American Institute of Mathematical Sciences

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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
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.  Google Scholar

[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.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, Rhode Island, 2002.  Google Scholar

[4]

A. Hill, Global dissipativity for A-stable methods, SIAM J. Numer. Anal., 34 (1997), 119-142. doi: 10.1137/S0036142994270971.  Google Scholar

[5]

A. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces, BIT, 37 (1997), 37-42. doi: 10.1007/BF02510171.  Google Scholar

[6]

P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105. 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), 423-433.  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), 986-995. doi: 10.1137/0723066.  Google Scholar

[9]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268.  Google Scholar

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 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), 2695-2719. 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), 513-525.  Google Scholar

[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.  Google Scholar

[14]

J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976.  Google Scholar

[15]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, vol. 2002, Springer-Verlag, Heidelberg, 2010. 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), 2505-2518. doi: 10.1137/140996719.  Google Scholar

[17]

A. M. Stuart and A. R. Humphries, Numerical Analysis and Dynamical Systems, Cambridge University Press, Cambridge, 1996.  Google Scholar

[18]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 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), 490-522. 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, 182. Springer, New York, 2013.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, Rhode Island, 2002.  Google Scholar

[4]

A. Hill, Global dissipativity for A-stable methods, SIAM J. Numer. Anal., 34 (1997), 119-142. doi: 10.1137/S0036142994270971.  Google Scholar

[5]

A. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces, BIT, 37 (1997), 37-42. doi: 10.1007/BF02510171.  Google Scholar

[6]

P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105. 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), 423-433.  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), 986-995. doi: 10.1137/0723066.  Google Scholar

[9]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268.  Google Scholar

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 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), 2695-2719. 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), 513-525.  Google Scholar

[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.  Google Scholar

[14]

J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976.  Google Scholar

[15]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, vol. 2002, Springer-Verlag, Heidelberg, 2010. 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), 2505-2518. doi: 10.1137/140996719.  Google Scholar

[17]

A. M. Stuart and A. R. Humphries, Numerical Analysis and Dynamical Systems, Cambridge University Press, Cambridge, 1996.  Google Scholar

[18]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.  Google Scholar

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