Motivated by pulse-replication phenomena observed in the FitzHugh--Nagumo equation, we investigate traveling pulses whose slow/fast profiles exhibit canard-like transitions. We show that spectra of PDE linearization about such may contain many point eigenvalues accumulate onto a union curves as slow scale parameter approaches zero. The limit sets are related to absolute spectrum homogeneous res...

Abstrad-Various control algorithms have been proposed in recent years to control chaotic systems. These methods are broadly classified into feedback and nonfeedback methods. In this paper, we make a critical analysis of nonfeedback methods such as (i) addition of constant bias, (ii) addition of second periodic force, (iii) addition of weak periodic pulse, and (iv) entrainment control. We apply ...

Excitable media support circulating waves on a one-dimensional ring. Continuity arguments, developed originally for resetting limit cycle oscillations in finite-dimensional dynamical systems, suggest that there must exist a range of phases and amplitudes for perturbations of the reentrant wave that lead to its annihilation. The annihilation is illustrated in the Fitzhugh-Nagumo equation of exci...

This paper deals with the modeling of a Hansen’s disease (HD) affected nerve using FEM based simulation tool. There exists an exponential delay in the propagation of action potential for a diseased nerve as compared to a healthy one. To include this exponential delay, the FitzHugh Nagumo equation has been modified and modeled using COMSOL Multiphysics 4.0a. Time domain and frequency domain anal...

A continuation approach to the computation of essential and absolute spectra of reaction-diffusion operators on the real line is presented. The advantage of this approach compared with direct eigenvalue computations for the discretized operator are the efficient and accurate computation of selected parts of the spectrum (typically those near the imaginary axis) and the option to compute nonline...

The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model are studied using a phase dynamics approach. A Cross-Newell phase equation describing slow and weak modulations of periodic stationary solutions is derived. The derivation applies to the bistable, excitable, and Turing unstable regimes. In the bistable case stability thresholds are obtained for the Eckhaus and zigzag ins...

We study the rotating wave patterns in an excitable medium in a disk. This wave pattern is rotating along the given disk boundary with a constant angular speed. To study this pattern we use the wave front interaction model proposed by Zykov in 2007. This model is derived from the FitzHugh-Nagumo equation and it can be described by two systems of ordinary differential equations for wave front an...

Populations of uncoupled limit-cycle oscillators receiving common random impulses show various types of phase-coherent states, which are characterized by the distribution of phase differences between pairs of oscillators. We develop a theory to predict the stationary distribution of pairwise phase differences from the phase response curve, which quantitatively encapsulates the oscillator dynami...

We have studied the synchronization in finite N -unit FitzHugh-Nagumo neuron ensembles subjected to additive and multiplicative noises, by using the augmented moment method (AMM) which is reformulated with the use of the Fokker-Planck equation. It has been shown that for diffusive couplings, the synchronization may be enhanced by multiplicative noises while additive noises are detrimental to th...

In this paper, a new alternating direction implicit Galerkin–Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank–Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived ...