Equivariant Hopf Bifurcation in a Ring of Identical Cells with Delay

Abstract

A kind of delay neural network withnelements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. (1988) and delay differential equations due to Wu (1998), and combining the normal form theory of functional differential equations due to Faria and Magalhaes (1995), the equivariant Hopf bifurcation is completely analyzed.

Equivariant Hopf Bifurcation in a Time-Delayed Ring of Antigenic Variants

“…We arrive at the first of our main results. (20) has no positive real roots for ∈ (0,̃], wherẽ> (0) . Since Φ(0) = − 2 < 0, it follows that 0 is asymptotically stable when < (0) and unstable when (0) < < min{̃, (1) }.…”

“…The extension of the theory of equivariant Hopf bifurcation to functional differential equations was established in the series of papers [18,21,22]. Much of the development in this section is in the spirit of the work of [16,19,20] that independently studied equivariant Hopf bifurcation in a ring of identical neurons characterised by signal transmission time delays. Defining the 3 × 1 column vector…”

“…With the apparatus developed above, we are now in a position to establish some lemmas [16] that will lead us to an equivariant Hopf bifurcation theorem. As a consequence of the fact that the matrix (54) is circulant and to facilitate the analysis to follow, we set [16,[18][19][20] V fl (1, , 2 , . .…”

Bifurcation Analysis and Spatiotemporal Patterns of Nonlinear Oscillations in a Ring Lattice of Identical Neurons with Delayed Coupling

“…Later, in a series of papers, Wu and coworkers [20][21][22] have extended the theory of equivariant Hopf bifurcations to functional differential equations. These theoretical advances have led to a vast literature on the mechanisms of spatiotemporal activity in neural networks with symmetry and delays (see, e.g., [22][23][24][25][26][27][28][29][30][31][32][33][34][35]). The majority of these studies have focused on a ring structure with bidirectional couplings between the neighboring elements.…”

Role of delay for the symmetry in the dynamics of networks

“…Equivariant theory in delayed systems has recently been developed and used in applied mathematics; results include a Lie group classification of second-order delay differential equations [19], an analysis of the bifurcation scenario in a ring of one-dimensional identical units [20], and delayed systems with translational [21] and mirror symmetry [22]. In networks of chaotic delayed-coupling elements it is known that different synchronization patterns appear depending on the coupling topology.…”