30,368 research outputs found
A delay differential model of ENSO variability: Parametric instability and the distribution of extremes
We consider a delay differential equation (DDE) model for El-Nino Southern
Oscillation (ENSO) variability. The model combines two key mechanisms that
participate in ENSO dynamics: delayed negative feedback and seasonal forcing.
We perform stability analyses of the model in the three-dimensional space of
its physically relevant parameters. Our results illustrate the role of these
three parameters: strength of seasonal forcing , atmosphere-ocean coupling
, and propagation period of oceanic waves across the Tropical
Pacific. Two regimes of variability, stable and unstable, are separated by a
sharp neutral curve in the plane at constant . The detailed
structure of the neutral curve becomes very irregular and possibly fractal,
while individual trajectories within the unstable region become highly complex
and possibly chaotic, as the atmosphere-ocean coupling increases. In
the unstable regime, spontaneous transitions occur in the mean ``temperature''
({\it i.e.}, thermocline depth), period, and extreme annual values, for purely
periodic, seasonal forcing. The model reproduces the Devil's bleachers
characterizing other ENSO models, such as nonlinear, coupled systems of partial
differential equations; some of the features of this behavior have been
documented in general circulation models, as well as in observations. We
expect, therefore, similar behavior in much more detailed and realistic models,
where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure
Asymptotic properties of the spectrum of neutral delay differential equations
Spectral properties and transition to instability in neutral delay
differential equations are investigated in the limit of large delay. An
approximation of the upper boundary of stability is found and compared to an
analytically derived exact stability boundary. The approximate and exact
stability borders agree quite well for the large time delay, and the inclusion
of a time-delayed velocity feedback improves this agreement for small delays.
Theoretical results are complemented by a numerically computed spectrum of the
corresponding characteristic equations.Comment: 14 pages, 6 figure
Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control
For many years it was believed that an unstable periodic orbit with an odd
number of real Floquet multipliers greater than unity cannot be stabilized by
the time-delayed feedback control mechanism of Pyragus. A recent paper by
Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a
counterexample to this theorem. Using the Lorenz equations as an example, we
demonstrate that the stabilization mechanism identified by Fiedler et al for
the Hopf normal form can also apply to unstable periodic orbits created by
subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our
analysis focuses on a particular codimension-two bifurcation that captures the
stabilization mechanism in the Hopf normal form example, and we show that the
same codimension-two bifurcation is present in the Lorenz equations with
appropriately chosen Pyragus-type time-delayed feedback. This example suggests
a possible strategy for choosing the feedback gain matrix in Pyragus control of
unstable periodic orbits that arise from a subcritical Hopf bifurcation of a
stable equilibrium. In particular, our choice of feedback gain matrix is
informed by the Fiedler et al example, and it works over a broad range of
parameters, despite the fact that a center-manifold reduction of the
higher-dimensional problem does not lead to their model problem.Comment: 21 pages, 8 figures, to appear in PR
Optimal linear stability condition for scalar differential equations with distributed delay
Linear scalar differential equations with distributed delays appear in the
study of the local stability of nonlinear differential equations with feedback,
which are common in biology and physics. Negative feedback loops tend to
promote oscillations around steady states, and their stability depends on the
particular shape of the delay distribution. Since in applications the mean
delay is often the only reliable information available about the distribution,
it is desirable to find conditions for stability that are independent from the
shape of the distribution. We show here that for a given mean delay, the linear
equation with distributed delay is asymptotically stable if the associated
differential equation with a discrete delay is asymptotically stable. We
illustrate this criterion on a compartment model of hematopoietic cell dynamics
to obtain sufficient conditions for stability
Delay-induced patterns in a two-dimensional lattice of coupled oscillators
We show how a variety of stable spatio-temporal periodic patterns can be
created in 2D-lattices of coupled oscillators with non-homogeneous coupling
delays. A "hybrid dispersion relation" is introduced, which allows studying the
stability of time-periodic patterns analytically in the limit of large delay.
The results are illustrated using the FitzHugh-Nagumo coupled neurons as well
as coupled limit cycle (Stuart-Landau) oscillators
Bifurcation structure of cavity soliton dynamics in a VCSEL with saturable absorber and time-delayed feedback
We consider a wide-aperture surface-emitting laser with a saturable absorber
section subjected to time-delayed feedback. We adopt the mean-field approach
assuming a single longitudinal mode operation of the solitary VCSEL. We
investigate cavity soliton dynamics under the effect of time- delayed feedback
in a self-imaging configuration where diffraction in the external cavity is
negligible. Using bifurcation analysis, direct numerical simulations and
numerical path continuation methods, we identify the possible bifurcations and
map them in a plane of feedback parameters. We show that for both the
homogeneous and localized stationary lasing solutions in one spatial dimension
the time-delayed feedback induces complex spatiotemporal dynamics, in
particular a period doubling route to chaos, quasiperiodic oscillations and
multistability of the stationary solutions
Control of unstable steady states in neutral time-delayed systems
We present an analysis of time-delayed feedback control used to stabilize an
unstable steady state of a neutral delay differential equation. Stability of
the controlled system is addressed by studying the eigenvalue spectrum of a
corresponding characteristic equation with two time delays. An analytic
expression for the stabilizing control strength is derived in terms of original
system parameters and the time delay of the control. Theoretical and numerical
results show that the interplay between the control strength and two time
delays provides a number of regions in the parameter space where the
time-delayed feedback control can successfully stabilize an otherwise unstable
steady state.Comment: 11 pages, 8 figure
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