5,757 research outputs found
Connecting period-doubling cascades to chaos
The appearance of infinitely-many period-doubling cascades is one of the most
prominent features observed in the study of maps depending on a parameter. They
are associated with chaotic behavior, since bifurcation diagrams of a map with
a parameter often reveal a complicated intermingling of period-doubling
cascades and chaos. Period doubling can be studied at three levels of
complexity. The first is an individual period-doubling bifurcation. The second
is an infinite collection of period doublings that are connected together by
periodic orbits in a pattern called a cascade. It was first described by
Myrberg and later in more detail by Feigenbaum. The third involves infinitely
many cascades and a parameter value of the map at which there is chaos.
We show that often virtually all (i.e., all but finitely many) ``regular''
periodic orbits at are each connected to exactly one cascade by a path
of regular periodic orbits; and virtually all cascades are either paired --
connected to exactly one other cascade, or solitary -- connected to exactly one
regular periodic orbit at . The solitary cascades are robust to large
perturbations. Hence the investigation of infinitely many cascades is
essentially reduced to studying the regular periodic orbits of . Examples discussed include the forced-damped pendulum and the
double-well Duffing equation.Comment: 29 pages, 13 figure
Sufficient Conditions for Polynomial Asymptotic Behaviour of the Stochastic Pantograph Equation
This paper studies the asymptotic growth and decay properties of solutions of
the stochastic pantograph equation with multiplicative noise. We give
sufficient conditions on the parameters for solutions to grow at a polynomial
rate in -th mean and in the almost sure sense. Under stronger conditions the
solutions decay to zero with a polynomial rate in -th mean and in the almost
sure sense. When polynomial bounds cannot be achieved, we show for a different
set of parameters that exponential growth bounds of solutions in -th mean
and an almost sure sense can be obtained. Analogous results are established for
pantograph equations with several delays, and for general finite dimensional
equations.Comment: 29 pages, to appear Electronic Journal of Qualitative Theory of
Differential Equations, Proc. 10th Coll. Qualitative Theory of Diff. Equ.
(July 1--4, 2015, Szeged, Hungary
Low-energy behavior of spin-liquid electron spectral functions
We calculate the electron spectral function for a spin-liquid with a spinon
Fermi surface and a Dirac spin-liquid. Calculations are based upon the
slave-rotor mean-field theory. We consider the effect of gauge fluctuations
using a simple model and find the behavior is not strongly modified. The
results, distinct from conventional Mott insulator or band theory predictions,
suggest that measuring the spectral function e.g. via ARPES could help in the
experimental verification and characterization of spin liquids.Comment: 7 pages, 7 figure
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