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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
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show all references

References:
[1]

J. Differential Equations, 257 (2014), 490-522. doi: 10.1016/j.jde.2014.04.008.  Google Scholar

[2]

Applied Mathematical Sciences, 182. Springer, New York, 2013.  Google Scholar

[3]

Amer. Math. Soc., Providence, Rhode Island, 2002.  Google Scholar

[4]

SIAM J. Numer. Anal., 34 (1997), 119-142. doi: 10.1137/S0036142994270971.  Google Scholar

[5]

BIT, 37 (1997), 37-42. doi: 10.1007/BF02510171.  Google Scholar

[6]

J. Differential Equations, 19 (1975), 91-105. doi: 10.1016/0022-0396(75)90021-2.  Google Scholar

[7]

Discrete Contin. Dyn. Systems., 10 (2004), 423-433.  Google Scholar

[8]

SIAM J. Numer. Analysis, 23 (1986), 986-995. doi: 10.1137/0723066.  Google Scholar

[9]

Proc. Amer. Mat. Soc., 144 (2016), 259-268.  Google Scholar

[10]

Amer. Math. Soc., Providence, 2011.  Google Scholar

[11]

Nonlinear Analysis TMA, 74 (2011), 2695-2719. doi: 10.1016/j.na.2010.12.025.  Google Scholar

[12]

J. Difference Eqns. Applns., 22 (2016), 513-525.  Google Scholar

[13]

in Proc. Inter. Symp. Diff. Equations and Dynamical Systems, Puerto Rico 1965, Academic Press, New York, (1967), 363-373.  Google Scholar

[14]

SIAM-CBMS, Philadelphia, 1976.  Google Scholar

[15]

Lecture Notes in Mathematics, vol. 2002, Springer-Verlag, Heidelberg, 2010. Google Scholar

[16]

SIAM J. Numer. Anal., 53 (2015), 2505-2518. doi: 10.1137/140996719.  Google Scholar

[17]

Cambridge University Press, Cambridge, 1996.  Google Scholar

[18]

Cambridge University Press, Cambridge, 1992.  Google Scholar

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