14,895 research outputs found
Boolean Delay Equations: A simple way of looking at complex systems
Boolean Delay Equations (BDEs) are semi-discrete dynamical models with
Boolean-valued variables that evolve in continuous time. Systems of BDEs can be
classified into conservative or dissipative, in a manner that parallels the
classification of ordinary or partial differential equations. Solutions to
certain conservative BDEs exhibit growth of complexity in time. They represent
therewith metaphors for biological evolution or human history. Dissipative BDEs
are structurally stable and exhibit multiple equilibria and limit cycles, as
well as more complex, fractal solution sets, such as Devil's staircases and
``fractal sunbursts``. All known solutions of dissipative BDEs have stationary
variance. BDE systems of this type, both free and forced, have been used as
highly idealized models of climate change on interannual, interdecadal and
paleoclimatic time scales. BDEs are also being used as flexible, highly
efficient models of colliding cascades in earthquake modeling and prediction,
as well as in genetics. In this paper we review the theory of systems of BDEs
and illustrate their applications to climatic and solid earth problems. The
former have used small systems of BDEs, while the latter have used large
networks of BDEs. We moreover introduce BDEs with an infinite number of
variables distributed in space (``partial BDEs``) and discuss connections with
other types of dynamical systems, including cellular automata and Boolean
networks. This research-and-review paper concludes with a set of open
questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular
the discussion on partial BDEs is updated and enlarge
Functional Inequalities: New Perspectives and New Applications
This book is not meant to be another compendium of select inequalities, nor
does it claim to contain the latest or the slickest ways of proving them. This
project is rather an attempt at describing how most functional inequalities are
not merely the byproduct of ingenious guess work by a few wizards among us, but
are often manifestations of certain natural mathematical structures and
physical phenomena. Our main goal here is to show how this point of view leads
to "systematic" approaches for not just proving the most basic functional
inequalities, but also for understanding and improving them, and for devising
new ones - sometimes at will, and often on demand.Comment: 17 pages; contact Nassif Ghoussoub (nassif @ math.ubc.ca) for a
pre-publication pdf cop
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
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
Phase reduction approach to synchronization of spatiotemporal rhythms in reaction-diffusion systems
Reaction-diffusion systems can describe a wide class of rhythmic
spatiotemporal patterns observed in chemical and biological systems, such as
circulating pulses on a ring, oscillating spots, target waves, and rotating
spirals. These rhythmic dynamics can be considered limit cycles of
reaction-diffusion systems. However, the conventional phase-reduction theory,
which provides a simple unified framework for analyzing synchronization
properties of limit-cycle oscillators subjected to weak forcing, has mostly
been restricted to low-dimensional dynamical systems. Here, we develop a
phase-reduction theory for stable limit-cycle solutions of infinite-dimensional
reaction-diffusion systems. By generalizing the notion of isochrons to
functional space, the phase sensitivity function - a fundamental quantity for
phase reduction - is derived. For illustration, several rhythmic dynamics of
the FitzHugh-Nagumo model of excitable media are considered. Nontrivial phase
response properties and synchronization dynamics are revealed, reflecting their
complex spatiotemporal organization. Our theory will provide a general basis
for the analysis and control of spatiotemporal rhythms in various
reaction-diffusion systems.Comment: 19 pages, 6 figures, see the journal for a full versio
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