Tempered exponential dichotomies and Lyapunov exponents for perturbations

2016 ◽  
Vol 18 (05) ◽  
pp. 1550058 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Lyapunov Exponents ◽  
Ergodic Theory ◽  
Lyapunov Theory ◽  
Type Result ◽  
Exponential Dichotomies ◽  
Linear Cocycle

We establish a Perron-type result for the perturbations of a linear cocycle in the context of ergodic theory. More precisely, we show that the Lyapunov exponents of a linear cocycle are preserved under sufficiently small nonautonomous perturbations. Our approach is based on the Lyapunov theory of regularity.

NONUNIFORM EXPONENTIAL BEHAVIOUR AND TOPOLOGICAL EQUIVALENCE

2015 ◽  
Vol 58 (2) ◽  
pp. 279-291
Author(s):  
LUIS BARREIRA ◽  
LIVIU HORIA POPESCU ◽  
CLAUDIA VALLS
Keyword(s):  
Asymptotic Behaviour ◽  
Ergodic Theory ◽  
Canonical Form ◽  
Exponential Dichotomy ◽  
Topological Equivalence ◽  
Exponential Dichotomies ◽  
Exponential Behaviour ◽  
Nonuniform Exponential Dichotomies ◽  
Nonuniform Exponential Dichotomy ◽  
Topological Conjugacies

AbstractWe show that any evolution family with a strong nonuniform exponential dichotomy can always be transformed by a topological equivalence to a canonical form that contracts and/or expands the same in all directions. We emphasize that strong nonuniform exponential dichotomies are ubiquitous in the context of ergodic theory. The main novelty of our work is that we are able to control the asymptotic behaviour of the topological conjugacies at the origin and at infinity.

Exponential Growth Rates of Periodic Asymmetric Oscillators

2008 ◽  
Vol 8 (4) ◽  
Author(s):  
Meirong Zhang ◽  
Zhe Zhou
Keyword(s):  
Lyapunov Exponents ◽  
Ergodic Theory ◽  
Growth Rates ◽  
Exponential Growth ◽  
Zero Solution ◽  
Unique Ergodicity ◽  
Topological Dynamics ◽  
Asymmetric Oscillator ◽  
Ergodicity Theorem

AbstractIn this paper we will study the dynamics of the periodic asymmetric oscillator xʺ + qdoes exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle di®eomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory.

scholarly journals Contributions to the geometric and ergodic theory of conservative flows

2012 ◽  
Vol 33 (6) ◽  
pp. 1709-1731 ◽  
Author(s):  
MÁRIO BESSA ◽  
JORGE ROCHA
Keyword(s):  
Lyapunov Exponents ◽  
Ergodic Theory ◽  
Vector Fields ◽  
Dominated Splitting ◽  
Residual Subset ◽  
Ergodic Flow ◽  
Volume Preserving

AbstractWe prove the following dichotomy for vector fields in a $C^1$-residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and $C^1$-stably ergodic flow can be $C^1$-approximated by another volume-preserving flow which is non-uniformly hyperbolic.

Ergodic Theory

1983 ◽  
Author(s):  
Karl E. Petersen
Keyword(s):  
Ergodic Theory

Lyapunov Exponents

2016 ◽  
Author(s):  
Arkady Pikovsky ◽  
Antonio Politi
Keyword(s):  
Lyapunov Exponents

Control of Vibrations due to Moving Loads on Suspension Bridges

2006 ◽  
Vol 11 (3) ◽  
pp. 293-318 ◽  
Author(s):  
M. Zribi ◽  
N. B. Almutairi ◽  
M. Abdel-Rohman
Keyword(s):  
Bridge Deck ◽  
Suspension Bridge ◽  
Lyapunov Theory ◽  
Dynamic Responses ◽  
Suspension Bridges ◽  
Control Force ◽  
Moving Loads ◽  
Hydraulic Actuator ◽  
Long Span ◽  
Suspended Cables

The flexibility and low damping of the long span suspended cables in suspension bridges makes them prone to vibrations due to wind and moving loads which affect the dynamic responses of the suspended cables and the bridge deck. This paper investigates the control of vibrations of a suspension bridge due to a vertical load moving on the bridge deck with a constant speed. A vertical cable between the bridge deck and the suspended cables is used to install a hydraulic actuator able to generate an active control force on the bridge deck. Two control schemes are proposed to generate the control force needed to reduce the vertical vibrations in the suspended cables and in the bridge deck. The proposed controllers, whose design is based on Lyapunov theory, guarantee the asymptotic stability of the system. The MATLAB software is used to simulate the performance of the controlled system. The simulation results indicate that the proposed controllers work well. In addition, the performance of the system with the proposed controllers is compared to the performance of the system controlled with a velocity feedback controller.

Sign in / Sign up

Export Citation Format

Share Document