# American Institute of Mathematical Sciences

January  2014, 1(1): 135-162. doi: 10.3934/jcd.2014.1.135

## A closing scheme for finding almost-invariant sets in open dynamical systems

 1 School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052 2 School of Mathematics and Physics, The University of Queensland, St Lucia QLD 4072, Australia 3 School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia

Received  September 2011 Revised  June 2012 Published  April 2014

We explore the concept of metastability or almost-invariance in open dynamical systems. In such systems, the loss of mass through a hole'' occurs in the presence of metastability. We extend existing techniques for finding almost-invariant sets in closed systems to open systems by introducing a closing operation that has a small impact on the system's metastability.
Citation: Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135
##### References:

show all references

##### References:
 [1] Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457 [2] Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003 [3] Martin Lustig, Caglar Uyanik. Perron-Frobenius theory and frequency convergence for reducible substitutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 355-385. doi: 10.3934/dcds.2017015 [4] Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007 [5] Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219 [6] Jiu Ding, Noah H. Rhee. A unified maximum entropy method via spline functions for Frobenius-Perron operators. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 235-245. doi: 10.3934/naco.2013.3.235 [7] Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007 [8] Paweł Góra, Abraham Boyarsky. Stochastic perturbations and Ulam's method for W-shaped maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1937-1944. doi: 10.3934/dcds.2013.33.1937 [9] Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569 [10] Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806 [11] Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016 [12] Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35 [13] R. Baier, M. Dellnitz, M. Hessel-von Molo, S. Sertl, I. G. Kevrekidis. The computation of convex invariant sets via Newton's method. Journal of Computational Dynamics, 2014, 1 (1) : 39-69. doi: 10.3934/jcd.2014.1.39 [14] Alexey Glutsyuk, Yury Kudryashov. No planar billiard possesses an open set of quadrilateral trajectories. Journal of Modern Dynamics, 2012, 6 (3) : 287-326. doi: 10.3934/jmd.2012.6.287 [15] Simone Paleari, Tiziano Penati. Equipartition times in a Fermi-Pasta-Ulam system. Conference Publications, 2005, 2005 (Special) : 710-719. doi: 10.3934/proc.2005.2005.710 [16] N. D. Cong, T. S. Doan, S. Siegmund. A Bohl-Perron type theorem for random dynamical systems. Conference Publications, 2011, 2011 (Special) : 322-331. doi: 10.3934/proc.2011.2011.322 [17] Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123 [18] Anushaya Mohapatra, William Ott. Memory loss for nonequilibrium open dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3747-3759. doi: 10.3934/dcds.2014.34.3747 [19] Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206 [20] Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817

Impact Factor: