scholarly journals Markov covers and finiteness of periodic attractors for diffeomorphisms with a dominated splitting

2000 ◽  
Vol 20 (1) ◽  
pp. 1-14
Author(s):  
MASAYUKI ASAOKA
Keyword(s):  
Dynamical System ◽  
Dominated Splitting ◽  
One Dimensional ◽  
Surface Diffeomorphisms ◽  
Periodic Attractors ◽  
Dimensional Dynamical System

In this paper, we show the existence of Markov covers for $C^2$ surface diffeomorphisms with a dominated splitting under some assumptions. Using a Markov cover, we can reduce the dynamics to a one-dimensional dynamical system having a good metric property. As an application, we show finiteness of periodic attractors for the above diffeomorphisms.

scholarly journals BISTABILITY, BIFURCATION AND CHAOS IN A LASER SYSTEM

1995 ◽  
Vol 05 (04) ◽  
pp. 1021-1031 ◽  
Author(s):  
F. O'CAIRBRE ◽  
A. G. O'FARRELL ◽  
A. O'REILLY
Keyword(s):  
Dynamical System ◽  
Laser Physics ◽  
Laser System ◽  
Bifurcation And Chaos ◽  
One Dimensional ◽  
Fabry Perot ◽  
Dimensional Dynamical System ◽  
Parameter Ranges

In this paper we study a one-dimensional dynamical system that provides a model for the evolution of a Fabry–Perot cavity in Laser Physics. We determine the parameter ranges for bistability and chaos in the system. We also examine the bifurcations of the system and the occurrence of various types of cycles.

scholarly journals Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems: 1. The case of negative Schwarzian derivative

1989 ◽  
Vol 9 (4) ◽  
pp. 737-749 ◽  
Author(s):  
M. Yu. Lyubich
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Limit Cycle ◽  
Critical Points ◽  
Schwarzian Derivative ◽  
Dimensional Manifold ◽  
Generic Point ◽  
One Dimensional ◽  
Manifold With Boundary ◽  
Dimensional Dynamical System

AbstractIt is proved that an arbitrary one dimensional dynamical system with negative Schwarzian derivative and non-degenerate critical points has no wandering intervals. This result implies a rather complete view of the dynamics of such a system. In particular, every minimal topological attractor is either a limit cycle, or a one dimensional manifold with boundary, or a solenoid. The orbit of a generic point tends to some minimal attractor.

scholarly journals Quantum Hall effect in a one-dimensional dynamical system

2011 ◽  
Vol 84 (11) ◽  
Author(s):  
J. P. Dahlhaus ◽  
J. M. Edge ◽  
J. Tworzydło ◽  
C. W. J. Beenakker
Keyword(s):  
Dynamical System ◽  
Hall Effect ◽  
Quantum Hall Effect ◽  
Quantum Hall ◽  
One Dimensional ◽  
Dimensional Dynamical System

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

On the equation ut = ∆uα + uβ

1993 ◽  
Vol 123 (2) ◽  
pp. 373-390 ◽  
Author(s):  
Yuan-Wei Qi
Keyword(s):  
Dynamical System ◽  
Blow Up ◽  
Uniqueness Result ◽  
Fast Diffusion ◽  
Negative Solution ◽  
Fast Diffusion Equation ◽  
Infinite Dimensional ◽  
The Cauchy Problem ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

SynopsisThe Cauchy problem of ut, = ∆uα + uβ, where 0 < α < l and α>1, is studied. It is proved that if 1< β<α + 2/n then every nontrivial non-negative solution is not global in time. But if β>α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.

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