# American Institute of Mathematical Sciences

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

## Asymptotic invariance and the discretisation of nonautonomous forward attracting sets

 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.
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.

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##### 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.
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