Change in criticality of synchronous Hopf bifurcation in a multiple-delayed neural system

Dynamics of an Inverted Pendulum with Delayed Feedback Control

“…It has been shown in [19,28] that when the lowest order terms in the nonlinearity of the delay equation are of third order (i.e., of the same order as the lowest order terms in the normal form of the Hopf bifurcation), the dynamical system of the center manifold is determined to third order by projecting the delay differential equation onto the subspace, P , of solutions of (4.4) corresponding to the eigenvalues λ = ±iω c . In fact, it can be shown that the dynamical system, accurate to O(W 3 ), is [19,28] …”

Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system

“…See e.g. [36] where this is done for the standard Hopf bifurcation and [53] where it is done for the equivariant Hopf.…”

“…This work was based on symmetric Hopf bifurcation theory in [51] and the normal form theory for DDEs developed in [18], and it was possible to depict the bifurcation surface and to describe the stability of phase-locked periodic solutions since the calculation of the normal form up to the 5th order was still feasible as only one time delay is involved. In the case where there are two distinct delays, the symbolic calculation of the normal form, even up to the third order in the case of single Hopf bifurcation of (synchronized) periodic solutions, is a formidable challenge, as shown in [36]. This motivates us to use singular perturbation techniques as an alternative approach.…”

Patterns of Oscillation in a Ring of Identical Cells With Delayed Coupling

“…a system which has the symmetries of a polygon with n sides of equal length. Most of these studies have concerned lower dimensional systems (e.g [Campbell et al, 2006;Guo et al, 2004;Ncube et al, 2003;Shayer and Campbell, 2000;Wu et al, 1999]) and/or systems with a single time delay [Guo, 2005;Guo and Huang, 2003, 2005Wu, 1998;Wu et al, 1999]. In previous work [Yuan and Campbell, 2004;Campbell et al, 2005] we studied the stability and bifurcations (both standard and equivariant) of the trivial solution for a ring of arbitrary size with two time delays.…”

“…Centre manifold analysis was used to determine the local stability of the synchronous oscillations in Ncube et al [2003]; Yuan and Campbell [2004] and of all branches of asynchronous oscillations in Campbell et al [2005]. Perturbation analysis was used to determine the local stability of the phase-locked oscillations in Campbell et al [2006].…”