Chapter 4
Nonautonomous Attractors
Peter E. Kloeden and Meihua Yang
Abstract The nature of time in a nonautonomous dynamical system is very different
from that in autonomous systems, which depend on the time that has elapsed since
starting rather on the actual time. This requires new concepts of invariant sets and
attractors. Pullback and forward attractors as well as forward omega-limit sets will
be reviewed here in the simpler setting of nonautonomous difference equations. Both
two-parameter semi-group and skew product flow formulations of nonautonomous
dynamical systems are considered.
Keywords Nonautonomous difference equations · Process · Two-parameter
semi-group · Skew product flows · Pullback attractor · Forward attractor · Omega
limit points
4.1 Introduction
Autonomous dynamical systems are now a very well established area of mathematics. Although nonautonomous systems have been investigated in an ad hoc way for
many years, a mathematical theory of nonautonomous dynamical systems has only
been developed systematically in recent decades. Time has a very different role in
nonautonomous dynamical systems than in autonomous systems, which depend only
on the time that has elapsed since starting rather on the actual time. This has some
profound consequences in terms of definitions and the interpretation of dynamical
Dedicated to Aneta Stefanovska on her 60th birthday.
P. E. Kloeden (B)
Mathematische Institut, Universität Tübingen, 72076 Tübingen, Germany
e-mail: kloeden@na.uni-tuebingen.de
M. Yang
School of Mathematics and Statistics, Huanzhong University of Science and Technology,
Wuhan, China
e-mail: yangmeih@hust.edu.cn
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_4
55
Nonautonomous Attractors
Peter E. Kloeden and Meihua Yang
Abstract The nature of time in a nonautonomous dynamical system is very different
from that in autonomous systems, which depend on the time that has elapsed since
starting rather on the actual time. This requires new concepts of invariant sets and
attractors. Pullback and forward attractors as well as forward omega-limit sets will
be reviewed here in the simpler setting of nonautonomous difference equations. Both
two-parameter semi-group and skew product flow formulations of nonautonomous
dynamical systems are considered.
Keywords Nonautonomous difference equations · Process · Two-parameter
semi-group · Skew product flows · Pullback attractor · Forward attractor · Omega
limit points
4.1 Introduction
Autonomous dynamical systems are now a very well established area of mathematics. Although nonautonomous systems have been investigated in an ad hoc way for
many years, a mathematical theory of nonautonomous dynamical systems has only
been developed systematically in recent decades. Time has a very different role in
nonautonomous dynamical systems than in autonomous systems, which depend only
on the time that has elapsed since starting rather on the actual time. This has some
profound consequences in terms of definitions and the interpretation of dynamical
Dedicated to Aneta Stefanovska on her 60th birthday.
P. E. Kloeden (B)
Mathematische Institut, Universität Tübingen, 72076 Tübingen, Germany
e-mail: kloeden@na.uni-tuebingen.de
M. Yang
School of Mathematics and Statistics, Huanzhong University of Science and Technology,
Wuhan, China
e-mail: yangmeih@hust.edu.cn
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_4
55