A closing scheme for finding almostinvariant sets in open dynamical systems
Pages: 135  162,
Issue 1,
June
2014
doi:10.3934/jcd.2014.1.135 Abstract
References
Full text (1584.7K)
Related Articles
Gary Froyland  School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia (email)
Philip K. Pollett  School of Mathematics and Physics, The University of Queensland, St Lucia QLD 4072, Australia (email)
Robyn M. Stuart  School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia (email)
1 
W. Bahsoun, Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), 2529. 

2 
M. S. Bartlett, Stochastic Population Models in Ecology and Epidemiology, Methuen, London, 1960. 

3 
L. Billings and I. B. Schwartz, Identifying almost invariant sets in stochastic dynamical systems, Chaos, 18 (2008), 023122. 

4 
C. Bose, G. Froyland, C. Gonzáles Tokman and R. Murray, Ulam's method for LasotaYorke maps with holes, To appear in SIAM J. Appl. Dynam. Syst. arXiv:1204.2329. 

5 
A. Boyarsky and P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension (Probability and its Applications), Springer, Berlin, 1997. 

6 
P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, SpringerVerlag, 1999. 

7 
H. Bruin, M. Demers and I. Melbourne, Existence and convergence properties of physical measures for certain dynamical systems with holes, Ergodic Theory and Dynamical Systems, 30 (2010), 687728. 

8 
D. Clancy and P. K. Pollett, A note on quasistationary distributions of birthdeath processes and the SIS logistic epidemic, Journal of Applied Probability, 40 (2003), 821825. 

9 
P. Collet, S. Martínez and V. MaumeDeschamps, On the existence of conditionally invariant probability measures in dynamical systems, Nonlinearity, 13 (2000), 12631274. 

10 
P. Collet, S. Martínez and B. Schmitt, The LasotaYorke measure and the asymptotic law in the limit Cantor set of expanding systems, Nonlinearity, 7 (1996), 14371443. 

11 
P. Collet, S. Martínez and B. Schmitt, On the enhancement of diffusion by chaos, escape rates and stochastic stability, Transactions of the American Mathematical Society, 351 (1999), 28752897. 

12 
M. Dellnitz, G. Froyland, C. Horenkamp, K. PadbergGehle and A. Sen Gupta, Seasonal variability of the subpolar gyres in the southern ocean: A numerical investigation based on transfer operators, Nonlinear Processes in Geophysics, 16 (2009), 655663. 

13 
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO  set oriented numerical methods for dynamical systems, In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), Springer (2001) 145174. 

14 
M. Dellnitz and O. Junge, Almostinvariant sets in Chua's circuit, International Journal of Bifurcation and Chaos Appl. Sci. Engrg., 7 (1997), 24752485. 

15 
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1999), 491515. 

16 
M. Demers, Markov extensions for dynamical systems with holes: An application to expanding maps of the interval, Israel J. Math., 146 (2005), 189221. 

17 
M. Demers and L.S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377397. 

18 
P. Deuflhard, M. Dellnitz, O. Junge and C. Schütte, Computation of essential molecular dynamics by subdivision techniques, in Computational Molecular Dynamics: Challenges, Methods, Ideas (eds. P. Deuflhard, J. Hermans, B. Leimkuhler, A.E. Mark, S. Reich, R.D. Skeel), Springer Berlin Heidelberg (1999), 98115. 

19 
P. Deuflhard, W. Huisinga, A. Fischer and C. Schütte, Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains, Linear Algebra and its Applications, 315 (2000), 3959. 

20 
J. Ding and A. Zhou, Finite element approximations of FrobeniusPerron operators  a solution to Ulam's conjecture for multidimensional transformations, Physica D, 92 (1996), 6168. 

21 
J. Ding and A. Zhou, Statistical Properties of Deterministic Systems, Springer, Berlin, 2009. 

22 
P.A. Ferrari, H. Kesten, S. Martínez and S. Picco, Existence of quasistationary distributions. a renewal dynamical approach, Annals of Probability, 23 (1995), 501521. 

