Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

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On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

Georgian Mathematical Journal ◽

10.1515/gmj.2009.597 ◽

2009 ◽

Vol 16 (4) ◽

pp. 597-616

Author(s):

Shota Akhalaia ◽

Malkhaz Ashordia ◽

Nestan Kekelia

Keyword(s):

Linear System ◽

Difference Equations ◽

Closed Interval ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Difference Systems ◽

The Stability ◽

Necessary And Sufficient ◽

Generalized Ordinary Differential Equations ◽

Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

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Topological conjugacy for multifunctions with finitely many set-valued points and an application to iterative roots

The Journal of Difference Equations and Applications ◽

10.1080/10236198.2021.1916483 ◽

2021 ◽

pp. 1-15

Author(s):

Jinghua Liu ◽

Lin Li

Keyword(s):

Topological Conjugacy ◽

Iterative Roots

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Reduction principle at work

Nonautonomous Dynamical Systems ◽

10.1515/msds-2020-0124 ◽

2021 ◽

Vol 8 (1) ◽

pp. 46-74

Author(s):

Christian Pötzsche ◽

Evamaria Russ

Keyword(s):

Center Manifold ◽

Reduction Principle ◽

Critical Stability ◽

Center Manifold Reduction ◽

Infinite Dimensional ◽

Linearized Stability ◽

Dichotomy Spectrum ◽

Nonautonomous Case ◽

Necessary And Sufficient ◽

Manifold Reduction

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

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