scholarly journals Infinite-Dimensional Linear Dynamical Systems with Chaoticity

1999 ◽  
Vol 9 (2) ◽  
pp. 197-211 ◽  
Author(s):  
X. -C. Fu ◽  
J. Duan
Keyword(s):  
Dynamical Systems ◽  
Linear Dynamical Systems ◽  
Infinite Dimensional

CHAOS FOR BACKWARD SHIFT OPERATORS

2002 ◽  
Vol 12 (08) ◽  
pp. 1703-1715 ◽  
Author(s):  
F. MARTÍNEZ-GIMÉNEZ ◽  
A. PERIS
Keyword(s):  
Dynamical Systems ◽  
Differential Operator ◽  
Applied Mathematics ◽  
Linear Dynamical Systems ◽  
Infinite Dimensional ◽  
Shift Operators ◽  
Rich Variety ◽  
Backward Shift ◽  
Backward Shifts ◽  
Spaces Of Analytic Functions

Backward shift operators provide a general class of linear dynamical systems on infinite dimensional spaces. Despite linearity, chaos is a phenomenon that occurs within this context. In this paper we give characterizations for chaos in the sense of Auslander and Yorke [1980] and in the sense of Devaney [1989] of weighted backward shift operators and perturbations of the identity by backward shifts on a wide class of sequence spaces. We cover and unify a rich variety of known examples in different branches of applied mathematics. Moreover, we give new examples of chaotic backward shift operators. In particular we prove that the differential operator I + D is Auslander–Yorke chaotic on the most usual spaces of analytic functions.

Average Shadowing Property and Asymptotic Average Shadowing Property of Linear Dynamical Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950170
Author(s):  
Lixin Jiao ◽  
Lidong Wang ◽  
Fengquan Li
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Invertible Operator ◽  
Linear Dynamical Systems ◽  
Shadowing Property ◽  
Infinite Dimensional ◽  
Asymptotic Average Shadowing ◽  
Necessary And Sufficient

This paper investigates the average shadowing property and the asymptotic average shadowing property of linear dynamical systems in Banach spaces. Firstly, necessary and sufficient conditions for an invertible operator [Formula: see text] on a Banach space to have the average shadowing property and the asymptotic average shadowing property are given, respectively. Then, it is concluded that both the average shadowing property and the asymptotic average shadowing property are preserved under iterations. Furthermore, if [Formula: see text] is hyperbolic, then [Formula: see text] has the (asymptotic) average shadowing property. However, the inverse implication fails in infinite-dimensional Banach spaces. Finally, it is proved that the (asymptotic) average shadowing property is equivalent to the hyperbolicity for dynamical systems in a finite-dimensional Banach space.

scholarly journals Initial-Value Privacy of Linear Dynamical Systems

2020 ◽  
Author(s):  
Lei Wang ◽  
Ian R. Manchester ◽  
Jochen Trumpf ◽  
Guodong Shi
Keyword(s):  
Dynamical Systems ◽  
Initial Value ◽  
Linear Dynamical Systems
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