scholarly journals Admissibility and exponential trichotomy of dynamical systems described by skew-product flows

2016 ◽  
Vol 260 (2) ◽  
pp. 1656-1689 ◽  
Author(s):  
Adina Luminiţa Sasu ◽  
Bogdan Sasu
Keyword(s):  
Dynamical Systems ◽  
Skew Product ◽  
Exponential Trichotomy ◽  
Skew Product Flows

scholarly journals On Exponential Dichotomy of Variational Difference Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Bogdan Sasu
Keyword(s):  
Difference Equations ◽  
Exponential Dichotomy ◽  
New Method ◽  
Skew Product ◽  
Skew Product Flows

We give very general characterizations for uniform exponential dichotomy of variational difference equations. We propose a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories. The obtained results are applied to difference equations and also to linear skew-product flows.

ON THE CONTINUITY OF LIAO QUALITATIVE FUNCTIONS OF DIFFERENTIAL SYSTEMS AND APPLICATIONS

2005 ◽  
Vol 07 (06) ◽  
pp. 747-768 ◽  
Author(s):  
XIONGPING DAI
Keyword(s):  
Riemannian Manifold ◽  
Stochastic Stability ◽  
Vector Fields ◽  
Finite Dimension ◽  
Skew Product ◽  
Ergodic System ◽  
Differential Systems ◽  
Lyapunov Spectra ◽  
Skew Product Flows ◽  
Global Linearization

Let 𝔛r(M), r ≥ 1, denote the space of all Cr vector fields over a compact, smooth and boundaryless Riemannian manifold M of finite dimension; let [Formula: see text], 1 ≤ ℓ ≤ dim M, be the bundle of orthonormal ℓ-frames of the tangent space TM of M. For any V ∈ 𝔛r(M), Liao defined functions [Formula: see text], k = 1, …, ℓ, on [Formula: see text], which are qualitatively equivalent to the Lyapunov exponents of the differential system V. In this paper, the author shows that every [Formula: see text] depends Cr-1-continuously upon [Formula: see text] and Cr-continuously on [Formula: see text] for any given V. In addition, applying the qualitative functions, the author generalizes Liao's global linearization along a given orbit of V and considers the stochastic stability of Lyapunov spectra of linear skew-product flows based on a given ergodic system.

scholarly journals Translation Invariant Spaces and Asymptotic Properties of Variational Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-36 ◽  
Author(s):  
Adina Luminiţa Sasu ◽  
Bogdan Sasu
Keyword(s):  
Control Systems ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Invariant Function ◽  
Function Spaces ◽  
Skew Product ◽  
Translation Invariant ◽  
Variational Equations ◽  
Skew Product Flows ◽  
New Perspective

We present a new perspective concerning the study of the asymptotic behavior of variational equations by employing function spaces techniques. We give a complete description of the dichotomous behaviors of the most general case of skew-product flows, without any assumption concerning the flow, the cocycle or the splitting of the state space, our study being based only on the solvability of some associated control systems between certain function spaces. The main results do not only point out new necessary and sufficient conditions for the existence of uniform and exponential dichotomy of skew-product flows, but also provide a clear chart of the connections between the classes of translation invariant function spaces that play the role of the input or output classes with respect to certain control systems. Finally, we emphasize the significance of each underlying hypothesis by illustrative examples and present several interesting applications.

The topological centre of skew product dynamical systems

2011 ◽  
Vol 82 (3) ◽  
pp. 497-503 ◽  
Author(s):  
A. Jabbari ◽  
H. R. Ebrahimi Vishki
Keyword(s):  
Dynamical Systems ◽  
Skew Product ◽  
Topological Centre

scholarly journals SKEW-PRODUCT DYNAMICAL SYSTEMS, ELLIS GROUPS AND TOPOLOGICAL CENTRE

2009 ◽  
Vol 79 (1) ◽  
pp. 129-145 ◽  
Author(s):  
A. JABBARI ◽  
H. R. E. VISHKI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Type Systems ◽  
General Construction ◽  
Skew Product ◽  
Topological Properties ◽  
Natural Extensions ◽  
Special Case ◽  
Topological Centre

AbstractIn this paper, a general construction of a skew-product dynamical system, for which the skew-product dynamical system studied by Hahn is a special case, is given. Then the ergodic and topological properties (of a special type) of our newly defined systems (called Milnes-type systems) are investigated. It is shown that the Milnes-type systems are actually natural extensions of dynamical systems corresponding to some special distal functions. Finally, the topological centre of Ellis groups of any skew-product dynamical system is calculated.

scholarly journals Discrete admissibility and exponential trichotomy of dynamical systems

2014 ◽  
Vol 34 (7) ◽  
pp. 2929-2962 ◽  
Author(s):  
Adina Luminiţa Sasu ◽  
◽  
Bogdan Sasu
Keyword(s):  
Dynamical Systems ◽  
Exponential Trichotomy
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