Integral Equations and Exponential Trichotomy of 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.

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Characterizations of dichotomy for linear skew-product flows

Mathematical Surveys and Monographs - Evolution Semigroups in Dynamical Systems and Differential Equations ◽

10.1090/surv/070/07 ◽

1999 ◽

pp. 205-260

Author(s):

Carmen Chicone ◽

Yuri Latushkin

Keyword(s):

Skew Product ◽

Skew Product Flows

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

Communications in Contemporary Mathematics ◽

10.1142/s0219199705001933 ◽

2005 ◽

Vol 07 (06) ◽

pp. 747-768 ◽

Cited By ~ 4

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.

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Skew product flows of quasi-processes and stabilities

Differential Equations with Applications to Biology ◽

10.1090/fic/021/22 ◽

1998 ◽

pp. 269-277

Author(s):

Yoshiyuki Hino ◽

Satoru Murikami

Keyword(s):

Skew Product ◽

Skew Product Flows

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Translation Invariant Spaces and Asymptotic Properties of Variational Equations

Abstract and Applied Analysis ◽

10.1155/2011/539026 ◽

2011 ◽

Vol 2011 ◽

pp. 1-36 ◽

Cited By ~ 1

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.

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