Asymptotic invariance and the discretisation of nonautonomous forward attracting sets

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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

1.	School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074

Received August 2016 Revised November 2016 Published January 2017

Abstract
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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.*
[2]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems,, Applied Mathematical Sciences, (2013).
[3]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002).
[4]	A. Hill, Global dissipativity for A-stable methods,, SIAM J. Numer. Anal., 34 (1997), 119. doi: 10.1137/S0036142994270971.
[5]	A. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces,, BIT, 37 (1997), 37. doi: 10.1007/BF02510171.
[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.
[7]	P. E. Kloeden and V. S. Kozyakin, Uniform nonautonomous attractors under discretization,, Discrete Contin. Dyn. Systems., 10 (2004), 423.
[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.
[9]	P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors,, Proc. Amer. Mat. Soc., 144 (2016), 259.
[10]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Amer. Math. Soc., (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. 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.
[13]	V. Lakshmikantham an S. Leela, Asymptotic self-invariant sets and conditional stability,, in Proc. Inter. Symp. Diff. Equations and Dynamical Systems, (1967), 363.
[14]	J. P. Lasalle, The Stability of Dynamical Systems,, SIAM-CBMS, (1976).
[15]	C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems,, Lecture Notes in Mathematics, (2002).
[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.
[17]	A. M. Stuart and A. R. Humphries, Numerical Analysis and Dynamical Systems,, Cambridge University Press, (1996).
[18]	M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations,, Cambridge University Press, (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. 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, (2013).
[3]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002).
[4]	A. Hill, Global dissipativity for A-stable methods,, SIAM J. Numer. Anal., 34 (1997), 119. doi: 10.1137/S0036142994270971.
[5]	A. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces,, BIT, 37 (1997), 37. doi: 10.1007/BF02510171.
[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.
[7]	P. E. Kloeden and V. S. Kozyakin, Uniform nonautonomous attractors under discretization,, Discrete Contin. Dyn. Systems., 10 (2004), 423.
[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.
[9]	P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors,, Proc. Amer. Mat. Soc., 144 (2016), 259.
[10]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Amer. Math. Soc., (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. 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.
[13]	V. Lakshmikantham an S. Leela, Asymptotic self-invariant sets and conditional stability,, in Proc. Inter. Symp. Diff. Equations and Dynamical Systems, (1967), 363.
[14]	J. P. Lasalle, The Stability of Dynamical Systems,, SIAM-CBMS, (1976).
[15]	C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems,, Lecture Notes in Mathematics, (2002).
[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.
[17]	A. M. Stuart and A. R. Humphries, Numerical Analysis and Dynamical Systems,, Cambridge University Press, (1996).
[18]	M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations,, Cambridge University Press, (1992).

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