462 research outputs found
Computing fractal dimension in supertransient systems directly, fast and reliable
Chaotic transients occur in many experiments including those in fluids, in
simulations of the plane Couette flow, and in coupled map lattices and they are
a common phenomena in dynamical systems. Superlong chaotic transients are
caused by the presence of chaotic saddles whose stable sets have fractal
dimensions that are close to phase-space dimension. For many physical systems
chaotic saddles have a big impact on laboratory measurements, and it is
important to compute the dimension of such stable sets including fractal basin
boundaries through a direct method. In this work, we present a new method to
compute the dimension of stable sets of chaotic saddles directly, fast, and
reliable.Comment: 6 pages, 3 figure
Bifurcation and Chaos in Coupled Ratchets exhibiting Synchronized Dynamics
The bifurcation and chaotic behaviour of unidirectionally coupled
deterministic ratchets is studied as a function of the driving force amplitude
() and frequency (). A classification of the various types of
bifurcations likely to be encountered in this system was done by examining the
stability of the steady state in linear response as well as constructing a
two-parameter phase diagram in the () plane. Numerical explorations
revealed varieties of bifurcation sequences including quasiperiodic route to
chaos. Besides, the familiar period-doubling and crises route to chaos
exhibited by the one-dimensional ratchet were also found. In addition, the
coupled ratchets display symmetry-breaking, saddle-nodes and bubbles of
bifurcations. Chaotic behaviour is characterized by using the sensitivity to
initial condition as well as the Lyapunov exponent spectrum; while a perusal of
the phase space projected in the Poincar cross-section confirms some
of the striking features.Comment: 7 pages; 8 figure
Does an Unstable Keynesian Unemployment Equilibrium in a Non-Walrasian Dynamic Macroeconomic Model Imply Chaos?
Hofstadter butterfly as Quantum phase diagram
The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely
many phases, labelled by their (integer) Hall conductance, and a fractal
structure. We describe various properties of this phase diagram: We establish
Gibbs phase rules; count the number of components of each phase, and
characterize the set of multiple phase coexistence.Comment: 4 prl pages 1 colored figure typos corrected, reference [26] added,
"Ten Martini" assumption adde
Axin2 as regulatory and therapeutic target in newborn brain injury and remyelination.
Permanent damage to white matter tracts, comprising axons and myelinating oligodendrocytes, is an important component of brain injuries of the newborn that cause cerebral palsy and cognitive disabilities, as well as multiple sclerosis in adults. However, regulatory factors relevant in human developmental myelin disorders and in myelin regeneration are unclear. We found that AXIN2 was expressed in immature oligodendrocyte progenitor cells (OLPs) in white matter lesions of human newborns with neonatal hypoxic-ischemic and gliotic brain damage, as well as in active multiple sclerosis lesions in adults. Axin2 is a target of Wnt transcriptional activation that negatively feeds back on the pathway, promoting Ī²-catenin degradation. We found that Axin2 function was essential for normal kinetics of remyelination. The small molecule inhibitor XAV939, which targets the enzymatic activity of tankyrase, acted to stabilize Axin2 levels in OLPs from brain and spinal cord and accelerated their differentiation and myelination after hypoxic and demyelinating injury. Together, these findings indicate that Axin2 is an essential regulator of remyelination and that it might serve as a pharmacological checkpoint in this process
Discrete embedded solitons
We address the existence and properties of discrete embedded solitons (ESs),
i.e., localized waves existing inside the phonon band in a nonlinear
dynamical-lattice model. The model describes a one-dimensional array of optical
waveguides with both the quadratic (second-harmonic generation) and cubic
nonlinearities. A rich family of ESs was previously known in the continuum
limit of the model. First, a simple motivating problem is considered, in which
the cubic nonlinearity acts in a single waveguide. An explicit solution is
constructed asymptotically in the large-wavenumber limit. The general problem
is then shown to be equivalent to the existence of a homoclinic orbit in a
four-dimensional reversible map. From properties of such maps, it is shown that
(unlike ordinary gap solitons), discrete ESs have the same codimension as their
continuum counterparts. A specific numerical method is developed to compute
homoclinic solutions of the map, that are symmetric under a specific reversing
transformation. Existence is then studied in the full parameter space of the
problem. Numerical results agree with the asymptotic results in the appropriate
limit and suggest that the discrete ESs may be semi-stable as in the continuous
case.Comment: A revtex4 text file and 51 eps figure files. To appear in
Nonlinearit
Behavior of Dynamical Systems in the Regime of Transient Chaos
The transient chaos regime in a two-dimensional system with discrete time
(Eno map) is considered. It is demonstrated that a time series corresponding to
this regime differs from a chaotic series constructed for close values of the
control parameters by the presence of "nonregular" regions, the number of which
increases with the critical parameter. A possible mechanism of this effect is
discussed.Comment: 4 pages, 2 figure
The mixmaster universe: A chaotic Farey tale
When gravitational fields are at their strongest, the evolution of spacetime
is thought to be highly erratic. Over the past decade debate has raged over
whether this evolution can be classified as chaotic. The debate has centered on
the homogeneous but anisotropic mixmaster universe. A definite resolution has
been lacking as the techniques used to study the mixmaster dynamics yield
observer dependent answers. Here we resolve the conflict by using observer
independent, fractal methods. We prove the mixmaster universe is chaotic by
exposing the fractal strange repellor that characterizes the dynamics. The
repellor is laid bare in both the 6-dimensional minisuperspace of the full
Einstein equations, and in a 2-dimensional discretisation of the dynamics. The
chaos is encoded in a special set of numbers that form the irrational Farey
tree. We quantify the chaos by calculating the strange repellor's Lyapunov
dimension, topological entropy and multifractal dimensions. As all of these
quantities are coordinate, or gauge independent, there is no longer any
ambiguity--the mixmaster universe is indeed chaotic.Comment: 45 pages, RevTeX, 19 Figures included, submitted to PR
Bubbling and bistability in two parameter discrete systems
We present a graphical analysis of the mechanisms underlying the occurrences
of bubbling sequences and bistability regions in the bifurcation scenario of a
special class of one dimensional two parameter maps. The main result of the
analysis is that whether it is bubbling or bistability is decided by the sign
of the third derivative at the inflection point of the map function.Comment: LaTeX v2.09, 14 pages with 4 PNG figure
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