scholarly journals Uniform Hyperbolicity of a Scattering Map with Lorentzian Potential

2019 ◽  
Vol 5 (1) ◽  
pp. 1 ◽  
Author(s):  
Hajime Yoshino ◽  
Ryota Kogawa ◽  
Akira Shudo
Keyword(s):  
Sufficient Condition ◽  
Two Dimensional ◽  
Topological Horseshoe ◽  
Parameter Region ◽  
Sector Condition ◽  
Physical Boundary Condition ◽  
Weyl Law ◽  
Uniform Hyperbolicity ◽  
Area Preserving ◽  
Physical Boundary

We show that a two-dimensional area-preserving map with Lorentzian potential is a topological horseshoe and uniformly hyperbolic in a certain parameter region. In particular, we closely examine the so-called sector condition, which is known to be a sufficient condition leading to the uniformly hyperbolicity of the system. The map will be suitable for testing the fractal Weyl law as it is ideally chaotic yet free from any discontinuities which necessarily invokes a serious effect in quantum mechanics such as diffraction or nonclassical effects. In addition, the map satisfies a reasonable physical boundary condition at infinity, thus it can be a good model describing the ionization process of atoms and molecules.

Design of an Observer-Based Repetitive Control System to Reject Periodic Disturbance

2016 ◽  
Vol 25 (06) ◽  
pp. 1650061 ◽  
Author(s):  
Zhen Shao ◽  
Zhengrong Xiang
Keyword(s):  
Control System ◽  
Disturbance Rejection ◽  
Lyapunov Functional ◽  
Repetitive Control ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Periodic Disturbance ◽  
Repetitive Learning ◽  
The Stability ◽  
2D Model

This paper concerns the design of an observer-based repetitive control system (RCS) to improve the periodic disturbance rejection performance. The periodic disturbance is estimated by a repetitive learning based estimator (RLE) and rejected by incorporation of the estimation into a repetitive control (RC) input. Firstly, the configuration of the observer-based RCS with the RLE is described. Then, a continuous–discrete two-dimensional (2D) model is built to describe the RCS. By choosing an appropriate Lyapunov functional, a sufficient condition is proposed to guarantee the stability of the RCS. Finally, a numerical example is given to verify the effectiveness of the proposed method.

In Search of Weaker Conditions to Insure Robust Stability in Linear Feedback Systems

1991 ◽  
Vol 113 (1) ◽  
pp. 168-170
Author(s):  
Yossi Chait ◽  
Nir Cohen ◽  
C. R. MacCluer
Keyword(s):  
Robust Stability ◽  
Approximation Error ◽  
Maximum Modulus ◽  
Linear Feedback ◽  
Sufficient Condition ◽  
Feedback Systems ◽  
Sector Condition ◽  
Plant Uncertainty ◽  
Weaker Conditions

This paper is concerned with the well-known condition for robust stability of systems with plant uncertainty. It is shown that the usual sufficient condition for robust stability, given in terms of the maximum modulus of the plant approximation error, can be relaxed to a sector condition. This sector condition, related to the phase of the uncertain portion of the plant, can increase the range of the allowed variations in the parameters of the uncertain plant sufficient for robust stability.

Conditions for the Separability of Objects in Two-Dimensional Velocity Fields

1992 ◽  
Vol 35 (4) ◽  
pp. 484-491
Author(s):  
Stephan Foldes
Keyword(s):  
Velocity Field ◽  
Directed Graph ◽  
Velocity Fields ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Jordan Curves ◽  
Directed Cycles

AbstractWe consider the directed graph representing the obstruction relation between objects moving along the streamlines of a two-dimensional velocity field. A collection of objects is sequentially separable if and only if the corresponding graph has no directed cycles. A sufficient condition for this is the permeability of closed Jordan curves.

scholarly journals On the Measure-Preserving Mappings with Three-Dimensions

1993 ◽  
Vol 132 ◽  
pp. 73-89
Author(s):  
Yi-Sui Sun
Keyword(s):  
Fixed Point ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Invariant Curve ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Kolmogorov Entropy ◽  
Numerical Exploration ◽  
Area Preserving ◽  
Dimensional Mapping

AbstractWe have systematically made the numerical exploration about the perturbation extension of area-preserving mappings to three-dimensional ones, in which the fixed points of area preserving are elliptic, parabolic or hyperbolic respectively. It has been observed that: (i) the invariant manifolds in the vicinity of the fixed point generally don’t exist (ii) when the invariant curve of original two-dimensional mapping exists the invariant tubes do also in the neighbourhood of the invariant curve (iii) for the perturbation extension of area-preserving mapping the invariant manifolds can only be generated in the subset of the invariant manifolds of original two-dimensional mapping, (iv) for the perturbation extension of area preserving mappings with hyperbolic or parabolic fixed point the ordered region near and far from the invariant curve will be destroyed by perturbation more easily than the other one, This is a result different from the case with the elliptic fixed point. In the latter the ordered region near invariant curve is solid. Some of the results have been demonstrated exactly.Finally we have discussed the Kolmogorov Entropy of the mappings and studied some applications.

Bifurcations of one- and two-dimensional maps

1984 ◽  
Vol 311 (1515) ◽  
pp. 43-102 ◽  
Keyword(s):  
Periodic Orbits ◽  
Period Doubling ◽  
Jacobian Determinant ◽  
Two Dimensional ◽  
Hénon Map ◽  
Stable And Unstable Manifolds ◽  
Henon Map ◽  
One Dimensional ◽  
The Family ◽  
Area Preserving

We study the qualitative dynamics of two-parameter families of planar maps of the form F^e(x, y) = (y, -ex+f(y)), where f :R -> R is a C 3 map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f(y) = u —y 2 or (in different coordinates) f(y) = Ay(1 —y), in which case F^ e is the Henon map. The maps F e have constant Jacobian determinant e and, as e -> 0, collapse to the family f^. The behaviour of such one-dimensional families is quite well understood, and we are able to use their bifurcation structures and information on their non-wandering sets to obtain results on both local and global bifurcations of F/ ue , for small e . Moreover, we are able to extend these results to the area preserving family F/u. 1 , thereby obtaining (partial) bifurcation sets in the (/u, e)-plane. Among our conclusions we find that the bifurcation sequence for periodic orbits, which is restricted by Sarkovskii’s theorem and the kneading theory for one-dimensional maps, is quite different for two-dimensional families. In particular, certain periodic orbits that appear at the end of the one-dimensional sequence appear at the beginning of the area preserving sequence, and infinitely many families of saddle node and period doubling bifurcation curves cross each other in the ( /u, e ) -parameter plane between e = 0 and e = 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F /u.e and of the associated sequences of periodic bifurcations that accumulate on them. We illustrate our results with some numerical computations for the orientation-preserving Henon map.

scholarly journals NEW DISCRETE STATES IN TWO-DIMENSIONAL SUPERGRAVITY

2007 ◽  
Vol 22 (07) ◽  
pp. 1375-1394 ◽  
Author(s):  
DIMITRI POLYAKOV
Keyword(s):  
Operator Algebra ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Field Theories ◽  
Two Dimensional ◽  
Discrete States ◽  
Structure Constants ◽  
Area Preserving ◽  
Volume Preserving ◽  
Lowering Operator

Two-dimensional string theory is known to contain the set of discrete states that are the SU (2) multiplets generated by the lowering operator of the SU (2) current algebra. Their structure constants are defined by the area preserving diffeomorphisms in two dimensions. In this paper we show that the interaction of d = 2 superstrings with the superconformal β - γ ghosts enlarges the actual algebra of the dimension 1 currents and hence the new ghost-dependent discrete states appear. Generally, these states are the SU (N) multiplets if the algebra includes the currents of ghost numbers n : -N ≤ n ≤ N - 2, not related by picture changing. We compute the structure constants of these ghost-dependent discrete states for N = 3 and express them in terms of SU (3) Clebsch–Gordan coefficients, relating this operator algebra to the volume preserving diffeomorphisms in d = 3. For general N, the operator algebra is conjectured to be isomorphic to SDiff (N). This points at possible holographic relations between two-dimensional superstrings and field theories in higher dimensions.

GEOMETRIC LIOUVILLE GRAVITY

1992 ◽  
Vol 07 (21) ◽  
pp. 1887-1893
Author(s):  
HOSEONG LA
Keyword(s):  
Scalar Field ◽  
Two Dimensional ◽  
Liouville Gravity ◽  
Curvature Constraint ◽  
Dimensional Gravity ◽  
Area Preserving ◽  
Geometric Formulation

A new geometric formulation of Liouville gravity based on the area preserving diffeomorphism is given and a possible alternative to reinterpret Liouville gravity is suggested, namely, a scalar field coupled to two-dimensional gravity with a curvature constraint.

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