Jump to ContentJump to Main Navigation
Show Summary Details
In This Section

Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose

1 Issue per year


Mathematical Citation Quotient (MCQ) 2015: 0.33


Emerging Science

Open Access
Online
ISSN
2353-0626
See all formats and pricing
In This Section

Pullback incremental attraction

Peter E. Kloeden
  • Corresponding author
  • Institut für Mathematik Goethe-Universität, 60054 Frankfurt am Main, Germany
  • Email:
/ Thomas Lorenz
  • Institut für Mathematik Goethe-Universität, 60054 Frankfurt am Main, Germany
  • Email:
Published Online: 2013-12-27 | DOI: https://doi.org/10.2478/msds-2013-0004

Abstract

A pullback incremental attraction, a nonautonomous version of incremental stability, is introduced for nonautonomous systems that may have unbounded limiting solutions. Its characterisation by a Lyapunov function is indicated

Keywords: Nonautonomous dynamical system; nonautonomous differential equation; pullback incremental stability; Lyapunov function; pullback attractors

References

  • [1] D. Angeli, A Lyapunov approach to the incremental stability properties, IEEE Trans. Automat. Control 47 (2002), 410-421. [Crossref]

  • [2] T. Caraballo, M.J. Garrido Atienza and B. Schmalfuß, Existence of exponentially attracting stationary solutions for delay evolution equations. Discrete Contin. Dyn. Syst. Ser. A 18 (2007), 271-293.

  • [3] T. Caraballo, P.E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl. Math. Optim. 50 (2004), 183-207. [Crossref]

  • [4] C.M. Dafermos, An invariance principle for compact processes, J. Differential Equations 9 (1971), 239-252. [Crossref]

  • [5] L. Grüne, P.E. Kloeden, S. Siegmund and F.R. Wirth, Lyapunov’s second method for nonautonomous differential equations, Discrete Contin. Dyn. Syst. Ser. A 18 (2007), 375-403.

  • [6] P.E. Kloeden, Lyapunov functions for cocycle attractors in nonautonomous difference equations, Izvetsiya Akad Nauk Rep Moldovia Mathematika 26 (1998), 32-42.

  • [7] P.E. Kloeden, A Lyapunov function for pullback attractors of nonautonomous differential equations, Electron. J. Differ. Equ. Conf. 05 (2000), 91-102.

  • [8] P.E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl. 28 (2010), 937-945.

  • [9] P.E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations 253 (2012), 1422-1438. [Crossref]

  • [10] P.E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011.

  • [11] B.S. Rüffer, N. van de Wouw and M. Mueller, Convergent systems vs. incremental stability, Systems Control Lett. 62 (2013), 277-285.

  • [12] E.D. Sontag, Comments on integral variants of ISS, Systems Control Lett. 34 (1998), 93-100.

  • [13] A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.

  • [14] Fuke Wu and P.E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B 18, No.6, (2013), 1715-1734.

  • [15] T. Yoshizawa, Stability Theory by Lyapunov’s Second Method. Math. Soc Japan, Tokyo, 1966.

About the article

Received: 2013-06-09

Accepted: 2013-11-11

Published Online: 2013-12-27

Published in Print: 2014-01-01



Citation Information: Nonautonomous Dynamical Systems, ISSN (Online) 2353-0626, DOI: https://doi.org/10.2478/msds-2013-0004. Export Citation

© 2013 Peter E. Kloeden, Thomas Lorenz. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Comments (0)

Please log in or register to comment.
Log in