1,004 research outputs found
Reversible skew laurent polynomial rings and deformations of poisson automorphisms
A skew Laurent polynomial ring S = R[x(+/- 1); alpha] is reversible if it has a reversing automorphism, that is, an automorphism theta of period 2 that transposes x and x(-1) and restricts to an automorphism gamma of R with gamma = gamma(-1). We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of the two most familiar examples of simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus, both of which are deformations of Poisson algebras over the base field F. Their reversing automorphisms are deformations of Poisson automorphisms of those Poisson algebras. In each case, the ring of invariants of the Poisson automorphism is the coordinate ring B of a surface in F-3 and the ring of invariants S-theta of the reversing automorphism is a deformation of B and is a factor of a deformation of F[x(1), x(2), x(3)] for a Poisson bracket determined by the appropriate surface
Discrete Dynamical Systems Embedded in Cantor Sets
While the notion of chaos is well established for dynamical systems on
manifolds, it is not so for dynamical systems over discrete spaces with
variables, as binary neural networks and cellular automata. The main difficulty
is the choice of a suitable topology to study the limit . By
embedding the discrete phase space into a Cantor set we provided a natural
setting to define topological entropy and Lyapunov exponents through the
concept of error-profile. We made explicit calculations both numerical and
analytic for well known discrete dynamical models.Comment: 36 pages, 13 figures: minor text amendments in places, time running
top to bottom in figures, to appear in J. Math. Phy
Calculations of Neutron Reflectivity in the eV Energy Range from Mirrors made of Heavy Nuclei with Neutron-Nucleus Resonances
We evaluate the reflectivity of neutron mirrors composed of certain heavy
nuclei which possess strong neutron-nucleus resonances in the eV energy range.
We show that the reflectivity of such a mirror for some nuclei can in principle
be high enough near energies corresponding to compound neutron-nucleus
resonances to be of interest for certain scientific applications in
non-destructive evaluation of subsurface material composition and in the theory
of neutron optics beyond the kinematic limit.Comment: 18 pages, 5 figures, 1 tabl
Nambu-Hamiltonian flows associated with discrete maps
For a differentiable map that has
an inverse, we show that there exists a Nambu-Hamiltonian flow in which one of
the initial value, say , of the map plays the role of time variable while
the others remain fixed. We present various examples which exhibit the map-flow
correspondence.Comment: 19 page
Cryptographic requirements for chaotic secure communications
In recent years, a great amount of secure communications systems based on
chaotic synchronization have been published. Most of the proposed schemes fail
to explain a number of features of fundamental importance to all cryptosystems,
such as key definition, characterization, and generation. As a consequence, the
proposed ciphers are difficult to realize in practice with a reasonable degree
of security. Likewise, they are seldom accompanied by a security analysis.
Thus, it is hard for the reader to have a hint about their security. In this
work we provide a set of guidelines that every new cryptosystems would benefit
from adhering to. The proposed guidelines address these two main gaps, i.e.,
correct key management and security analysis, to help new cryptosystems be
presented in a more rigorous cryptographic way. Also some recommendations are
offered regarding some practical aspects of communications, such as channel
noise, limited bandwith, and attenuation.Comment: 13 pages, 3 figure
A scalar nonlocal bifurcation of solitary waves for coupled nonlinear Schroedinger systems
An explanation is given for previous numerical results which suggest a
certain bifurcation of `vector solitons' from scalar (single-component)
solitary waves in coupled nonlinear Schroedinger (NLS) systems. The bifurcation
in question is nonlocal in the sense that the vector soliton does not have a
small-amplitude component, but instead approaches a solitary wave of one
component with two infinitely far-separated waves in the other component. Yet,
it is argued that this highly nonlocal event can be predicted from a purely
local analysis of the central solitary wave alone. Specifically the
linearisation around the central wave should contain asymptotics which grow at
precisely the speed of the other-component solitary waves on the two wings.
This approximate argument is supported by both a detailed analysis based on
matched asymptotic expansions, and numerical experiments on two example
systems. The first is the usual coupled NLS system involving an arbitrary ratio
between the self-phase and cross-phase modulation terms, and the second is a
coupled NLS system with saturable nonlinearity that has recently been
demonstrated to support stable multi-peaked solitary waves. The asymptotic
analysis further reveals that when the curves which define the proposed
criterion for scalar nonlocal bifurcations intersect with boundaries of certain
local bifurcations, the nonlocal bifurcation could turn from scalar to
non-scalar at the intersection. This phenomenon is observed in the first
example. Lastly, we have also selectively tested the linear stability of
several solitary waves just born out of scalar nonlocal bifurcations. We found
that they are linearly unstable. However, they can lead to stable solitary
waves through parameter continuation.Comment: To appear in Nonlinearit
Renormalization Group Functional Equations
Functional conjugation methods are used to analyze the global structure of
various renormalization group trajectories, and to gain insight into the
interplay between continuous and discrete rescaling. With minimal assumptions,
the methods produce continuous flows from step-scaling {\sigma} functions, and
lead to exact functional relations for the local flow {\beta} functions, whose
solutions may have novel, exotic features, including multiple branches. As a
result, fixed points of {\sigma} are sometimes not true fixed points under
continuous changes in scale, and zeroes of {\beta} do not necessarily signal
fixed points of the flow, but instead may only indicate turning points of the
trajectories.Comment: A physical model with a limit cycle added as section IV, along with
reference
Complex Analysis of a Piece of Toda Lattice
We study a small piece of two dimensional Toda lattice as a complex dynamical
system. In particular the Julia set, which appears when the piece is deformed,
is shown analytically how it disappears as the system approaches to the
integrable limit.Comment: 17 pages, LaTe
Statistical properties of Lorenz like flows, recent developments and perspectives
We comment on mathematical results about the statistical behavior of Lorenz
equations an its attractor, and more generally to the class of singular
hyperbolic systems. The mathematical theory of such kind of systems turned out
to be surprisingly difficult. It is remarkable that a rigorous proof of the
existence of the Lorenz attractor was presented only around the year 2000 with
a computer assisted proof together with an extension of the hyperbolic theory
developed to encompass attractors robustly containing equilibria. We present
some of the main results on the statisitcal behavior of such systems. We show
that for attractors of three-dimensional flows, robust chaotic behavior is
equivalent to the existence of certain hyperbolic structures, known as
singular-hyperbolicity. These structures, in turn, are associated to the
existence of physical measures: \emph{in low dimensions, robust chaotic
behavior for flows ensures the existence of a physical measure}. We then give
more details on recent results on the dynamics of singular-hyperbolic
(Lorenz-like) attractors.Comment: 40 pages; 10 figures; Keywords: sensitive dependence on initial
conditions, physical measure, singular-hyperbolicity, expansiveness, robust
attractor, robust chaotic flow, positive Lyapunov exponent, large deviations,
hitting and recurrence times. Minor typos corrected and precise
acknowledgments of financial support added. To appear in Int J of Bif and
Chaos in App Sciences and Engineerin
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