16 research outputs found
Delay-Induced Transient Oscillations in a Two-Neuron Network
Finite transmission times between neurons, referred to as delays, may appear in hardware implementation of neural networks. We analyze the dynamics of a two-neuron network in which the delay modifies the transient and not the long-term behavior of the network. We show that the delay causes some trajectories to oscillate transiently before reaching stationary behavior and the duration of these transients increases exponentially with the delay. Such a phenomeno deteriorates network performance
Separatrix splitting at a Hamiltonian bifurcation
We discuss the splitting of a separatrix in a generic unfolding of a
degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We
assume that the unperturbed fixed point has two purely imaginary eigenvalues
and a double zero one. It is well known that an one-parametric unfolding of the
corresponding Hamiltonian can be described by an integrable normal form. The
normal form has a normally elliptic invariant manifold of dimension two. On
this manifold, the truncated normal form has a separatrix loop. This loop
shrinks to a point when the unfolding parameter vanishes. Unlike the normal
form, in the original system the stable and unstable trajectories of the
equilibrium do not coincide in general. The splitting of this loop is
exponentially small compared to the small parameter. This phenomenon implies
non-existence of single-round homoclinic orbits and divergence of series in the
normal form theory. We derive an asymptotic expression for the separatrix
splitting. We also discuss relations with behaviour of analytic continuation of
the system in a complex neighbourhood of the equilibrium
Metastable periodic patterns in singularly perturbed delayed equations.
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