scholarly journals Lower semicontinuity of attractors for non-autonomous dynamical systems

2009 ◽  
Vol 29 (6) ◽  
pp. 1765-1780 ◽  
Author(s):  
ALEXANDRE N. CARVALHO ◽  
JOSÉ A. LANGA ◽  
JAMES C. ROBINSON
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Banach Spaces ◽  
Lower Semicontinuity ◽  
Exponential Dichotomy ◽  
Time Dependent ◽  
Unstable Manifolds ◽  
Autonomous Differential Equations ◽  
Autonomous Dynamical Systems

AbstractThis paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

scholarly journals State space decomposition for non-autonomous dynamical systems

2011 ◽  
Vol 141 (5) ◽  
pp. 957-974 ◽  
Author(s):  
Xiaopeng Chen ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
State Space ◽  
Decomposition Theorem ◽  
Navier Stokes ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
The Lorenz System ◽  
A Chain ◽  
Autonomous Dynamical Systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

(μ, ν)-Pseudo almost automorphic solutions for some non-autonomous differential equations

2015 ◽  
Vol 26 (11) ◽  
pp. 1550090 ◽  
Author(s):  
El Hadi Ait Dads ◽  
Khalil Ezzinbi ◽  
Mohsen Miraoui
Keyword(s):  
Differential Equations ◽  
Exponential Dichotomy ◽  
Linear Part ◽  
Nonlinear Part ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
Pseudo Almost Automorphic ◽  
Autonomous Differential Equations ◽  
Automorphic Functions

The aim of this work is to study the new concept of the (μ, ν)-pseudo almost automorphic functions for some non-autonomous differential equations. We suppose that the linear part has an exponential dichotomy. The nonlinear part is assumed to be (μ, ν)-pseudo almost automorphic. We show some results regarding the completness and the invariance of the space consisting in (μ, ν)-pseudo almost automorphic functions. Then we propose to study the existence of (μ, ν)-pseudo almost automorphic solutions for some differential equations involving reflection of the argument.

scholarly journals Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

2016 ◽  
Vol 13 (02) ◽  
pp. 1650014 ◽  
Author(s):  
Tiberiu Harko ◽  
Praiboon Pantaragphong ◽  
Sorin V. Sabau
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Cold Dark Matter ◽  
Evolution Equations ◽  
Dynamical Evolution ◽  
Finsler Spaces ◽  
Stability And Instability ◽  
The Relationship ◽  
Autonomous Dynamical Systems

The Kosambi–Cartan–Chern (KCC) theory represents a powerful mathematical method for the investigation of the properties of dynamical systems. The KCC theory introduces a geometric description of the time evolution of a dynamical system, with the solution curves of the dynamical system described by methods inspired by the theory of geodesics in a Finsler spaces. The evolution of a dynamical system is geometrized by introducing a nonlinear connection, which allows the construction of the KCC covariant derivative, and of the deviation curvature tensor. In the KCC theory, the properties of any dynamical system are described in terms of five geometrical invariants, with the second one giving the Jacobi stability of the system. Usually, the KCC theory is formulated by reducing the dynamical evolution equations to a set of second-order differential equations. In this paper, we introduce and develop the KCC approach for dynamical systems described by systems of arbitrary [Formula: see text]-dimensional first-order differential equations. We investigate in detail the properties of the [Formula: see text]-dimensional autonomous dynamical systems, as well as the relationship between the linear stability and the Jacobi stability. As a main result we find that only even-dimensional dynamical systems can exhibit both Jacobi stability and instability behaviors, while odd-dimensional dynamical systems are always Jacobi unstable, no matter their Lyapunov stability. As applications of the developed formalism we consider the geometrization and the study of the Jacobi stability of the complex dynamical networks, and of the [Formula: see text]-Cold Dark Matter ([Formula: see text]CDM) cosmological models, respectively.

Discussion on the Complicated Topological Dynamic System of Ordinary Differential Equation (ODE)

2014 ◽  
Vol 696 ◽  
pp. 30-37
Author(s):  
Yun Xia Wang
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Dynamic System ◽  
Closed Form ◽  
Topological Dynamic ◽  
Autonomous Differential Equations

The dynamical system of ODEs is about closed-form researches into the field of ODEs from the perspective of dynamical systems. This paper, starting with the research of path in autonomous differential equations and discussion on Poincaré’s viewpoints, probes into the complicated topological dynamic system of ODEs.

scholarly journals Uniform Dichotomy Concepts for Discrete-Time Skew Evolution Cocycles in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2177
Author(s):  
Ariana Găină ◽  
Mihail Megan ◽  
Carmen Florinela Popa
Keyword(s):  
Dynamical Systems ◽  
Banach Spaces ◽  
Discrete Time ◽  
Exponential Dichotomy ◽  
Uniform Dichotomy

In the present paper, we consider the problem of dichotomic behaviors of dynamical systems described by discrete-time skew evolution cocycles in Banach spaces. We study two concepts of uniform dichotomy: uniform exponential dichotomy and uniform polynomial dichotomy. Some characterizations of these notions and connections between these concepts are given.

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