1,254 research outputs found
Knotted Strange Attractors and Matrix Lorenz Systems
A generalization of the Lorenz equations is proposed where the variables take
values in a Lie algebra. The finite dimensionality of the representation
encodes the quantum fluctuations, while the non-linear nature of the equations
can describe chaotic fluctuations. We identify a criterion, for the appearance
of such non-linear terms. This depends on whether an invariant, symmetric
tensor of the algebra can vanish or not. This proposal is studied in detail for
the fundamental representation of . We find a knotted
structure for the attractor, a bimodal distribution for the largest Lyapunov
exponent and that the dynamics takes place within the Cartan subalgebra, that
does not contain only the identity matrix, thereby can describe the quantum
fluctuations.Comment: 10 pages Revtex, 3 figure
Chapman-Enskog expansion about nonequilibrium states: the sheared granular fluid
The Chapman-Enskog method of solution of kinetic equations, such as the
Boltzmann equation, is based on an expansion in gradients of the deviations fo
the hydrodynamic fields from a uniform reference state (e.g., local
equilibrium). This paper presents an extension of the method so as to allow for
expansions about \emph{arbitrary}, far-from equilibrium reference states. The
primary result is a set of hydrodynamic equations for studying variations from
the arbitrary reference state which, unlike the usual Navier-Stokes
hydrodynamics, does not restrict the reference state in any way. The method is
illustrated by application to a sheared granular gas which cannot be studied
using the usual Navier-Stokes hydrodynamics.Comment: 23 pages, no figures. Submited to PRE Replaced to correct misc.
errors Replaced to correct misc. errors, make notation more consistant,
extend discussio
Oscillators and relaxation phenomena in Pleistocene climate theory
Ice sheets appeared in the northern hemisphere around 3 million years ago and
glacial-interglacial cycles have paced Earth's climate since then. Superimposed
on these long glacial cycles comes an intricate pattern of millennial and
sub-millennial variability, including Dansgaard-Oeschger and Heinrich events.
There are numerous theories about theses oscillations. Here, we review a number
of them in order to draw a parallel between climatic concepts and dynamical
system concepts, including, in particular, the relaxation oscillator,
excitability, slow-fast dynamics and homoclinic orbits. Namely, almost all
theories of ice ages reviewed here feature a phenomenon of synchronisation
between internal climate dynamics and the astronomical forcing. However, these
theories differ in their bifurcation structure and this has an effect on the
way the ice age phenomenon could grow 3 million years ago. All theories on
rapid events reviewed here rely on the concept of a limit cycle in the ocean
circulation, which may be excited by changes in the surface freshwater surface
balance. The article also reviews basic effects of stochastic fluctuations on
these models, including the phenomenon of phase dispersion, shortening of the
limit cycle and stochastic resonance. It concludes with a more personal
statement about the potential for inference with simple stochastic dynamical
systems in palaeoclimate science.
Keywords: palaeoclimates, dynamical systems, limit cycle, ice ages,
Dansgaard-Oeschger eventsComment: Published in the Transactions of the Philosophical Transactions of
the Royal Society (Series A, Physical Mathematical and Engineering Sciences),
as a contribution to the Proceedings of the workshop on Stochastic Methods in
Climate Modelling, Newton Institute (23-27 August). Philosophical
Transactions of the Royal Society (Series A, Physical Mathematical and
Engineering Sciences), vol. 370, pp. xx-xx (2012); Source codes available on
request to author and on http://www.uclouvain.be/ito
Ratio control in a cascade model of cell differentiation
We propose a kind of reaction-diffusion equations for cell differentiation,
which exhibits the Turing instability. If the diffusivity of some variables is
set to be infinity, we get coupled competitive reaction-diffusion equations
with a global feedback term. The size ratio of each cell type is controlled by
a system parameter in the model. Finally, we extend the model to a cascade
model of cell differentiation. A hierarchical spatial structure appears as a
result of the cell differentiation. The size ratio of each cell type is also
controlled by the system parameter.Comment: 13 pages, 7 figure
Modes of Growth in Dynamic Systems
Regardless of a system's complexity or scale, its growth can be considered to
be a spontaneous thermodynamic response to a local convergence of down-gradient
material flows. Here it is shown how growth can be constrained to a few
distinct modes that depend on the availability of material and energetic
