We investigate the behaviour of a neural network model consisting of three neurons with delayed self and nearest-neighbour connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D3-equivariant, Hopf and pitchfork bifurcation points. We use ...

We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. We describe the case of the collision of two stable cycles of any period and any symbolic sequen...

We consider a bifurcation that occurs in some two-dimensional vector fields, namely a codimension-one bifurcation in which there is simultaneously the creation of a pair of equilibria via a steady state bifurcation and the destruction of a large amplitude periodic orbit. We show that this phenomenon may occur in an unfolding of the saddle-node/pitchfork normal form equations, although not near ...

In this paper, a bifurcation phenomenon which arises from the interaction of static and dynamic modes in the vicinity of a compound critical point of a nonlinear autonomous system is considered. The critical point is characterized by a double zero and a pair of pure imaginary eigenvalues of the Jacobian. A set of simplified differential equations is obtained by applying the unification techniqu...

Layek and Pati (Phys. Lett. A, 2017) studied a nonlinear system of five coupled equations, which describe thermal relaxation in Rayleigh-Benard convection Boussinesq fluid layer, heated from below. Here we return to that paper use techniques dynamical systems theory analyse the codimension-one Hopf bifurcation codimension-two double-zero Bogdanov-Takens bifurcation....

In this paper, we consider analytically sliding bifurcations of periodic orbits in the dry friction oscillator. The system depends on two parameters; F , which corresponds to the intensity of the friction and ω, the frequency of the forcing. We prove the existence of infinitely many codimension-2 bifurcation points and we focus our attention on two of them; A1 := (ω , F ) = (2, 1/3) and B1 := (...

The complex, so called meandering, dynamics of spiral waves in excitable media is examined from the point of view of bifurcation theory. A computational bifurcation analysis is made of spiral dynamics. It is shown that spiral meandering is organized in parameter space around a codimension-two point where a Hopf bifurcation from rotating waves interacts with symmetries on the plane. A simple mod...

Hyperpolarization-activated cyclic nucleotide-gated cation current (Ih ) plays important roles in the achievement of many physiological/pathological functions in the nervous system by modulating the electrophysiological activities, such as the rebound (spike) to hyperpolarization stimulations, subthreshold membrane resonance to sinusoidal currents, and spike-timing precision to stochastic facto...

We contribute to the analysis of codimension-two bifurcations in discontinuous systems by studying all equilibrium bifurcations of 2D Filippov systems that involve a sliding limit cycle. There are only two such local bifurcations: (1) a degenerate boundary focus, which we call the homoclinic boundary focus (HBF), and (2) the boundary Hopf (BH). We prove that—besides local bifurcations of equili...

This paper presents a study of the effects of symmetry on the generic bifurcation at a double-zero eigenvalue that was first investigated by Bogdanov and Takens. Two different symmetry groups are considered: Huygens symmetry and odd-Huygens symmetry. Here Huygens symmetry means that the system is equivariant under permutation of the two state variables. Using Hilbert-Weyl theory, normal forms a...