23 
G. Froyland, Finite approximation of SinaiBowenRuelle measures of Anosov systems in two dimensions, Random Comput. Dynamics, 3 (1995), 251264. 

24 
G. Froyland, Ulam's method for random interval maps, Nonlinearity, 12 (1999), 10291052. 

25 
G. Froyland, Extracting dynamical behaviour via Markov models, in Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, Cambridge, 1998, (ed. Alistair Mees), Birkhauser (2001), 283324. 

26 
G. Froyland, Statistically optimal almostinvariant sets, Physica D, 200 (2005), 205219. 

27 
G. Froyland and M. Dellnitz, Detecting and locating nearoptimal almostinvariant sets and cycles, SIAM Journal Sci. Comput., 24 (2003), 18391863. 

28 
G. Froyland and K. Padberg, Almostinvariant sets and invariant manifolds: Connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D, 238 (2009), 15071523. 

29 
G. Froyland, K. Padberg, M. England and A.M. Treguier, Detection of coherent oceanic structures via transfer operators, Physics Review Letters, 98 (2007), 224503. 

30 
G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in timedependent dynamical systems: Finitetime coherent sets, Chaos, 20 (2010), 043116. 

31 
G. Froyland and O. Stancevic, Escape rates and PerronFrobenius operators: Open and closed dynamical systems, Disc. Cont. Dynam. Sys. B, 14 (2010), 457472. 

32 
C. González Tokman, B.R. Hunt and P. Wright, Approximating invariant densities of metastable systems, Ergodic Theory and Dynamical Systems, 31 (2010), 13451361. 

33 
G. Keller and C. Liverani, Rare events, escape rates and quasistationarity: Some exact formulae, Journal of Statistical Physics, 135 (2009), 519534. 

34 
T.Y. Li, Finite approximation for the PerronFrobenius operator: a solution to Ulam's conjecture, Journal of Approximation Theory, 17 (1976), 177186. 

35 
C. Liverani and V. MaumeDeschamps, LasotaYorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set, Ann. Inst. H. Poinc. Probab. Statist., 39 (2003), 385412. 

36 
R. Murray, Discrete Approximation of Invariant Densities, Ph.D thesis, University of Cambridge, 1997. 

37 
R. Murray, Ulam's method for some nonuniformly expanding maps, Discrete and continuous dynamical systems, 26 (2010),10071018. 

38 
G. Pianigiani and J. Yorke, Expanding maps on sets which are almost invariant: Decay and chaos, Transactions of the American Mathematical Society, 252 (1979), 351366. 

39 
V. RomKedar, A. Leonard and S. Wiggins, An analytical study of transport, mixing and chaos in an unsteady vortical flow, Journal of Fluid Mechanics, 214 (1990), 347394. 

40 
V. RomKedar and S. Wiggins, Transport in twodimensional maps, Archive for Rational Mechanics and Analysis, 109 (1990), 239298. 

41 
C. Schütte, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules, Ph.D thesis, Freie Universität Berlin, Department of Mathematics and Computer Science, 1999. 

42 
S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finitetime Lyapunov exponents in twodimensional aperiodic flows, Physica D, 212 (2005), 217304. 

43 
A. Sinclair and M. Jerrum, Approximate counting, uniform generation and rapidly mixing Markov chains, Information and Computation, 82 (1989), 93133. 

44 
M. A. Stremler, S. D. Ross, P. Grover and P. Kumar, Topological chaos and periodic braiding of almostcyclic sets, Phys. Rev. Lett., 106 (2011), 114101. 

45 
S. Ulam, A Collection of Mathematical Problems, Interscience, 1979. 

46 
P. Walters, An introduction to Ergodic Theory, SpringerVerlag, 1982. 

47 
S. Wiggins, Chaotic Transport in Dynamical Systems, Springer, New York, 1992. 

Go to top
