double-zero with order two symmetry) and a codimension three bifurcation (degenerated Bogdanov-Takens of order two). In addition, some non-generic codimension two bifurcations generated by the coexistence of two codimension one bifurcations are shown. In our study we used the normal form theory [3], [6] and the center manifold theory [2].