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Computing coherent sets using the Fokker-Planck equation
Asymptotic invariance and the discretisation of nonautonomous forward attracting sets
1. | School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074 |
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. |
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. |
[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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