scholarly journals Invariant measures for some one-dimensional attractors

1982 ◽  
Vol 2 (3-4) ◽  
pp. 317-337 ◽  
Author(s):  
M. V. Jacobson
Keyword(s):  
Invariant Measures ◽  
Small Perturbations ◽  
One Dimensional ◽  
Quadratic Maps ◽  
Hyperbolic Attractors

AbstractWe consider certain non-invertible maps of the square which are extensions of the quadratic maps of the interval and their small perturbations. We show that several maps of the type possess attractors which are not hyperbolic but have invariant measures similar to Bowen-Ruelle measures for hyperbolic attractors.

ROBUSTNESS OF COMPLETE STABILITY FOR 1-D CIRCULAR CNNs

2006 ◽  
Vol 16 (08) ◽  
pp. 2177-2190
Author(s):  
MAURO DI MARCO ◽  
CHIARA GHILARDI
Keyword(s):  
Neural Networks ◽  
Equilibrium Point ◽  
Nearest Neighbor ◽  
Perturbation Parameter ◽  
Relative Magnitude ◽  
Loss Of Stability ◽  
Unique Equilibrium ◽  
Small Perturbations ◽  
Structure Preserving ◽  
One Dimensional

This paper investigates the issue of robustness of complete stability of standard Cellular Neural Networks (CNNs) with respect to small perturbations of the nominally symmetric interconnections. More specifically, a class of circular one-dimensional (1-D) CNNs with nearest-neighbor interconnections only, is considered. The class has sparse interconnections and is subject to perturbations which preserve the interconnecting structure. Conditions assuring that the perturbed CNN has a unique equilibrium point at the origin, which is unstable, are provided in terms of relative magnitude of the perturbations with respect to the nominal interconnection weights. These conditions allow one to characterize regions in the perturbation parameter space where there is loss of stability for the perturbed CNN. In turn, this shows that even for sparse interconnections and structure preserving perturbations, robustness of complete stability is not guaranteed in the general case.

scholarly journals Interaction instability of localization in quasiperiodic systems

2018 ◽  
Vol 115 (18) ◽  
pp. 4595-4600 ◽  
Author(s):  
Marko Žnidarič ◽  
Marko Ljubotina
Keyword(s):  
Transport Properties ◽  
Quantum Systems ◽  
Theoretical Physics ◽  
Solvable Models ◽  
Many Body ◽  
Small Perturbations ◽  
One Dimensional ◽  
Quasiperiodic Potential ◽  
Quasiperiodic Systems ◽  
Small Interactions

Integrable models form pillars of theoretical physics because they allow for full analytical understanding. Despite being rare, many realistic systems can be described by models that are close to integrable. Therefore, an important question is how small perturbations influence the behavior of solvable models. This is particularly true for many-body interacting quantum systems where no general theorems about their stability are known. Here, we show that no such theorem can exist by providing an explicit example of a one-dimensional many-body system in a quasiperiodic potential whose transport properties discontinuously change from localization to diffusion upon switching on interaction. This demonstrates an inherent instability of a possible many-body localization in a quasiperiodic potential at small interactions. We also show how the transport properties can be strongly modified by engineering potential at only a few lattice sites.

An approach to the ordering of one-dimensional quadratic maps

1996 ◽  
Vol 7 (4) ◽  
pp. 565-584 ◽  
Author(s):  
G. Pastor ◽  
M. Romera ◽  
F. Montoya
Keyword(s):  
One Dimensional ◽  
Quadratic Maps

Symbolic sequences of one-dimensional quadratic maps points

1998 ◽  
Vol 256 (3-4) ◽  
pp. 369-382 ◽  
Author(s):  
G. Pastor ◽  
M. Romera ◽  
J.C. Sanz-Martı́n ◽  
F. Montoya
Keyword(s):  
One Dimensional ◽  
Quadratic Maps ◽  
Symbolic Sequences

scholarly journals On preservation of an exponentially stable invariant torus

2015 ◽  
Vol 63 (1) ◽  
pp. 215-222 ◽  
Author(s):  
Mykola Perestyuk ◽  
Petro Feketa
Keyword(s):  
Simple Structure ◽  
Linear Extensions ◽  
Qualitative Behaviour ◽  
Small Perturbations ◽  
One Dimensional ◽  
Exponentially Stable ◽  
Invariant Toroidal Manifold ◽  
Dimensional Dynamical System ◽  
Stable Invariant ◽  
Stable Invariant Torus

Abstract New conditions of the preservation of an exponentially stable invariant toroidal manifold of linear extension of one-dimensional dynamical system on torus under small perturbations in ω-limit set are established. This approach is applied to the investigation of the qualitative behaviour of solutions of linear extensions of dynamical systems with simple structure of limit sets.

scholarly journals Estimating invariant measures and Lyapunov exponents

1996 ◽  
Vol 16 (4) ◽  
pp. 735-749 ◽  
Author(s):  
Brian R. Hunt
Keyword(s):  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Recent Interest ◽  
Absolutely Continuous Invariant Measure ◽  
Time Averages ◽  
Absolutely Continuous Invariant

AbstractThis paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius—Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.

Higher Dimensional Asymptotic Cycles

2003 ◽  
Vol 55 (3) ◽  
pp. 636-648 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Invariant Measure ◽  
Compact Manifold ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Stationary Points ◽  
Dimensional Case ◽  
One Dimensional ◽  
Smooth Flow ◽  
Higher Dimensional ◽  
The One

AbstractGiven a p-dimensional oriented foliation of an n-dimensional compact manifold Mn and a transversal invariant measure τ, Sullivan has defined an element of Hp(Mn; R). This generalized the notion of a μ-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure μ. In this one-dimensional case there was a natural 1—1 correspondence between transversal invariant measures τ and invariant measures μ when one had a smooth flow without stationary points.For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

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