27,042 research outputs found
Derivation of Delay Equation Climate Models Using the Mori-Zwanzig Formalism
Models incorporating delay have been frequently used to understand climate
variability phenomena, but often the delay is introduced through an ad-hoc
physical reasoning, such as the propagation time of waves. In this paper, the
Mori-Zwanzig formalism is introduced as a way to systematically derive delay
models from systems of partial differential equations and hence provides a
better justification for using these delay-type models. The Mori-Zwanzig
technique gives a formal rewriting of the system using a projection onto a set
of resolved variables, where the rewritten system contains a memory term. The
computation of this memory term requires solving the orthogonal dynamics
equation, which represents the unresolved dynamics. For nonlinear systems, it
is often not possible to obtain an analytical solution to the orthogonal
dynamics and an approximate solution needs to be found. Here, we demonstrate
the Mori-Zwanzig technique for a two-strip model of the El Nino Southern
Oscillation (ENSO) and explore methods to solve the orthogonal dynamics. The
resulting nonlinear delay model contains an additional term compared to
previously proposed ad-hoc conceptual models. This new term leads to a larger
ENSO period, which is closer to that seen in observations.Comment: Submitted to Proceedings of the Royal Society A, 25 pages, 10 figure
Oscillations and temporal signalling in cells
The development of new techniques to quantitatively measure gene expression
in cells has shed light on a number of systems that display oscillations in
protein concentration. Here we review the different mechanisms which can
produce oscillations in gene expression or protein concentration, using a
framework of simple mathematical models. We focus on three eukaryotic genetic
regulatory networks which show "ultradian" oscillations, with time period of
the order of hours, and involve, respectively, proteins important for
development (Hes1), apoptosis (p53) and immune response (NFkB). We argue that
underlying all three is a common design consisting of a negative feedback loop
with time delay which is responsible for the oscillatory behaviour
Relaxation oscillations, pulses, and travelling waves in the diffusive Volterra delay-differential equation
The diffusive Volterra equation with discrete or continuous delay is studied in the limit of long delays using matched asymptotic expansions. In the case of continuous delay, the procedure was explicitly carried out for general normalized kernels of the form Sigma/sub n=p//sup N/ g/sub n/(t/sup n//T/sup n+1/)e/sup -t/T/, pges2, in the limit in which the strength of the delayed regulation is much greater than that of the instantaneous one, and also for g/sub n/=delta/sub n2/ and any strength ratio. Solutions include homogeneous relaxation oscillations and travelling waves such as pulses, periodic wavetrains, pacemakers and leading centers, so that the diffusive Volterra equation presents the main features of excitable media
Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
A chimera state is a spatio-temporal pattern in a network of identical
coupled oscillators in which synchronous and asynchronous oscillation coexist.
This state of broken symmetry, which usually coexists with a stable spatially
symmetric state, has intrigued the nonlinear dynamics community since its
discovery in the early 2000s. Recent experiments have led to increasing
interest in the origin and dynamics of these states. Here we review the history
of research on chimera states and highlight major advances in understanding
their behaviour.Comment: 26 pages, 3 figure
Hopf bifurcations in time-delay systems with band-limited feedback
We investigate the steady-state solution and its bifurcations in time-delay
systems with band-limited feedback. This is a first step in a rigorous study
concerning the effects of AC-coupled components in nonlinear devices with
time-delayed feedback. We show that the steady state is globally stable for
small feedback gain and that local stability is lost, generically, through a
Hopf bifurcation for larger feedback gain. We provide simple criteria that
determine whether the Hopf bifurcation is supercritical or subcritical based on
the knowledge of the first three terms in the Taylor-expansion of the
nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the
steady state is shown, which indicates possible quasiperiodic and chaotic
dynamics in these systems. As a result of this investigation, we find that
AC-coupling introduces fundamental differences to systems of Ikeda-type [Ikeda
et al., Physica D 29 (1987) 223-235] already at the level of steady-state
bifurcations, e.g. bifurcations exist in which limit cycles are created with
periods other than the fundamental ``period-2'' mode found in Ikeda-type
systems.Comment: 32 pages, 5 figures, accepted for publication in Physica D: Nonlinear
Phenomen
Oscillations in I/O monotone systems under negative feedback
Oscillatory behavior is a key property of many biological systems. The
Small-Gain Theorem (SGT) for input/output monotone systems provides a
sufficient condition for global asymptotic stability of an equilibrium and
hence its violation is a necessary condition for the existence of periodic
solutions. One advantage of the use of the monotone SGT technique is its
robustness with respect to all perturbations that preserve monotonicity and
stability properties of a very low-dimensional (in many interesting examples,
just one-dimensional) model reduction. This robustness makes the technique
useful in the analysis of molecular biological models in which there is large
uncertainty regarding the values of kinetic and other parameters. However,
verifying the conditions needed in order to apply the SGT is not always easy.
This paper provides an approach to the verification of the needed properties,
and illustrates the approach through an application to a classical model of
circadian oscillations, as a nontrivial ``case study,'' and also provides a
theorem in the converse direction of predicting oscillations when the SGT
conditions fail.Comment: Related work can be retrieved from second author's websit
Delayed Feedback Control near Hopf Bifurcation
The stability of functional differential equations under delayed feedback is
investigated near a Hopf bifurcation. Necessary and sufficient conditions are
derived for the stability of the equilibrium solution using averaging theory.
The results are used to compare delayed versus undelayed feedback, as well as
discrete versus distributed delays. Conditions are obtained for which delayed
feedback with partial state information can yield stability where undelayed
feedback is ineffective. Furthermore, it is shown that if the feedback is
stabilizing (respectively, destabilizing), then a discrete delay is locally the
most stabilizing (resp., destabilizing) one among delay distributions having
the same mean. The result also holds globally if one considers delays that are
symmetrically distributed about their mean
- …