6,879 research outputs found
Complex oscillations in the delayed Fitzhugh-Nagumo equation
Motivated by the dynamics of neuronal responses, we analyze the dynamics of
the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system
provides a canonical example of a canard explosion for sufficiently small
delays. Beyond this regime, delays significantly enrich the dynamics, leading
to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a
delay-induced subcritical Bogdanov-Takens instability arising at the fold
points of the S-shaped critical manifold. Underlying the transition from
canard-induced to delay-induced dynamics is an abrupt switch in the nature of
the Hopf bifurcation
Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling
Small lattices of nearest neighbor coupled excitable FitzHugh-Nagumo
systems, with time-delayed coupling are studied, and compared with systems of
FitzHugh-Nagumo oscillators with the same delayed coupling. Bifurcations of
equilibria in N=2 case are studied analytically, and it is then numerically
confirmed that the same bifurcations are relevant for the dynamics in the case
. Bifurcations found include inverse and direct Hopf and fold limit cycle
bifurcations. Typical dynamics for different small time-lags and coupling
intensities could be excitable with a single globally stable equilibrium,
asymptotic oscillatory with symmetric limit cycle, bi-stable with stable
equilibrium and a symmetric limit cycle, and again coherent oscillatory but
non-symmetric and phase-shifted. For an intermediate range of time-lags inverse
sub-critical Hopf and fold limit cycle bifurcations lead to the phenomenon of
oscillator death. The phenomenon does not occur in the case of FitzHugh-Nagumo
oscillators with the same type of coupling.Comment: accepted by Phys.Rev.
Control of Dynamic Hopf Bifurcations
The slow passage through a Hopf bifurcation leads to the delayed appearance
of large amplitude oscillations. We construct a smooth scalar feedback control
which suppresses the delay and causes the system to follow a stable equilibrium
branch. This feature can be used to detect in time the loss of stability of an
ageing device. As a by-product, we obtain results on the slow passage through a
bifurcation with double zero eigenvalue, described by a singularly perturbed
cubic Lienard equation.Comment: 25 pages, 4 figure
Canard explosion in delayed equations with multiple timescales
We analyze canard explosions in delayed differential equations with a
one-dimensional slow manifold. This study is applied to explore the dynamics of
the van der Pol slow-fast system with delayed self-coupling. In the absence of
delays, this system provides a canonical example of a canard explosion. We show
that as the delay is increased a family of `classical' canard explosions ends
as a Bogdanov-Takens bifurcation occurs at the folds points of the S-shaped
critical manifold.Comment: arXiv admin note: substantial text overlap with arXiv:1404.584
Hydro-dynamical models for the chaotic dripping faucet
We give a hydrodynamical explanation for the chaotic behaviour of a dripping
faucet using the results of the stability analysis of a static pendant drop and
a proper orthogonal decomposition (POD) of the complete dynamics. We find that
the only relevant modes are the two classical normal forms associated with a
Saddle-Node-Andronov bifurcation and a Shilnikov homoclinic bifurcation. This
allows us to construct a hierarchy of reduced order models including maps and
ordinary differential equations which are able to qualitatively explain prior
experiments and numerical simulations of the governing partial differential
equations and provide an explanation for the complexity in dripping. We also
provide a new mechanical analogue for the dripping faucet and a simple
rationale for the transition from dripping to jetting modes in the flow from a
faucet.Comment: 16 pages, 14 figures. Under review for Journal of Fluid Mechanic
Negative-coupling resonances in pump-coupled lasers
We consider coupled lasers, where the intensity deviations from the steady
state, modulate the pump of the other lasers. Most of our results are for two
lasers where the coupling constants are of opposite sign. This leads to a Hopf
bifurcation to periodic output for weak coupling. As the magnitude of the
coupling constants is increased (negatively) we observe novel amplitude effects
such as a weak coupling resonance peak and, strong coupling subharmonic
resonances and chaos. In the weak coupling regime the output is predicted by a
set of slow evolution amplitude equations. Pulsating solutions in the strong
coupling limit are described by discrete map derived from the original model.Comment: 29 pages with 8 figures Physica D, in pres
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