scholarly journals A dynamical system on ${\bf R}\sp 3$ with uniformly bounded trajectories and no compact trajectories

1989 ◽  
Vol 106 (4) ◽  
pp. 1095-1095
Author(s):  
K. M. Kuperberg ◽  
Coke S. Reed
Keyword(s):  
Dynamical System ◽  
Uniformly Bounded ◽  
Bounded Trajectories

Parallelisability in Banach spaces: a parallelisable dynamical system with uniformly bounded trajectories

1988 ◽  
Vol 108 (3-4) ◽  
pp. 371-378
Author(s):  
B. M. Garay
Keyword(s):  
Dynamical System ◽  
Banach Space ◽  
Banach Spaces ◽  
Supremum Norm ◽  
Uniformly Bounded ◽  
Bounded Trajectories

SynopsisIn the Banach space of real sequences which converge to zero with the supremum norm, we construct a parallelisable dynamical system with uniformly-bounded trajectories.

Dynamics of Morse-Smale urn processes

1995 ◽  
Vol 15 (6) ◽  
pp. 1005-1030 ◽  
Author(s):  
Michel Benaïm ◽  
Morris W. Hirsch
Keyword(s):  
Dynamical System ◽  
Sample Path ◽  
Asymptotically Stable ◽  
Martingale Differences ◽  
Stable Periodic Orbit ◽  
Sample Paths ◽  
Stable Periodic ◽  
Image Position ◽  
Deterministic Dynamical System ◽  
Uniformly Bounded

AbstractWe consider stochastic processes {xn}n≥0 of the formwhere F: ℝm → ℝm is C2, {λi}i≥1 is a sequence of positive numbers decreasing to 0 and {Ui}i≥1 is a sequence of uniformly bounded ℝm-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Γ, the probability of sample paths approaching Γ is positive. This gives the generic behavior of three-color urn models.

CHAOTIFICATION OF DISCRETE-TIME DYNAMICAL SYSTEMS: AN EXTENSION OF THE CHEN–LAI ALGORITHM

2005 ◽  
Vol 15 (01) ◽  
pp. 109-117 ◽  
Author(s):  
DEJIAN LAI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Design Method ◽  
Control Design ◽  
Asymptotically Stable ◽  
Controlled System ◽  
Dimensional Dynamical System ◽  
Discrete Time Dynamical Systems ◽  
Uniformly Bounded

In this article, we propose and study an extension of the Chen–Lai algorithm for chaotification of discrete-time dynamical systems. The proposed method is a simple but mathematically rigorous feedback control design method that can gradually make all the Lyapunov exponents of the controlled system strictly positive for any given n-dimensional dynamical system that has a uniformly bounded Jacobian but otherwise could be originally nonchaotic or even asymptotically stable.

scholarly journals On topological rank of factors of Cantor minimal systems

2021 ◽  
pp. 1-24
Author(s):  
NASSER GOLESTANI ◽  
MARYAM HOSSEINI
Keyword(s):  
Dynamical System ◽  
Finite Rank ◽  
Affirmative Answer ◽  
Cantor Set ◽  
Minimal System ◽  
Full Generality ◽  
Cantor Minimal Systems ◽  
Minimal Dynamical System ◽  
Uniformly Bounded

Abstract A Cantor minimal system is of finite topological rank if it has a Bratteli–Vershik representation whose number of vertices per level is uniformly bounded. We prove that if the topological rank of a minimal dynamical system on a Cantor set is finite, then all its minimal Cantor factors have finite topological rank as well. This gives an affirmative answer to a question posed by Donoso, Durand, Maass, and Petite in full generality. As a consequence, we obtain the dichotomy of Downarowicz and Maass for Cantor factors of finite-rank Cantor minimal systems: they are either odometers or subshifts.

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