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]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122
[2]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465
[3]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[4]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[5]

Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017
[6]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[7]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[8]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292
[9]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218
[10]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304
[11]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[12]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[13]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[14]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318
[15]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[16]

Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293
[17]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384
[18]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[19]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[20]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

 Impact Factor: 

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences