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.

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.

scholarly journals BISTABILITY, BIFURCATION AND CHAOS IN A LASER SYSTEM

1995 ◽  
Vol 05 (04) ◽  
pp. 1021-1031 ◽  
Author(s):  
F. O'CAIRBRE ◽  
A. G. O'FARRELL ◽  
A. O'REILLY
Keyword(s):  
Dynamical System ◽  
Laser Physics ◽  
Laser System ◽  
Bifurcation And Chaos ◽  
One Dimensional ◽  
Fabry Perot ◽  
Dimensional Dynamical System ◽  
Parameter Ranges

In this paper we study a one-dimensional dynamical system that provides a model for the evolution of a Fabry–Perot cavity in Laser Physics. We determine the parameter ranges for bistability and chaos in the system. We also examine the bifurcations of the system and the occurrence of various types of cycles.

scholarly journals Markov covers and finiteness of periodic attractors for diffeomorphisms with a dominated splitting

2000 ◽  
Vol 20 (1) ◽  
pp. 1-14
Author(s):  
MASAYUKI ASAOKA
Keyword(s):  
Dynamical System ◽  
Dominated Splitting ◽  
One Dimensional ◽  
Surface Diffeomorphisms ◽  
Periodic Attractors ◽  
Dimensional Dynamical System

In this paper, we show the existence of Markov covers for $C^2$ surface diffeomorphisms with a dominated splitting under some assumptions. Using a Markov cover, we can reduce the dynamics to a one-dimensional dynamical system having a good metric property. As an application, we show finiteness of periodic attractors for the above diffeomorphisms.

Model Development for a Composite Core-Pulling Mechanism

2014 ◽  
Vol 494-495 ◽  
pp. 387-390 ◽  
Author(s):  
Yan Fang Yin ◽  
Hua Shun Hu
Keyword(s):  
Mathematical Model ◽  
Mechanical System ◽  
Dimensional Analysis ◽  
Simple Structure ◽  
Simulation Analysis ◽  
Model Development ◽  
Plastic Part ◽  
One Dimensional ◽  
Safety And Reliability ◽  
Plastic Parts

The main purpose of this work was to develop a mathematical model and computer programs for design analysis in composite core-pulling mechanism. The injection mould with slanted guide pillar for two-step core-pulling from the side of the product effects significantly the quality and efficiency of plastic part. The study is a one-dimensional analysis of secondaryejector mechanism, based on a forward marching technique of solution for the simple structure and high safety and reliability. Using the simulation analysis of mechanical system based on Pro/E, the mechanism can solve the problem that plastic parts are difficult to side core pulling in different directions by the use of bevel pillar.

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.

scholarly journals Forced Oscillations of a Damped Korteweg-de Vries Equation in a Quarter Plane

2003 ◽  
Vol 05 (03) ◽  
pp. 369-400 ◽  
Author(s):  
Jerry L. Bona ◽  
S. M. Sun ◽  
Bing-Yu Zhang
Keyword(s):  
Dynamical System ◽  
Limit Cycle ◽  
Small Amplitude ◽  
Vries Equation ◽  
Initial Condition ◽  
Infinite Dimensional ◽  
De Vries ◽  
Exponentially Stable ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in the context of a damped Korteweg-de Vries equation. Thus, consideration is given to the initial-boundary-value problem [Formula: see text] For this problem, it is shown that if the small amplitude, boundary forcing h is periodic of period T, say, then the solution u of (*) is eventually periodic of period T. More precisely, we show for each x > 0, that u(x, t + T) - u(x, t) converges to zero exponentially as t → ∞. Viewing (*) (without the initial condition) as an infinite dimensional dynamical system in the Hilbert space Hs(R+) for suitable values of s, we also demonstrate that for a given, small amplitude time-periodic boundary forcing, the system (*) admits a unique limit cycle, or forced oscillation (a solution of (*) without the initial condition that is exactly periodic of period T). Furthermore, we show that this limit cycle is globally exponentially stable. In other words, it comprises an exponentially stable attractor for the infinite-dimensional dynamical system described by (*).

The growth, saturation, and scaling behaviour of one- and two-dimensional disturbances in fluidized beds

1998 ◽  
Vol 362 ◽  
pp. 83-119 ◽  
Author(s):  
M. F. GÖZ ◽  
S. SUNDARESAN
Keyword(s):  
Fluidized Bed ◽  
Growth Rates ◽  
Fluidized Beds ◽  
Secondary Growth ◽  
Two Dimensional ◽  
Small Perturbations ◽  
One Dimensional ◽  
Gas Fluidized Beds ◽  
The Waves ◽  
Primary Instability

It is well-known that fluidized beds are usually unstable to small perturbations and that this leads to the primary bifurcation of vertically travelling plane wavetrains. These one-dimensional periodic waves have been shown recently to be unstable to two-dimensional perturbations of large transverse wavelength in gas-fluidized beds. Here, this result is generalized to include liquid-fluidized beds and to compare typical beds fluidized with either air or water. It is shown that the instability mechanism remains the same but there are big differences in the ratio of the primary and secondary growth rates in the two cases. The tendency is that the secondary growth rates, scaled with the amplitude of a fully developed plane wave, are of similar magnitude for both gas- and liquid-fluidized beds, while the primary growth rate is much larger in the gas-fluidized bed. This means that the secondary instability is accordingly stronger than the primary instability in the liquid-fluidized bed, and consequently sets in at a much smaller amplitude of the primary wave. However, since the waves in the liquid-fluidized bed develop on a larger time and length scale, the primary perturbations need longer time and thereby travel farther until they reach the critical amplitude. Which patterns are more amenable to being visually recognized depends on the magnitude of the initially imposed disturbance and the dimensions of the apparatus. This difference in scale plays a key role in bringing about the differences between gas- and liquid-fluidized beds; it is produced mainly by the different values of the Froude number.

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