ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 13, pp. 1703–1714.
Pleiades Publishing, Ltd., 2017.
Existence and Dimension Properties
of a Global B-Pullback Attractor for a Cocycle
Generated by a Discrete Control System
A. A. Maltseva
and V. Reitmann
Faculty of Mathematics and Mechanics, Saint Petersburg State University,
Peterhof, 198504 Russia
Abstract— We consider cocycles on ﬁnite-dimensional manifolds generated by discrete-time
control systems. Frequency conditions for the existence of a global B-pullback attractor for
such cocycles considered over a general base system on a metric space are given. Upper bounds
for the Hausdorﬀ dimension of the global B-pullback attractor of a discrete cocycle are obtained
using the transfer function of the linear part of the cocycle and the discrete Kalman–Yakubovich–
Popov frequency theorem.
This paper deals with discrete-time cocycles over general base systems. Such systems are gen-
erated, in particular, by discrete-time control systems on a ﬁnite-dimensional linear space or on
a manifold for which the class of admissible controls is a metric space. Dissipativity and the ex-
istence of a global attractor are important properties of cocycles. Kloeden–Schmalfuss  and
other authors obtained suﬃcient conditions for the existence of a global B-pullback attractor for
such cocycles of the general type. Based on these results, frequency conditions for a dissipativity of
a discrete cocycle deﬁned on a cylinder were obtained in . In the present paper, frequency dissi-
pativity conditions for a discrete cocycle are given for a general control system in a ﬁnite-dimensional
space for which the class of admissible controls is described with the use of quadratic forms.
The Hausdorﬀ dimension is an important characteristic of a global B-pullback attractor. The pa-
per  was the ﬁrst to give upper bounds for the Hausdorﬀ dimension of the invariant set of a cocycle
acting on a linear space. For cocycles deﬁned on ﬁnite-dimensional manifolds, similar bounds were
obtained in . Finding upper bounds for the Hausdorﬀ dimension in terms of Lyapunov expo-
nents and Lyapunov dimension is a topical trend in dimension theory [11, 12, 13, 14]. In the
present paper, based on the results in [4, 19], we derive upper bounds for the Hausdorﬀ dimension
of an invariant compact set of a discrete cocycle, frequency conditions being used to estimate the
singular numbers of the linearized cocycle.
Now let us brieﬂy describe the content of the paper. In Section 1, we present general deﬁnitions
and properties of discrete cocycles acting on a metric space over a certain base system also deﬁned
on some metric space. Section 2 deals with discrete cocycles generated by discrete control systems.
In Section 3, a theorem on the dissipativity of a cocycle is proved using the frequency characteristic
of the linear part of the cocycle and the discrete Kalman–Yakubovich–Popov frequency theorem.
We also prove the existence of a global B-pullback and the existence of a cocycle trajectory bounded
. The latter characterizes the internal structure of the global B-pullback attractor in this case.
In Section 4, we prove a theorem on upper bounds for the Hausdorﬀ dimension of an invariant
set of a cocycle acting on a Riemannian manifold. As an example, we consider the well-known
H´enon system for the case in which, instead of constant parameters, parameters depending on
a base system on a metric space are allowed. We study the cocycle generated by these systems