4 Nonautonomous Attractors
69
4.7 Concluding Remark
The asymptotic behaviour of nonautonomous dynamical systems is much more complicated than for autonomous dynamical systems. Pullback attractors have attracted
a lot of attention in the past three decades and play an important role, but they do not
not completely characterize the asymptotic behaviour of nonautonomous dynamical
systems. They are just part of the picture.
When a pullback attractor is also forward attracting, then we have the ideal situation. But this situation usually does not hold. Then we still have the forward omega
limit sets, which are more general than the Vishik uniform attractor since they do not
require a rate of attraction that is uniform in the initial time, which is unrealistic in
many contexts. Each of these concepts provides us with useful information about the
asymptotic behaviour of a nonautonomous dynamical system. Taken together they
give us the complete picture.
Acknowledgements This work was partly supported by Chinese NSF grant 11971184.
References
1. M.L. Barabash Y.F. Suprunenko, A. Stefanovska, Chronotaxic systems: a simple paradigm
to treat time-dependent oscillatory dynamics stable under continuous perturbation. Nonlinear
Phenom. Comp. Syst. 8, 392–400 (2015)
2. A.N. Carvalho, J.A. Langa, J.C. Robinson, Attractors of Infinite Dimensional Nonautonomous
Dynamical Systems (Springer, Berlin, 2012)
3. D. Cheban, P.E. Kloeden, B. Schmalfuß, The relationship between pullback, forwards and
global attractors of nonautonomous dynamical systems. Nonlinear Dynam. Syst. Theory 2,
9–28 (2002)
4. V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics (American
Mathematical Society, Providence, Rhode Island, 2002)
5. P. Clemson, B. Petkoski, T. Stankovski, A. Stefanovska, Coupled nonautonomous oscillators,
in Nonautonomous Dynamical Systems in the Life Sciences, vol. 2102, ed. by P.E. Kloeden, C.
Pötzsche (Springer LMS, 2014), pp. 163–197
6. H. Crauel, P.E. Kloeden, Nonautonomous and random Attractors. Jahresbericht der Deutschen
Mathematiker-Vereinigung 117, 173–206 (2015)
7. C.M. Dafermos, An invariance principle for compact processes. J. Different. Eqs. 9, 239–252
(1971)
8. J.K. Hale, Asymptotic Behavior of Dissipative Systems (American Mathematical Society, Providence, 1988)
9. A. Haraux, Systemes dynamiques dissipatifs et applications, Research in Applied Mathematics,
vol. 17 (Masson, Paris, 1991)
10. P.E. Kloeden, Asymptotic invariance and limit sets of general control systems. J. Different.
Eqs. 19, 91–105 (1975)
11. P.E. Kloeden, Asymptotic invariance and the discretisation of nonautonomous forward attracting sets. J. Comput. Dynam. 3, 179–189 (2016)
12. P.E. Kloeden, T. Lorenz, Pullback and forward attractors of nonautonomous difference equations, in Proceedings of ICDEA Wuhan, vol. 2015, ed. by M. Bohner, Y. Ding, O. Dosly
(Springer, Heidelberg, 2014), pp. 37–48
69
4.7 Concluding Remark
The asymptotic behaviour of nonautonomous dynamical systems is much more complicated than for autonomous dynamical systems. Pullback attractors have attracted
a lot of attention in the past three decades and play an important role, but they do not
not completely characterize the asymptotic behaviour of nonautonomous dynamical
systems. They are just part of the picture.
When a pullback attractor is also forward attracting, then we have the ideal situation. But this situation usually does not hold. Then we still have the forward omega
limit sets, which are more general than the Vishik uniform attractor since they do not
require a rate of attraction that is uniform in the initial time, which is unrealistic in
many contexts. Each of these concepts provides us with useful information about the
asymptotic behaviour of a nonautonomous dynamical system. Taken together they
give us the complete picture.
Acknowledgements This work was partly supported by Chinese NSF grant 11971184.
References
1. M.L. Barabash Y.F. Suprunenko, A. Stefanovska, Chronotaxic systems: a simple paradigm
to treat time-dependent oscillatory dynamics stable under continuous perturbation. Nonlinear
Phenom. Comp. Syst. 8, 392–400 (2015)
2. A.N. Carvalho, J.A. Langa, J.C. Robinson, Attractors of Infinite Dimensional Nonautonomous
Dynamical Systems (Springer, Berlin, 2012)
3. D. Cheban, P.E. Kloeden, B. Schmalfuß, The relationship between pullback, forwards and
global attractors of nonautonomous dynamical systems. Nonlinear Dynam. Syst. Theory 2,
9–28 (2002)
4. V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics (American
Mathematical Society, Providence, Rhode Island, 2002)
5. P. Clemson, B. Petkoski, T. Stankovski, A. Stefanovska, Coupled nonautonomous oscillators,
in Nonautonomous Dynamical Systems in the Life Sciences, vol. 2102, ed. by P.E. Kloeden, C.
Pötzsche (Springer LMS, 2014), pp. 163–197
6. H. Crauel, P.E. Kloeden, Nonautonomous and random Attractors. Jahresbericht der Deutschen
Mathematiker-Vereinigung 117, 173–206 (2015)
7. C.M. Dafermos, An invariance principle for compact processes. J. Different. Eqs. 9, 239–252
(1971)
8. J.K. Hale, Asymptotic Behavior of Dissipative Systems (American Mathematical Society, Providence, 1988)
9. A. Haraux, Systemes dynamiques dissipatifs et applications, Research in Applied Mathematics,
vol. 17 (Masson, Paris, 1991)
10. P.E. Kloeden, Asymptotic invariance and limit sets of general control systems. J. Different.
Eqs. 19, 91–105 (1975)
11. P.E. Kloeden, Asymptotic invariance and the discretisation of nonautonomous forward attracting sets. J. Comput. Dynam. 3, 179–189 (2016)
12. P.E. Kloeden, T. Lorenz, Pullback and forward attractors of nonautonomous difference equations, in Proceedings of ICDEA Wuhan, vol. 2015, ed. by M. Bohner, Y. Ding, O. Dosly
(Springer, Heidelberg, 2014), pp. 37–48