Asymptotic invariance and the discretisation of nonautonomous forward attracting sets

Previous Article
Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation
JCD Home
This Issue
Next Article
Computing coherent sets using the Fokker-Planck equation

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

Abstract
Related Papers

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), 490-522. doi: 10.1016/j.jde.2014.04.008.
[2]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
[3]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, Rhode Island, 2002.
[4]	A. Hill, Global dissipativity for A-stable methods, SIAM J. Numer. Anal., 34 (1997), 119-142. doi: 10.1137/S0036142994270971.
[5]	A. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces, BIT, 37 (1997), 37-42. doi: 10.1007/BF02510171.
[6]	P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105. doi: 10.1016/0022-0396(75)90021-2.
[7]	P. E. Kloeden and V. S. Kozyakin, Uniform nonautonomous attractors under discretization, Discrete Contin. Dyn. Systems., 10 (2004), 423-433.
[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), 986-995. doi: 10.1137/0723066.
[9]	P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268.
[10]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011.
[11]	P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic, Nonlinear Analysis TMA, 74 (2011), 2695-2719. doi: 10.1016/j.na.2010.12.025.
[12]	P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Eqns. Applns., 22 (2016), 513-525.
[13]	V. Lakshmikantham an S. Leela, Asymptotic self-invariant sets and conditional stability, in Proc. Inter. Symp. Diff. Equations and Dynamical Systems, Puerto Rico 1965, Academic Press, New York, (1967), 363-373.
[14]	J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976.
[15]	C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, vol. 2002, Springer-Verlag, Heidelberg, 2010.
[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), 2505-2518. doi: 10.1137/140996719.
[17]	A. M. Stuart and A. R. Humphries, Numerical Analysis and Dynamical Systems, Cambridge University Press, Cambridge, 1996.
[18]	M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.

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), 490-522. doi: 10.1016/j.jde.2014.04.008.
[2]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
[3]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, Rhode Island, 2002.
[4]	A. Hill, Global dissipativity for A-stable methods, SIAM J. Numer. Anal., 34 (1997), 119-142. doi: 10.1137/S0036142994270971.
[5]	A. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces, BIT, 37 (1997), 37-42. doi: 10.1007/BF02510171.
[6]	P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105. doi: 10.1016/0022-0396(75)90021-2.
[7]	P. E. Kloeden and V. S. Kozyakin, Uniform nonautonomous attractors under discretization, Discrete Contin. Dyn. Systems., 10 (2004), 423-433.
[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), 986-995. doi: 10.1137/0723066.
[9]	P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268.
[10]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011.
[11]	P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic, Nonlinear Analysis TMA, 74 (2011), 2695-2719. doi: 10.1016/j.na.2010.12.025.
[12]	P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Eqns. Applns., 22 (2016), 513-525.
[13]	V. Lakshmikantham an S. Leela, Asymptotic self-invariant sets and conditional stability, in Proc. Inter. Symp. Diff. Equations and Dynamical Systems, Puerto Rico 1965, Academic Press, New York, (1967), 363-373.
[14]	J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976.
[15]	C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, vol. 2002, Springer-Verlag, Heidelberg, 2010.
[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), 2505-2518. doi: 10.1137/140996719.
[17]	A. M. Stuart and A. R. Humphries, Numerical Analysis and Dynamical Systems, Cambridge University Press, Cambridge, 1996.
[18]	M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.

[1]	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
[2]	Christoph Kawan. Upper and lower estimates for invariance entropy. Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899
[5]	David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499
[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]	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 and 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]	Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309
[10]	Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521
[11]	Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete and 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 and 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 and 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 and 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 and Control Theory, 2018, 7 (3) : 373-401. doi: 10.3934/eect.2018019
[16]	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 and Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
[17]	Doria Affane, Meriem Aissous, Mustapha Fateh Yarou. Almost mixed semi-continuous perturbation of Moreau's sweeping process. Evolution Equations and Control Theory, 2020, 9 (1) : 27-38. doi: 10.3934/eect.2020015
[18]	Xiaojin Zheng, Zhongyi Jiang. Tighter quadratically constrained convex reformulations for semi-continuous quadratic programming. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2331-2343. doi: 10.3934/jimo.2020071
[19]	Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic and Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030
[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

Other articles
by authors

on AIMS
Peter E. Kloeden
on Google Scholar
Peter E. Kloeden

[Back to Top]