41,610 research outputs found
Transient spatiotemporal chaos on complex networks
Thesis (M.S.) University of Alaska Fairbanks, 2004Some of today's most important questions regard complex dynamical systems with many interacting components. Network models provide a means to gain insight into such systems. This thesis focuses on a network model based upon the Gray-Scott cubic autocatalytic reaction-diffusion system that manifests transient spatiotemporal chaos. Motivated by recent studies on the small-world topology discovered by Watts and Strogatz, the network's original regular ring topology was modified by the addition of a few irregular connections. The effects of these added connections on the system's transience as well on the dynamics local to the added connections were examined. It was found that the addition of a single connection can significantly effect the transient time of spatiotemporal chaos and that the addition of two connections can transform the system's spatiotemporal chaos from transient to asymptotic. These findings suggest that small modifications to a network's topology can greatly affect its behavior.Introduction -- Background -- Transient spatiotemporal chaos in the presence of one shortcut -- Local dynamics in the vicinity of a shortcut -- Conclusion -- Bibliography
Relation of Origins of Primitive Chaos
A new concept, primitive chaos, was proposed, as a concept closely related to
the fundamental problems of sciences themselves such as determinism, causality,
free will, predictability, and time asymmetry [{\em J. Phys. Soc. Jpn.} {\bf
2014}, {\em 83}, 1401]. This concept is literally a primitive chaos in such a
sense that it leads to the characteristic properties of the conventional chaos
under natural conditions. Then, two contrast concepts, nondegenerate Peano
continuum and Cantor set, are known as the origins of the primitive chaos. In
this study, the relation of these origins is investigated with the aid of a
mathematical method, topology. Then, we can see the emergence of interesting
concepts such as the relation of whole and part, and coarse graining, which
imply the essence of our intrinsic recognition for phenomena
Algorithms for recognizing knots and 3-manifolds
This is a survey paper on algorithms for solving problems in 3-dimensional
topology. In particular, it discusses Haken's approach to the recognition of
the unknot, and recent variations.Comment: 17 Pages, 7 figures, to appear in Chaos, Fractals and Soliton
Curvature Fields, Topology, and the Dynamics of Spatiotemporal Chaos
The curvature field is measured from tracer particle trajectories in a
two-dimensional fluid flow that exhibits spatiotemporal chaos, and is used to
extract the hyperbolic and elliptic points of the flow. These special points
are pinned to the forcing when the driving is weak, but wander over the domain
and interact in pairs at stronger driving, changing the local topology of the
flow. Their behavior reveals a two-stage transition to spatiotemporal chaos: a
gradual loss of spatial and temporal order followed by an abrupt onset of
topological changes.Comment: 5 pages, 5 figure
Homology and symmetry breaking in Rayleigh-Benard convection: Experiments and simulations
Algebraic topology (homology) is used to analyze the weakly turbulent state
of spiral defect chaos in both laboratory experiments and numerical simulations
of Rayleigh-Benard convection.The analysis reveals topological asymmetries that
arise when non-Boussinesq effects are present.Comment: 21 pages with 6 figure
Dynamical topology and statistical properties of spatiotemporal chaos
For spatiotemporal chaos described by partial differential equations, there
are generally locations where the dynamical variable achieves its local
extremum or where the time partial derivative of the variable vanishes
instantaneously. To a large extent, the location and movement of these
topologically special points determine the qualitative structure of the
disordered states. We analyze numerically statistical properties of the
topologically special points in one-dimensional spatiotemporal chaos. The
probability distribution functions for the number of point, the lifespan, and
the distance covered during their lifetime are obtained from numerical
simulations. Mathematically, we establish a probabilistic model to describe the
dynamics of these topologically special points. In despite of the different
definitions in different spatiotemporal chaos, the dynamics of these special
points can be described in a uniform approach.Comment: 6 pages, 5 figure
Synchronization in discrete-time networks with general pairwise coupling
We consider complete synchronization of identical maps coupled through a
general interaction function and in a general network topology where the edges
may be directed and may carry both positive and negative weights. We define
mixed transverse exponents and derive sufficient conditions for local complete
synchronization. The general non-diffusive coupling scheme can lead to new
synchronous behavior, in networks of identical units, that cannot be produced
by single units in isolation. In particular, we show that synchronous chaos can
emerge in networks of simple units. Conversely, in networks of chaotic units
simple synchronous dynamics can emerge; that is, chaos can be suppressed
through synchrony
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