resources. These modes include a law of diminishing returns, logistic behavior
and, if resources are expanding very rapidly, super-exponential growth. For a
case where a system has a resolved sink as well as a source, growth and decay
can be characterized in terms of a slightly modified form of the predator-prey
equations commonly employed in ecology, where the perturbation formulation of
these equations is equivalent to a damped simple harmonic oscillator. Thus, the
framework presented here suggests a common theoretical under-pinning for
emergent behaviors in the physical and life sciences. Specific examples are
described for phenomena as seemingly dissimilar as the development of rain and
the evolution of fish stocks.Comment: 16 pages, 6 figures, including appendi
Nonequilibrium stochastic processes: Time dependence of entropy flux and entropy production
Based on the Fokker-Planck and the entropy balance equations we have studied
the relaxation of a dissipative dynamical system driven by external
Ornstein-Uhlenbeck noise processes in absence and presence of nonequilibrium
constraint in terms of the thermodynamically inspired quantities like entropy
flux and entropy production. The interplay of nonequilibrium constraint,
dissipation and noise reveals some interesting extremal nature in the time
dependence of entropy flux and entropy production.Comment: RevTex, 17 pages, 9 figures. To appear in Phys. Rev.
Chiral Defect Fermions and the Layered Phase
Chiral defect fermions on the lattice in 4+1 dimensions are analyzed using
mean field theory. The fermion propagator has a localized chiral mode in weak
coupling but loses it when the coupling in the unphysical fifth direction
becomes too large. A layered phase \`a la Fu-Nielsen appears where the theory
is vector-like in every layer.Comment: 9 pages + postscript figure (We have correctly quoted the work of our
reference 10 in the paper
Dynamical estimates of chaotic systems from Poincar\'e recurrences
We show that the probability distribution function that best fits the
distribution of return times between two consecutive visits of a chaotic
trajectory to finite size regions in phase space deviates from the exponential
statistics by a small power-law term, a term that represents the deterministic
manifestation of the dynamics, which can be easily experimentally detected and
theoretically estimated. We also provide simpler and faster ways to calculate
the positive Lyapunov exponents and the short-term correlation function by
either realizing observations of higher probable returns or by calculating the
eigenvalues of only one very especial unstable periodic orbit of low-period.
Finally, we discuss how our approaches can be used to treat data coming from
complex systems.Comment: subm. for publication. Accepted fpr publication in Chao
Magic number 7 2 in networks of threshold dynamics
Information processing by random feed-forward networks consisting of units
with sigmoidal input-output response is studied by focusing on the dependence
of its outputs on the number of parallel paths M. It is found that the system
leads to a combination of on/off outputs when , while for , chaotic dynamics arises, resulting in a continuous distribution of
outputs. This universality of the critical number is explained by
combinatorial explosion, i.e., dominance of factorial over exponential
increase. Relevance of the result to the psychological magic number
is briefly discussed.Comment: 6 pages, 5 figure
Thermodynamics of Chemical Waves
Chemical waves constitute a known class of dissipative structures emerging in
reaction-diffusion systems. They play a crucial role in biology, spreading
information rapidly to synchronize and coordinate biological events. We develop
a rigorous thermodynamic theory of reaction-diffusion systems to characterize
chemical waves. Our main result is the definition of the proper thermodynamic
potential of the local dynamics as a nonequilibrium free energy density and
establishing its balance equation. This enables us to identify the dynamics of
the free energy, of the dissipation, and of the work spent to sustain the wave
propagation. Two prototypical classes of chemical waves are examined. From a
thermodynamic perspective, the first is sustained by relaxation towards
equilibrium and the second by nonconservative forces generated by chemostats.
We analytically study step-like waves, called wavefronts, using the
Fisher-Kolmogorov equation as representative of the first class and oscillating
waves in the Brusselator model as representative of the second. Given the
fundamental role of chemical waves as message carriers in biosystems, our
thermodynamic theory constitutes an important step toward an understanding of
information transfers and processing in biology.Comment: 12 pages, 2 figure
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