668 research outputs found
Qualitative dynamics and inflationary attractors in loop cosmology
Qualitative dynamics of three different loop quantizations of spatially flat
isotropic and homogeneous models is studied using effective spacetime
description of the underlying quantum geometry. These include the standard loop
quantum cosmology (LQC), its recently revived modification (referred to as
mLQC-I), and another related modification of LQC (mLQC-II) whose dynamics is
studied in detail for the first time. Various features of LQC, including
quantum bounce and pre-inflationary dynamics, are found to be shared with the
mLQC-I and mLQC-II models. We study universal properties of dynamics for
chaotic inflation, fractional monodromy inflation, Starobinsky potential,
non-minimal Higgs inflation, and an exponential potential. We find various
critical points and study their stability, which reveal various qualitative
similarities in the post-bounce phase for all these models. The pre-bounce
qualitative dynamics of LQC and mLQC-II turns out to be very similar, but is
strikingly different from that of mLQC-I. In the dynamical analysis, some of
the fixed points turn out to be degenerate for which center manifold theory is
used. For all these potentials, non-perturbative quantum gravitational effects
always result in a non-singular inflationary scenario with a phase of
super-inflation succeeded by the conventional inflation. We show the existence
of inflationary attractors, and obtain scaling solutions in the case of the
exponential potential. Since all of the models agree with general relativity at
late times, our results are also of use in classical theory where qualitative
dynamics of some of the potentials has not been studied earlier.Comment: 29 pages, 18 figures. Minor changes. To appear in Phys. Rev.
Nonintegrability, Chaos, and Complexity
Two-dimensional driven dissipative flows are generally integrable via a
conservation law that is singular at equilibria. Nonintegrable dynamical
systems are confined to n*3 dimensions. Even driven-dissipative deterministic
dynamical systems that are critical, chaotic or complex have n-1 local
time-independent conservation laws that can be used to simplify the geometric
picture of the flow over as many consecutive time intervals as one likes. Those
conserevation laws generally have either branch cuts, phase singularities, or
both. The consequence of the existence of singular conservation laws for
experimental data analysis, and also for the search for scale-invariant
critical states via uncontrolled approximations in deterministic dynamical
systems, is discussed. Finally, the expectation of ubiquity of scaling laws and
universality classes in dynamics is contrasted with the possibility that the
most interesting dynamics in nature may be nonscaling, nonuniversal, and to
some degree computationally complex
Global adaptation in networks of selfish components: emergent associative memory at the system scale
In some circumstances complex adaptive systems composed of numerous self-interested agents can self-organise into structures that enhance global adaptation, efficiency or function. However, the general conditions for such an outcome are poorly understood and present a fundamental open question for domains as varied as ecology, sociology, economics, organismic biology and technological infrastructure design. In contrast, sufficient conditions for artificial neural networks to form structures that perform collective computational processes such as associative memory/recall, classification, generalisation and optimisation, are well-understood. Such global functions within a single agent or organism are not wholly surprising since the mechanisms (e.g. Hebbian learning) that create these neural organisations may be selected for this purpose, but agents in a multi-agent system have no obvious reason to adhere to such a structuring protocol or produce such global behaviours when acting from individual self-interest. However, Hebbian learning is actually a very simple and fully-distributed habituation or positive feedback principle. Here we show that when self-interested agents can modify how they are affected by other agents (e.g. when they can influence which other agents they interact with) then, in adapting these inter-agent relationships to maximise their own utility, they will necessarily alter them in a manner homologous with Hebbian learning. Multi-agent systems with adaptable relationships will thereby exhibit the same system-level behaviours as neural networks under Hebbian learning. For example, improved global efficiency in multi-agent systems can be explained by the inherent ability of associative memory to generalise by idealising stored patterns and/or creating new combinations of sub-patterns. Thus distributed multi-agent systems can spontaneously exhibit adaptive global behaviours in the same sense, and by the same mechanism, as the organisational principles familiar in connectionist models of organismic learning
On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular
The main part of this contribution to the special issue of EJM-B/Fluids
dedicated to Patrick Huerre outlines the problem of the subcritical transition
to turbulence in wall-bounded flows in its historical perspective with emphasis
on plane Couette flow, the flow generated between counter-translating parallel
planes. Subcritical here means discontinuous and direct, with strong
hysteresis. This is due to the existence of nontrivial flow regimes between the
global stability threshold Re_g, the upper bound for unconditional return to
the base flow, and the linear instability threshold Re_c characterized by
unconditional departure from the base flow. The transitional range around Re_g
is first discussed from an empirical viewpoint ({\S}1). The recent
determination of Re_g for pipe flow by Avila et al. (2011) is recalled. Plane
Couette flow is next examined. In laboratory conditions, its transitional range
displays an oblique pattern made of alternately laminar and turbulent bands, up
to a third threshold Re_t beyond which turbulence is uniform. Our current
theoretical understanding of the problem is next reviewed ({\S}2): linear
theory and non-normal amplification of perturbations; nonlinear approaches and
dynamical systems, basin boundaries and chaotic transients in minimal flow
units; spatiotemporal chaos in extended systems and the use of concepts from
statistical physics, spatiotemporal intermittency and directed percolation,
large deviations and extreme values. Two appendices present some recent
personal results obtained in plane Couette flow about patterning from numerical
simulations and modeling attempts.Comment: 35 pages, 7 figures, to appear in Eur. J. Mech B/Fluid
Unstable Attractors: Existence and Robustness in Networks of Oscillators With Delayed Pulse Coupling
We consider unstable attractors; Milnor attractors such that, for some
neighbourhood of , almost all initial conditions leave . Previous
research strongly suggests that unstable attractors exist and even occur
robustly (i.e. for open sets of parameter values) in a system modelling
biological phenomena, namely in globally coupled oscillators with delayed pulse
interactions.
In the first part of this paper we give a rigorous definition of unstable
attractors for general dynamical systems. We classify unstable attractors into
two types, depending on whether or not there is a neighbourhood of the
attractor that intersects the basin in a set of positive measure. We give
examples of both types of unstable attractor; these examples have
non-invertible dynamics that collapse certain open sets onto stable manifolds
of saddle orbits.
In the second part we give the first rigorous demonstration of existence and
robust occurrence of unstable attractors in a network of oscillators with
delayed pulse coupling. Although such systems are technically hybrid systems of
delay differential equations with discontinuous `firing' events, we show that
their dynamics reduces to a finite dimensional hybrid system system after a
finite time and hence we can discuss Milnor attractors for this reduced finite
dimensional system. We prove that for an open set of phase resetting functions
there are saddle periodic orbits that are unstable attractors.Comment: 29 pages, 8 figures,submitted to Nonlinearit
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