scholarly journals Equicontinuous geodesic flows

2009 ◽  
Vol 29 (6) ◽  
pp. 1951-1963
Author(s):  
CHRISTIAN PRIES
Keyword(s):  
Geodesic Flow ◽  
Chaotic Behavior ◽  
Conjugate Points ◽  
Necessary Condition ◽  
Riemannian Surface ◽  
Geodesic Flows ◽  
Higher Dimensional ◽  
The Stability ◽  
Flows On Surfaces ◽  
Sufficient And Necessary Condition

AbstractThis article is about the interplay between topological dynamics and differential geometry. One could ask how much information about the geometry is carried in the dynamics of the geodesic flow. It was proved in Paternain [Expansive geodesic flows on surfaces. Ergod. Th. & Dynam. Sys.13 (1993), 153–165] that an expansive geodesic flow on a surface implies that there exist no conjugate points. Instead of considering concepts that relate to chaotic behavior (such as expansiveness), we focus on notions for describing the stability of orbits in dynamical systems, specifically, equicontinuity and distality. In this paper we give a new sufficient and necessary condition for a compact Riemannian surface to have all geodesics closed; this is the idea of a P-manifold: (M,g) is a P-manifold if and only if the geodesic flow SM×ℝ→SM is equicontinuous. We also prove a weaker theorem for flows on manifolds of dimension three. Finally, we discuss some properties of equicontinuous geodesic flows on non-compact surfaces and on higher-dimensional manifolds.

Expansive geodesic flows on surfaces

1993 ◽  
Vol 13 (1) ◽  
pp. 153-165 ◽  
Author(s):  
Miguel Paternain
Keyword(s):  
Geodesic Flow ◽  
Compact Surface ◽  
Conjugate Points ◽  
Riemannian Surface ◽  
Geodesic Flows ◽  
Flows On Surfaces

AbstractWe prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.

Consensus Problem of Singular Multi-Agent Systems with Continuous-Time and Fixed Topology

2013 ◽  
Vol 278-280 ◽  
pp. 1687-1691
Author(s):  
Tong Qiang Jiang ◽  
Jia Wei He ◽  
Yan Ping Gao
Keyword(s):  
Continuous Time ◽  
Spanning Tree ◽  
Undirected Graph ◽  
Necessary Condition ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Multi Agent ◽  
Fixed Topology ◽  
The Stability ◽  
Sufficient And Necessary Condition

The consensus problems of two situations for singular multi-agent systems with fixed topology are discussed: directed graph without spanning tree and the disconnected undirected graph. A sufficient and necessary condition is obtained by applying the stability theory and the system is reachable asymptotically. But for normal systems, this can’t occur in upper two situations. Finally a simulation example is provided to verify the effectiveness of our theoretical result.

A Simplified Stability Criterion for Nonconservative Systems

1979 ◽  
Vol 46 (2) ◽  
pp. 423-426 ◽  
Author(s):  
I. Fawzy
Keyword(s):  
Dynamic Stability ◽  
Stability Criterion ◽  
Degrees Of Freedom ◽  
Necessary Condition ◽  
Matrix Methods ◽  
Nonconservative System ◽  
Nonconservative Systems ◽  
The Stability ◽  
Lyapunov’S Theorem ◽  
Sufficient And Necessary Condition

Dynamic stability of a general nonconservative system of n degrees of freedom is investigated. A sufficient and necessary condition for the stability of such a system is developed. It represents a simplified criterion based on the famous Lyapunov’s theorem which is proved afresh using λ-matrix methods only. When this criterion is adopted, it reduces the number of equations in Lyapunov’s method to less than half. A systematic procedure is then suggested for the stability investigation and its use is illustrated through a numerical example at the end of the paper.

Coning motion stability of spinning missiles with strapdown seekers

2016 ◽  
Vol 120 (1232) ◽  
pp. 1566-1577 ◽  
Author(s):  
S. He ◽  
D. Lin ◽  
J. Wang
Keyword(s):  
Stability Analysis ◽  
Scaling Factor ◽  
Guidance System ◽  
Necessary Condition ◽  
Stability Conditions ◽  
Motion Stability ◽  
Model Derivation ◽  
The Stability ◽  
Stability Issues ◽  
Sufficient And Necessary Condition

ABSTRACTThis paper investigates the problem of coning motion stability of spinning missiles equipped with strapdown seekers. During model derivation, it is found that the scaling factor error between the strapdown seeker and the onboard gyro introduces an undesired parasitic loop in the guidance system and, therefore, results in stability issues. Through stability analysis, a sufficient and necessary condition for the stability of spinning missiles with strapdown seekers is proposed analytically. Theoretical and numerical results reveal that the scaling factor error, spinning rate and navigation ratio play important roles in stable regions of the guidance system. Consequently, autopilot gains must be checked carefully to satisfy the stability conditions.

