American Institute of Mathematical Sciences

Advanced
May  2005, 5(2): 215-238. doi: 10.3934/dcdsb.2005.5.215

Approximation of attractors of nonautonomous dynamical systems

Bernd Aulbach 1, , Martin Rasmussen 1, and  Stefan Siegmund 2,

1. 

Department of Mathematics, University of Augsburg, D-86135 Augsburg, Germany, Germany
2. 

Department of Mathematics, University of Frankfurt, D-60325 Frankfurt, Germany

Received  December 2003 Revised  July 2004 Published  February 2005

This paper is devoted to the numerical approximation of attractors. For general nonautonomous dynamical systems we first introduce a new type of attractor which includes some classes of noncompact attractors such as unbounded unstable manifolds. We then adapt two cell mapping algorithms to the nonautonomous setting and use the computer program GAIO for the analysis of an explicit example, a two-dimensional system of nonautonomous difference equations. Finally we present numerical data which indicate a bifurcation of nonautonomous attractors in the Duffing-van der Pol oscillator.
Keywords: Continuation algorithm, Subdivision algorithm, Nonautonomous diference equation, Invariant manifold, Numerical approximation, Forward attractor, Nonautonomous bifurcation., Pullback attractor, Nonautonomous dynamical system.
Mathematics Subject Classification: 34C30, 34D45, 37B55, 37G35, 39A10, 65L2.
Citation: Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215
[1]

Chunqiu Li, Desheng Li, Xuewei Ju. On the forward dynamical behavior of nonautonomous systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 473-487. doi: 10.3934/dcdsb.2019190
[2]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[3]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[4]

Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065
[5]

Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143
[6]

Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058
[7]

Christian Pötzsche. Nonautonomous continuation of bounded solutions. Communications on Pure and Applied Analysis, 2011, 10 (3) : 937-961. doi: 10.3934/cpaa.2011.10.937
[8]

Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663
[9]

Xiaoying Han, Peter E. Kloeden. Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces. Discrete and Continuous Dynamical Systems - S, 2022, 15 (10) : 2909-2927. doi: 10.3934/dcdss.2021143
[10]

David Julitz. Numerical approximation of atmospheric-ocean models with subdivision algorithm. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 429-447. doi: 10.3934/dcds.2007.18.429
[11]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[12]

Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011
[13]

Peter E. Kloeden. Asymptotic invariance and the discretisation of nonautonomous forward attracting sets. Journal of Computational Dynamics, 2016, 3 (2) : 179-189. doi: 10.3934/jcd.2016009
[14]

Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589
[15]

Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036
[16]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727
[17]

David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96
[18]

Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189
[19]

Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 855-882. doi: 10.3934/dcds.2009.24.855
[20]

Jintao Wang, Xiaoqian Zhang. Invariant sample measures and random Liouville type theorem for a nonautonomous stochastic $ p $-Laplacian equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022193

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (125)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy