Existence and dynamics of chaotic pulses on 1D lattice are discussed. Traveling pulses arise typically in reaction diffusion systems like the FitzHugh-Nagumo equations. Such pulses annihilate when they collide each other. A new type of traveling pulse has been found recently in many systems where pulses bounce off like elastic balls. We consider the behavior of such a localized pattern on 1D la...

The wavefront profile and the propagation velocity of waves in an experimentally observed Belousov-Zhabotinskii reaction are analyzed and a revised FitzHumgh-Nagumo(FHN) model of these systems is identified. The ratio between the excitation period and the recovery period, for a solitary wave are studied, and included within the model. Averaged travelling velocities at different spatial position...

We study several aspects of FitzHugh-Nagumo’s (FH-N) equations without diffusion. Some global stability results as well as the boundedness of solutions are derived by using a suitably defined Lyapunov functional. We show the existence of both supercritical and subcritical Hopf bifurcations. We demonstrate that the number of all bifurcation diagrams is 8 but that the possible sequential occurren...

We present a 3D numerical model of the heart ventricles which couples electrical and biomechanical activities. We have adopted a simpler formulation of previously published models to achieve a trade-off between biological accuracy and computational efficiency. To carry out computations, the FitzHugh-Nagumo equations are solved along with a constitutive law based on the Hill-Maxwell rheological ...

We report the occurrence of vibrational resonance in excitable systems. Namely, we show that an optimal amplitude of the high-frequency driving enhances the response of an excitable system to a low-frequency signal. The phenomenon is confirmed in an excitable electronic circuit and in the FitzHugh–Nagumo model. In this last case we also analyze the influence of additive noise and the interplay ...

Thermo-optical pulsing in semiconductor amplifiers is experimentally shown to correspond to a very common excitable scenario (the van der Pol-Fitzhugh-Nagumo system). Self-sustained oscillations appear in the sequence predicted by this simple dynamical model as we change either the injection level or the bias current. Periodic modulation of these parameters leads to the characteristic phase-loc...

Feedback control for the monodomain equations is studied. The dynamics of interest are governed by a coupled PDE-ODE reaction diffusion system with non-monotone nonlinearity of FitzHugh-Nagumo type. A localized distributed control is used to locally stabilize the nonlinear system. This is achieved by a Riccati-based feedback law, determined by the linearized system. It is shown that the Riccati...

An open-loop control for a system coupling a reaction-diffusion system and an ordinary differential equation is proposed in this study. We use a flatness-like property, indeed, the solution can be expressed in terms of an infinite series depending on a flat output, its derivatives and its integrals. This series is shown to be convergent if the flat output is Gevrey of order 1 < a ≤ 2. Approxima...

Adaptive numerical methods in space and time are introduced and studied for multiscale cardiac reaction-diffusion models in three dimensions. The evolution of a complete heartbeat, from the excitation to the recovery phase, is simulated with both the anisotropic Bidomain and Monodomain models, coupled with either a variant of the simple FitzHugh-Nagumo model or the more complex phase-I Luo-Rudy...

The FitzHugh-Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in an appropriate singular limit, where a parameter ε goes to zero. These pulses are known to be nonlinearly stable with respect to the underlying PDE. Recently, the existence of fast pulses with oscillatory tails was proved for the FitzHugh-N...