Dynamical System Topology Preserved in the Presence of Noise

Judy A. KENNEDY

Makale detayları

Dynamical System Topology Preserved in the Presence of Noise

Yazarlar:

Judy A. KENNEDY

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 1998 Vol.22 Issue 4, pp.379-414

Belge Tipi:

Makale

Yayın Tarihi:

3/1998

Belge Dili:

Türkçe

Document ID (Org No.):

2000749331

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

Anahtar kelimeler:

Özet:

 

We first give a precise definition of the terms  topological horseshoe'' and generalized quadrilateral'' and then  examine the behavior of a homeomorphism F on a locally compact,  separable, locally connected metric space X (X is usually a manifold  in applications) such that F restricted to some generalized  quadrilateral Q in X is a topological horseshoe map.  For a set  Q \subset X we define and describe (1) the permanent set Z of Q to be  \{x\in X:Fn(x)\in Q for all integers n\}, and (2) the entrainment set  of Q to be E(Q)=\{x\in X: F-n (x)\in Q for all sufficiently large  n\}.  We give conditions under which various closed sets of  \overline{E(Q)} are associated, in a strong way, with indecomposable,  closed, connected spaces invariant under F. (A connected set A is   indecomposable if it is not the union of two proper connected sets,  each of which is closed relative to A.) Next we show that even when  small amounts of noise are added to the dynamical system, there are  associated indecomposable sets.  These sets are not, in general,  invariant sets for our process with noise, but they are the physically  observable sets, while invariant Cantor sets are not, and they are the  sets that can be measured.

 

We first give a precise definition of the terms  topological horseshoe'' and generalized quadrilateral'' and then  examine the behavior of a homeomorphism F on a locally compact,  separable, locally connected metric space X (X is usually a manifold  in applications) such that F restricted to some generalized  quadrilateral Q in X is a topological horseshoe map.  For a set  Q \subset X we define and describe (1) the permanent set Z of Q to be  \{x\in X:Fn(x)\in Q for all integers n\}, and (2) the entrainment set  of Q to be E(Q)=\{x\in X: F-n (x)\in Q for all sufficiently large  n\}.  We give conditions under which various closed sets of  \overline{E(Q)} are associated, in a strong way, with indecomposable,  closed, connected spaces invariant under F. (A connected set A is   indecomposable if it is not the union of two proper connected sets,  each of which is closed relative to A.) Next we show that even when  small amounts of noise are added to the dynamical system, there are  associated indecomposable sets.  These sets are not, in general,  invariant sets for our process with noise, but they are the physically  observable sets, while invariant Cantor sets are not, and they are the  sets that can be measured.

Makale detayları

Dynamical System Topology Preserved in the Presence of Noise

Yazarlar:

Judy A. KENNEDY

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 1998 Vol.22 Issue 4, pp.379-414

Belge Tipi:

Makale

Yayın Tarihi:

3/1998

Belge Dili:

Türkçe

Document ID (Org No.):

2000749331

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

topological horseshoes

generalized quadrilateral

indecomposable continua

connected indecomposable sets

shift dynamics

noisy dynamical systems

permanent set

entrainment set

destination set.

Sayı

Makale detayları

Yazarlar: Judy A. KENNEDY

Yayıncı: TUBITAK

Dergi: Turkish Journal of Mathematics 1998 Vol.22 Issue 4, pp.379-414

Belge Tipi: Makale

Yayın Tarihi: 3/1998

Belge Dili: Türkçe

Document ID (Org No.): 2000749331

Sıklık: İki Aylık

eISSN: 1303-6149

pISSN: 1300-0098

Sayı

Yazarlar:

Judy A. KENNEDY

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 1998 Vol.22 Issue 4, pp.379-414

Belge Tipi:

Makale

Yayın Tarihi:

3/1998

Belge Dili:

Türkçe

Document ID (Org No.):

2000749331

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098