Carmen Rocşoreanu

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].

Global Bifurcation Diagram and Phase Dynamics for the FitzHugh-Nagumo Model

The partial results obtained in Chapters 2–4 are synthesized in the global bifurcation diagram for the F-N model (Section 5.1). It is obtained by putting together, as in a huge puzzle, all local bifurcation diagrams obtained in the previous chapters. This completes the treatment of the bifurcation for the F-N model.

Static Bifurcation and Linearization of the FitzHugh-Nagumo Model

In this chapter we start with a study of the system (1.1.17), assuming x and y to be functions of the time t and a ≥ 0. The nullclines, the curves of points of inflection of phase trajectories (Section 2.1), the surface of equilibrium points, and the curves of static bifurcation values in the parameters plane (b, a) (Section 2.2) are deduced.

Dynamic Bifurcation for the FitzHugh-Nagumo Model

In this chapter the dynamic bifurcation for the F-N system (1.1.17) is studied. In the (b, a)-parameter plane the oscillatory regimes are related to the Hopf bifurcation.

Models and Dynamics

In Section 1.1 the physiological and mathematical context of the FitzHugh-Nagumo (F-N) model is presented. The first four subsections contain the physiological basis and the models based on two first order ordinary differential equations (ODE) of initiation of cardiac impulse. Thus, in Section 1.1.1 the types and characteristics of the electrical circuits of the heart are described. The famous Van der Pol model is presented in Section 1.1.2, together with its main types of oscillations. The relaxation oscillations are defined. The Van der Pol generalized model and its particular case, the F-N model, are presented in Section 1.1.3. Different forms of the F-N model are given in Section 1.1.4, together with certain general properties of some of them. The study of one of the forms of the F-N model, namely (1.1.17), is the aim of this book. In the last subsection (1.1.5) the connection with the propagation by rotating waves in the whole cardiac muscle of the electric impulse generated in the sinusal node (Figure 1.1.1) is provided. Models of coupled oscillators, based on systems of ordinary differential equations (SODE) of first order and models with forcing terms related to the F-N model are also mentioned.

Models of Asymptotic Approximation for the FitzHugh-Nagumo System as c → ∞

After presenting all types of asymptotic behaviour of interest for the solutions of the F-N model (Section 4.1), we emphasize only the case c → ∞ and its periodic evolutions. In order to do this the outer and the inner asymptotic expansions of first order (Section 4.2) and higher orders (Section 4.3) are deduced and matched. In addition, in Section 4.4, the time necessary to cover different segments of the limit cycle for some values of c and for some models of asymptotic approximation, as c → ∞, is computed. In Section 4.5, relaxation oscillations, ducks and non-hyperbolic limit cycle bifurcation are discussed. Some results on the asymptotics involved into their investigation are presented. They form a basis for our numerical results on these oscillations and phase trajectories.