American Institute of Mathematical Sciences

Advanced
June  2016, 3(2): 179-189. doi: 10.3934/jcd.2016009

Asymptotic invariance and the discretisation of nonautonomous forward attracting sets

Peter E. Kloeden 1,

1. 

School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074

Received  August 2016 Revised  November 2016 Published  January 2017

The $\omega$-limit set $\omega_B$ of a nonautonomous dynamical system generated by a nonautonomous ODE with a positive invariant compact absorbing set $B$ is shown to be asymptotic positive invariant in general and asymptotic negative invariant if, in addition, the vector field is uniformly continuous in time on the absorbing set. This set has been called the forward attracting set of the nonautonomous dynamical system and is related to Vishik's concept of a uniform attractor. If $\omega_B$ is also assumed to be uniformly attracting, then its upper semi continuity in a parameter and the upper semi continuous convergence of its counterparts under discretisation by the implicit Euler scheme are established.
Keywords: asymptotic positive invariance, asymptotic negative invariance, implicit Euler scheme, forward attracting set, upper semi continuous convergence., uniform attractor, Nonautonomous dynamical system, upper semi continuous dependence on parameters, omega limit points.
Mathematics Subject Classification: Primary: 34D45; Secondary: 37B55, 65P4.
Citation: 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
References:
[1]

M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows,, J. Differential Equations, 257 (2014), 490.  doi: 10.1016/j.jde.2014.04.008.  Google Scholar

[2]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems,, Applied Mathematical Sciences, (2013).   Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002).   Google Scholar

[4]

A. Hill, Global dissipativity for A-stable methods,, SIAM J. Numer. Anal., 34 (1997), 119.  doi: 10.1137/S0036142994270971.  Google Scholar

[5]

A. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces,, BIT, 37 (1997), 37.  doi: 10.1007/BF02510171.  Google Scholar

[6]

P. E. Kloeden, Asymptotic invariance and limit sets of general control systems,, J. Differential Equations, 19 (1975), 91.  doi: 10.1016/0022-0396(75)90021-2.  Google Scholar

[7]

P. E. Kloeden and V. S. Kozyakin, Uniform nonautonomous attractors under discretization,, Discrete Contin. Dyn. Systems., 10 (2004), 423.   Google Scholar

[8]

P. E. Kloeden and J. Lorenz, Stable attracting sets in dynamical systems and in their one-step discretizations,, SIAM J. Numer. Analysis, 23 (1986), 986.  doi: 10.1137/0723066.  Google Scholar

[9]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors,, Proc. Amer. Mat. Soc., 144 (2016), 259.   Google Scholar

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Amer. Math. Soc., (2011).   Google Scholar

[11]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic,, Nonlinear Analysis TMA, 74 (2011), 2695.  doi: 10.1016/j.na.2010.12.025.  Google Scholar

[12]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations,, J. Difference Eqns. Applns., 22 (2016), 513.   Google Scholar

[13]

V. Lakshmikantham an S. Leela, Asymptotic self-invariant sets and conditional stability,, in Proc. Inter. Symp. Diff. Equations and Dynamical Systems, (1967), 363.   Google Scholar

[14]

J. P. Lasalle, The Stability of Dynamical Systems,, SIAM-CBMS, (1976).   Google Scholar

[15]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems,, Lecture Notes in Mathematics, (2002).   Google Scholar

[16]

S. Sato, T. Matsuo , H. Suzuki and D. Furihata, A Lyapunov-type theorem for dissipative numerical integrators with adaptive time-stepping,, SIAM J. Numer. Anal., 53 (2015), 2505.  doi: 10.1137/140996719.  Google Scholar

[17]

A. M. Stuart and A. R. Humphries, Numerical Analysis and Dynamical Systems,, Cambridge University Press, (1996).   Google Scholar

[18]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations,, Cambridge University Press, (1992).   Google Scholar

show all references

References:
[1]

M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows,, J. Differential Equations, 257 (2014), 490.  doi: 10.1016/j.jde.2014.04.008.  Google Scholar

[2]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems,, Applied Mathematical Sciences, (2013).   Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002).   Google Scholar