On a conjecture about expansive geodesic flows

1996 ◽  
Vol 16 (3) ◽  
pp. 545-553 ◽  
Author(s):  
Rafael Oswaldo Ruggierot
Keyword(s):  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Conjugate Points ◽  
Geodesic Flows

AbstractWe show that near the geodesic flow of a compact Riemannian manifold with no conjugate points which is expansive, every expansive geodesic flow has no conjugate points. We also prove that in the above hypotheses the geodesic flow istopologically stable.

scholarly journals Hierarchical B-spline complexes of discrete differential forms

2018 ◽  
Vol 40 (1) ◽  
pp. 422-473 ◽  
Author(s):  
John A Evans ◽  
Michael A Scott ◽  
Kendrick M Shepherd ◽  
Derek C Thomas ◽  
Rafael Vázquez Hernández
Keyword(s):  
Stokes Problem ◽  
Differential Forms ◽  
Spatial Dimension ◽  
Necessary Condition ◽  
Two Dimensional ◽  
B Spline ◽  
Discrete Differential Forms ◽  
The Stability ◽  
Local Exactness ◽  
Sufficient And Necessary Condition

Abstract In this paper we introduce the hierarchical B-spline complex of discrete differential forms for arbitrary spatial dimension. This complex may be applied to the adaptive isogeometric solution of problems arising in electromagnetics and fluid mechanics. We derive a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, and we derive a set of local, easy-to-compute and sufficient exactness conditions for the two-dimensional setting. We examine the stability properties of the hierarchical B-spline complex, and we find that it yields stable approximations of both the Maxwell eigenproblem and Stokes problem provided that the local exactness conditions are satisfied. We conclude by providing numerical results showing the promise of the hierarchical B-spline complex in an adaptive isogeometric solution framework.

The accessibility property of expansive geodesic flows without conjugate points

2008 ◽  
Vol 28 (1) ◽  
pp. 229-244
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Conjugate Points ◽  
Stable Set ◽  
Geodesic Flows ◽  
Continuous Path ◽  
Two Dimensional ◽  
Unstable Set ◽  
Unit Tangent Bundle ◽  
Continuous Curves

AbstractLet (M,g) be a compact, smooth Riemannian manifold without conjugate points whose geodesic flow is expansive. We show that the geodesic flow of (M,g) has the accessibility property, namely, given two pointsθ1,θ2in the unit tangent bundle, there exists a continuous path joiningθ1,θ2formed by the union of a finite number of continuous curves, each of which is contained either in a strong stable set or in a strong unstable set of the dynamics. Since expansive geodesic flows of compact surfaces have no conjugate points, the accessibility property holds for every two-dimensional expansive geodesic flow.

scholarly journals Stability of 1-Bit Compressed Sensing in Sparse Data Reconstruction

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Yuefang Lian ◽  
Jinchuan Zhou ◽  
Jingyong Tang ◽  
Zhongfeng Sun
Keyword(s):  
Optimization Problems ◽  
Inequality Constraints ◽  
Necessary Condition ◽  
Sensing Matrix ◽  
Quadratic Constraint ◽  
Linear Equality ◽  
Equality And Inequality Constraints ◽  
Measurement Vector ◽  
The Stability ◽  
Sufficient And Necessary Condition

1-bit compressing sensing (CS) is an important class of sparse optimization problems. This paper focuses on the stability theory for 1-bit CS with quadratic constraint. The model is rebuilt by reformulating sign measurements by linear equality and inequality constraints, and the quadratic constraint with noise is approximated by polytopes to any level of accuracy. A new concept called restricted weak RSP of a transposed sensing matrix with respect to the measurement vector is introduced. Our results show that this concept is a sufficient and necessary condition for the stability of 1-bit CS without noise and is a sufficient condition if the noise is available.

scholarly journals Sharkovskii's order and the stability of periodic points of maps of the interval

2000 ◽  
Vol 68 (2) ◽  
pp. 244-251 ◽  
Author(s):  
Guang Yuan Zhang ◽  
Qing Zhong Li
Keyword(s):  
Positive Integer ◽  
Periodic Points ◽  
Necessary Condition ◽  
The Stability ◽  
Sufficient And Necessary Condition

AbstractLet f be a Cr (r ≥ 0) map from the interval [0, 1] into itself and m be a positive integer. This paper gives a sufficient and necessary condition under which the set of periodic points of period m disappears after a certain small Cr-perturbation.

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