Asymptotic invariance and  the discretisation of nonautonomous forward attracting sets

Peter E. Kloeden

doi:10.3934/jcd.2016009

Article Contents

2016, Volume 3, Issue 2: 179-189. Doi: 10.3934/jcd.2016009

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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: July 31, 2016

Revised: October 31, 2016

Published: May 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 34D45; Secondary: 37B55, 65P40.

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