[4]

A. Hill, Global dissipativity for A-stable methods,, SIAM J. Numer. Anal., 34 (1997), 119.  doi: 10.1137/S0036142994270971.  Google Scholar

[5]

A. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces,, BIT, 37 (1997), 37.  doi: 10.1007/BF02510171.  Google Scholar

[6]

P. E. Kloeden, Asymptotic invariance and limit sets of general control systems,, J. Differential Equations, 19 (1975), 91.  doi: 10.1016/0022-0396(75)90021-2.  Google Scholar

[7]

P. E. Kloeden and V. S. Kozyakin, Uniform nonautonomous attractors under discretization,, Discrete Contin. Dyn. Systems., 10 (2004), 423.   Google Scholar

[8]

P. E. Kloeden and J. Lorenz, Stable attracting sets in dynamical systems and in their one-step discretizations,, SIAM J. Numer. Analysis, 23 (1986), 986.  doi: 10.1137/0723066.  Google Scholar

[9]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors,, Proc. Amer. Mat. Soc., 144 (2016), 259.   Google Scholar

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Amer. Math. Soc., (2011).   Google Scholar

[11]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic,, Nonlinear Analysis TMA, 74 (2011), 2695.  doi: 10.1016/j.na.2010.12.025.  Google Scholar

[12]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations,, J. Difference Eqns. Applns., 22 (2016), 513.   Google Scholar

[13]

V. Lakshmikantham an S. Leela, Asymptotic self-invariant sets and conditional stability,, in Proc. Inter. Symp. Diff. Equations and Dynamical Systems, (1967), 363.   Google Scholar

[14]

J. P. Lasalle, The Stability of Dynamical Systems,, SIAM-CBMS, (1976).   Google Scholar

[15]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems,, Lecture Notes in Mathematics, (2002).   Google Scholar

[16]

S. Sato, T. Matsuo , H. Suzuki and D. Furihata, A Lyapunov-type theorem for dissipative numerical integrators with adaptive time-stepping,, SIAM J. Numer. Anal., 53 (2015), 2505.  doi: 10.1137/140996719.  Google Scholar

[17]

A. M. Stuart and A. R. Humphries, Numerical Analysis and Dynamical Systems,, Cambridge University Press, (1996).   Google Scholar

[18]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations,, Cambridge University Press, (1992).   Google Scholar

[1]

Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065
[2]

Christoph Kawan. Upper and lower estimates for invariance entropy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 169-186. doi: 10.3934/dcds.2011.30.169
[3]

Christian Licht, Thibaut Weller. Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1709-1741. doi: 10.3934/dcdss.2019114
[4]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899
[5]

Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058
[6]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499
[7]

Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017
[8]

Messoud Efendiev, Etsushi Nakaguchi, Wolfgang L. Wendland. Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system. Conference Publications, 2007, 2007 (Special) : 334-343. doi: 10.3934/proc.2007.2007.334
[9]

Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521
[10]

Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309
[11]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557
[12]

Paola Goatin, Philippe G. LeFloch. $L^1$ continuous dependence for the Euler equations of compressible fluids dynamics. Communications on Pure & Applied Analysis, 2003, 2 (1) : 107-137. doi: 10.3934/cpaa.2003.2.107
[13]

Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2487-2503. doi: 10.3934/dcdsb.2013.18.2487
[14]

Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025
[15]

Aicha Balhag, Zaki Chbani, Hassan Riahi. Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems. Evolution Equations & Control Theory, 2018, 7 (3) : 373-401. doi: 10.3934/eect.2018019
[16]

Xiaojin Zheng, Zhongyi Jiang. Tighter quadratically constrained convex reformulations for semi-continuous quadratic programming. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020071
[17]

Doria Affane, Meriem Aissous, Mustapha Fateh Yarou. Almost mixed semi-continuous perturbation of Moreau's sweeping process. Evolution Equations & Control Theory, 2020, 9 (1) : 27-38. doi: 10.3934/eect.2020015
[18]

Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic & Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030
[19]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
[20]

Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807

 Impact Factor: 

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences