January  2016, 3(1): 51-79. doi: 10.3934/jcd.2016003

On the numerical approximation of the Perron-Frobenius and Koopman operator

1. 

Department of Mathematics and Computer Science, Freie Universität Berlin, Germany, Germany

2. 

Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin

Received  December 2015 Revised  July 2016 Published  September 2016

Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review di erent methods that have been developed over the last decades to compute nite-dimensional approximations of these in nite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and di erences between these approaches. The results will be illustrated using simple stochastic di erential equations and molecular dynamics examples.
Citation: 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
References:
[1]

J. R. Baxter and J. S. Rosenthal, Rates of convergence for everywhere-positive Markov chains,, Statistics & Probability Letters, 22 (1995), 333.  doi: 10.1016/0167-7152(94)00085-M.  Google Scholar

[2]

A. Bittracher, P. Koltai and O. Junge, Pseudogenerators of spatial transfer operators,, SIAM Journal on Applied Dynamical Systems, 14 (2015), 1478.  doi: 10.1137/14099872X.  Google Scholar

[3]

E. M. Bollt and N. Santitissadeekorn, Applied and Computational Measurable Dynamics,, Society for Industrial and Applied Mathematics, (2013).  doi: 10.1137/1.9781611972641.  Google Scholar

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C. J. Bose and R. Murray, The exact rate of approximation in Ulam's method,, Discrete and Continuous Dynamical Systems, 7 (2001), 219.   Google Scholar

[5]

C. J. Bose and R. Murray, Dynamical conditions for convergence of a maximum entropy method for Frobenius-Perron operator equations,, Applied Mathematics and Computation, 182 (2006), 210.  doi: 10.1016/j.amc.2006.01.089.  Google Scholar

[6]

C. J. Bose and R. Murray, Minimum 'energy' approximations of invariant measures for nonsingular transformations,, Discrete and Continuous Dynamical Systems, 14 (2006), 597.   Google Scholar

[7]

C. J. Bose and R. Murray, Duality and the computation of approximate invariant densities for nonsingular transformations,, SIAM Journal on Optimization, 18 (2007), 691.  doi: 10.1137/060658163.  Google Scholar

[8]

A. Boyarsky and P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension,, Probability and its Applications. Birkhäuser Boston, (1997).  doi: 10.1007/978-1-4612-2024-4.  Google Scholar

[9]

J. P. Boyd, Chebyshev and Fourier Spectral Methods,, 2nd edition, (2001).   Google Scholar

[10]

M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 ().   Google Scholar

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H.-J. Bungartz and M. Griebel, Sparse grids,, Acta Numerica, 13 (2004), 147.  doi: 10.1017/S0962492904000182.  Google Scholar

[12]

D. A. Case, J. T. Berryman, R. M. Betz, D. S. Cerutti, T. E. Cheatham, T. A. Darden, R. E. Duke, T. J. Giese, H. Gohlke, A. W. Goetz, N. Homeyer, S. Izadi, P. Janowski, J. Kaus, A. Kovalenko, T. S. Lee, S. LeGrand, P. Li, T. Luchko, R. Luo, B. Madej, K. M. Merz, G. Monard, P. Needham, H. Nguyen, H. T. Nguyen, I. Omelyan, A. Onufriev, D. R. Roe, A. Roitberg, R. Salomon-Ferrer, C. L. Simmerling, W. Smith, J. Swails, R. C. Walker, J. Wang, R. M. Wolf, X. Wu, D. M. York and P. A. Kollman, AMBER 2015,, University of California, (2015).   Google Scholar

[13]

M. D. Chekroun, J. D. Neelin, D. Kondrashov, J. C. McWilliams and M. Ghil, Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances,, Proceedings of the National Academy of Sciences, 111 (2014), 1684.  doi: 10.1073/pnas.1321816111.  Google Scholar

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M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic theory, (2001), 145.   Google Scholar

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M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.  doi: 10.1137/S0036142996313002.  Google Scholar

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J. Ding, A maximum entropy method for solving Frobenius-Perron operator equations,, Applied Mathematics and Computation, 93 (1998), 155.  doi: 10.1016/S0096-3003(97)10061-3.  Google Scholar

[18]

J. Ding, Q. Du and T.-Y. Li, High order approximation of the Frobenius-Perron operator,, Applied Mathematics and Computation, 53 (1993), 151.  doi: 10.1016/0096-3003(93)90099-Z.  Google Scholar

[19]

J. Ding and T.-Y. Li, Markov finite approximation of the Frobenius-Perron operator,, Nonlinear Analysis: Theory, 17 (1991), 759.  doi: 10.1016/0362-546X(91)90211-I.  Google Scholar

[20]

J. Ding and A. Zhou, Finite approximations of Frobenius-Perron operators. A solution of Ulam's conjucture on multi-dimensional transformations,, Physica D, 92 (1996), 61.  doi: 10.1016/0167-2789(95)00292-8.  Google Scholar

[21]

H. Federer, Geometric Measure Theory,, Springer New York, (1969).   Google Scholar

[22]

G. Froyland, C. González-Tokman and T. M. Watson, Optimal mixing enhancement by local perturbation,, Preprint., ().   Google Scholar

[23]

G. Froyland, G. Gottwald and A. Hammerlindl, A computational method to extract macroscopic variables and their dynamics in multiscale systems,, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1816.  doi: 10.1137/130943637.  Google Scholar

[24]

G. Froyland, R. M. Stuart and E. van Sebille, How well-connected is the surface of the global ocean?,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014).  doi: 10.1063/1.4892530.  Google Scholar

[25]

G. Froyland, Approximating physical invariant measures of mixing dynamical systems,, Nonlinear Analysis, 32 (1998), 831.  doi: 10.1016/S0362-546X(97)00527-0.  Google Scholar

[26]

G. Froyland and O. Junge, On fast computation of finite-time coherent sets using radial basis functions,, Chaos, 25 (2015).  doi: 10.1063/1.4927640.  Google Scholar

[27]

G. Froyland, O. Junge and P. Koltai, Estimating long term behavior of flows without trajectory integration: The infinitesimal generator approach,, SIAM Journal on Numerical Analysis, 51 (2013), 223.  doi: 10.1137/110819986.  Google Scholar

[28]

P. R. Halmos, Lectures on Ergodic Theory, vol. 142,, American Mathematical Soc., (1956).   Google Scholar

[29]

E. Hopf, The general temporally discrete Markoff process,, Journal of Rational Mechanics and Analysis, 3 (1954), 13.   Google Scholar

[30]

P. Huber, Dünngitter-Spektralmethoden zur Approximation des Frobenius-Perron-Operators,, Diploma thesis (in German), (2009).   Google Scholar

[31]

M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition,, Physics of Fluids, 26 ().   Google Scholar

[32]

O. Junge and P. Koltai, Discretization of the Frobenius-Perron operator using a sparse Haar tensor basis: The Sparse Ulam method,, SIAM Journal on Numerical Analysis, 47 (2009), 3464.  doi: 10.1137/080716864.  Google Scholar

[33]

P. Koltai, Efficient Approximation Methods for the Global Long-Term Behavior of Dynamical Systems - Theory, Algorithms and Examples,, PhD thesis, (2010).   Google Scholar

[34]

B. O. Koopman, Hamiltonian systems and transformation in Hilbert space,, Proceedings of the National Academy of Sciences of the United States of America, 17 (1931), 315.  doi: 10.1073/pnas.17.5.315.  Google Scholar

[35]

U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics,, Walter de Gruyter & Co., (1985).  doi: 10.1515/9783110844641.  Google Scholar

[36]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, vol. 97 of Applied Mathematical Sciences,, 2nd edition, (1994).  doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[37]

T.-Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture,, Journal of Approximation Theory, 17 (1976), 177.  doi: 10.1016/0021-9045(76)90037-X.  Google Scholar

[38]

J. C. Mattingly and A. M. Stuart, Geometric ergodicity of some hypo-elliptic diffusions for particle motions,, Markov Process. Related Fields, 8 (2002), 199.   Google Scholar

[39]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability,, Springer Science & Business Media, (2012).   Google Scholar

[40]

R. Murray, Discrete Approximation of Invariant Densities,, PhD thesis, (1997).   Google Scholar

[41]

R. Murray, Optimal partition choice for invariant measure approximation for one-dimensional maps,, Nonlinearity, 17 (2004), 1623.  doi: 10.1088/0951-7715/17/5/004.  Google Scholar

[42]

F. Noé and F. Nüske, A variational approach to modeling slow processes in stochastic dynamical systems,, Multiscale Modeling & Simulation, 11 (2013), 635.  doi: 10.1137/110858616.  Google Scholar

[43]

F. Nüske, B. G. Keller, G. Pérez-Hernández, A. S. J. S. Mey and F. Noé, Variational approach to molecular kinetics,, Journal of Chemical Theory and Computation, 10 (2014), 1739.   Google Scholar

[44]

F. Nüske, R. Schneider, F. Vitalini and F. Noé, Variational tensor approach for approximating the rare-event kinetics of macromolecular systems,, The Journal of Chemical Physics, 144 ().   Google Scholar

[45]

S. Ober-Blöbaum and K. Padberg-Gehle, Multiobjective optimal control of fluid mixing,, PAMM, 15 (2015), 639.   Google Scholar

[46]

D. Ornstein, Bernoulli shifts with the same entropy are isomorphic,, Advances in Mathematics, 4 (1970), 337.  doi: 10.1016/0001-8708(70)90029-0.  Google Scholar

[47]

R. Preis, M. Dellnitz, M. Hessel, C. Schütte and E. Meerbach, Dominant Paths Between Almost Invariant Sets of Dynamical Systems,, DFG Schwerpunktprogramm 1095, (1095).   Google Scholar

[48]

P. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data,, in 61st Annual Meeting of the APS Division of Fluid Dynamics, (2008).   Google Scholar

[49]

Schrödinger, LLC, The PyMOL molecular graphics system,, Version 1.7.4, (2014).   Google Scholar

[50]

C. Schütte, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules, 1999,, Habilitation Thesis., ().   Google Scholar

[51]

C. Schütte and M. Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches,, no. 24 in Courant Lecture Notes, (2013).   Google Scholar

[52]

Y. G. Sinai, On the notion of entropy of dynamical systems,, in Doklady Akademii Nauk, 124 (1959), 768.   Google Scholar

[53]

A. Tantet, V. Lucarini, F. Lunkeit and H. A. Dijkstra, Crisis of the chaotic attractor of a climate model: A transfer operator approach,, Preprint, ().   Google Scholar

[54]

A. Tantet, F. R. van der Burgt and H. A. Dijkstra, An early warning indicator for atmospheric blocking events using transfer operators,, Chaos, 25 (2015).  doi: 10.1063/1.4908174.  Google Scholar

[55]

J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and J. N. Kutz, On dynamic mode decomposition: Theory and applications,, ArXiv e-prints., ().   Google Scholar

[56]

S. M. Ulam, A Collection of Mathematical Problems,, Interscience Publisher NY, (1960).   Google Scholar

[57]

U. Vaidya, P. G. Mehta and U. V. Shanbhag, Nonlinear stabilization via control Lyapunov measure,, IEEE Transactions on Automatic Control, 55 (2010), 1314.  doi: 10.1109/TAC.2010.2042226.  Google Scholar

[58]

M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition,, J. Nonlinear Sci., 25 (2015), 1307.  doi: 10.1007/s00332-015-9258-5.  Google Scholar

[59]

M. O. Williams, C. W. Rowley and I. G. Kevrekidis, A kernel-based approach to data-driven Koopman spectral analysis,, J. Comput. Dyn., 2 (2015), 247.  doi: 10.3934/jcd.2015005.  Google Scholar

[60]

M. O. Williams, I. I. Rypina and C. W. Rowley, Identifying finite-time coherent sets from limited quantities of Lagrangian data,, Chaos, 25 (2015).  doi: 10.1063/1.4927424.  Google Scholar

show all references

References:
[1]

J. R. Baxter and J. S. Rosenthal, Rates of convergence for everywhere-positive Markov chains,, Statistics & Probability Letters, 22 (1995), 333.  doi: 10.1016/0167-7152(94)00085-M.  Google Scholar

[2]

A. Bittracher, P. Koltai and O. Junge, Pseudogenerators of spatial transfer operators,, SIAM Journal on Applied Dynamical Systems, 14 (2015), 1478.  doi: 10.1137/14099872X.  Google Scholar

[3]

E. M. Bollt and N. Santitissadeekorn, Applied and Computational Measurable Dynamics,, Society for Industrial and Applied Mathematics, (2013).  doi: 10.1137/1.9781611972641.  Google Scholar

[4]

C. J. Bose and R. Murray, The exact rate of approximation in Ulam's method,, Discrete and Continuous Dynamical Systems, 7 (2001), 219.   Google Scholar

[5]

C. J. Bose and R. Murray, Dynamical conditions for convergence of a maximum entropy method for Frobenius-Perron operator equations,, Applied Mathematics and Computation, 182 (2006), 210.  doi: 10.1016/j.amc.2006.01.089.  Google Scholar

[6]

C. J. Bose and R. Murray, Minimum 'energy' approximations of invariant measures for nonsingular transformations,, Discrete and Continuous Dynamical Systems, 14 (2006), 597.   Google Scholar

[7]

C. J. Bose and R. Murray, Duality and the computation of approximate invariant densities for nonsingular transformations,, SIAM Journal on Optimization, 18 (2007), 691.  doi: 10.1137/060658163.  Google Scholar

[8]

A. Boyarsky and P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension,, Probability and its Applications. Birkhäuser Boston, (1997).  doi: 10.1007/978-1-4612-2024-4.  Google Scholar

[9]

J. P. Boyd, Chebyshev and Fourier Spectral Methods,, 2nd edition, (2001).   Google Scholar

[10]

M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 ().   Google Scholar

[11]

H.-J. Bungartz and M. Griebel, Sparse grids,, Acta Numerica, 13 (2004), 147.  doi: 10.1017/S0962492904000182.  Google Scholar

[12]

D. A. Case, J. T. Berryman, R. M. Betz, D. S. Cerutti, T. E. Cheatham, T. A. Darden, R. E. Duke, T. J. Giese, H. Gohlke, A. W. Goetz, N. Homeyer, S. Izadi, P. Janowski, J. Kaus, A. Kovalenko, T. S. Lee, S. LeGrand, P. Li, T. Luchko, R. Luo, B. Madej, K. M. Merz, G. Monard, P. Needham, H. Nguyen, H. T. Nguyen, I. Omelyan, A. Onufriev, D. R. Roe, A. Roitberg, R. Salomon-Ferrer, C. L. Simmerling, W. Smith, J. Swails, R. C. Walker, J. Wang, R. M. Wolf, X. Wu, D. M. York and P. A. Kollman, AMBER 2015,, University of California, (2015).   Google Scholar

[13]

M. D. Chekroun, J. D. Neelin, D. Kondrashov, J. C. McWilliams and M. Ghil, Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances,, Proceedings of the National Academy of Sciences, 111 (2014), 1684.  doi: 10.1073/pnas.1321816111.  Google Scholar

[14]

G. Chen and T. Ueta (eds.), Chaos in Circuits and Systems,, World Scientific Series on Nonlinear Science, (2002).  doi: 10.1142/9789812705303.  Google Scholar

[15]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic theory, (2001), 145.   Google Scholar

[16]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.  doi: 10.1137/S0036142996313002.  Google Scholar

[17]

J. Ding, A maximum entropy method for solving Frobenius-Perron operator equations,, Applied Mathematics and Computation, 93 (1998), 155.  doi: 10.1016/S0096-3003(97)10061-3.  Google Scholar

[18]

J. Ding, Q. Du and T.-Y. Li, High order approximation of the Frobenius-Perron operator,, Applied Mathematics and Computation, 53 (1993), 151.  doi: 10.1016/0096-3003(93)90099-Z.  Google Scholar

[19]

J. Ding and T.-Y. Li, Markov finite approximation of the Frobenius-Perron operator,, Nonlinear Analysis: Theory, 17 (1991), 759.  doi: 10.1016/0362-546X(91)90211-I.  Google Scholar

[20]

J. Ding and A. Zhou, Finite approximations of Frobenius-Perron operators. A solution of Ulam's conjucture on multi-dimensional transformations,, Physica D, 92 (1996), 61.  doi: 10.1016/0167-2789(95)00292-8.  Google Scholar

[21]

H. Federer, Geometric Measure Theory,, Springer New York, (1969).   Google Scholar

[22]

G. Froyland, C. González-Tokman and T. M. Watson, Optimal mixing enhancement by local perturbation,, Preprint., ().   Google Scholar

[23]

G. Froyland, G. Gottwald and A. Hammerlindl, A computational method to extract macroscopic variables and their dynamics in multiscale systems,, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1816.  doi: 10.1137/130943637.  Google Scholar

[24]

G. Froyland, R. M. Stuart and E. van Sebille, How well-connected is the surface of the global ocean?,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014).  doi: 10.1063/1.4892530.  Google Scholar

[25]

G. Froyland, Approximating physical invariant measures of mixing dynamical systems,, Nonlinear Analysis, 32 (1998), 831.  doi: 10.1016/S0362-546X(97)00527-0.  Google Scholar

[26]

G. Froyland and O. Junge, On fast computation of finite-time coherent sets using radial basis functions,, Chaos, 25 (2015).  doi: 10.1063/1.4927640.  Google Scholar

[27]

G. Froyland, O. Junge and P. Koltai, Estimating long term behavior of flows without trajectory integration: The infinitesimal generator approach,, SIAM Journal on Numerical Analysis, 51 (2013), 223.  doi: 10.1137/110819986.  Google Scholar

[28]

P. R. Halmos, Lectures on Ergodic Theory, vol. 142,, American Mathematical Soc., (1956).   Google Scholar

[29]

E. Hopf, The general temporally discrete Markoff process,, Journal of Rational Mechanics and Analysis, 3 (1954), 13.   Google Scholar

[30]

P. Huber, Dünngitter-Spektralmethoden zur Approximation des Frobenius-Perron-Operators,, Diploma thesis (in German), (2009).   Google Scholar

[31]

M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition,, Physics of Fluids, 26 ().   Google Scholar

[32]

O. Junge and P. Koltai, Discretization of the Frobenius-Perron operator using a sparse Haar tensor basis: The Sparse Ulam method,, SIAM Journal on Numerical Analysis, 47 (2009), 3464.  doi: 10.1137/080716864.  Google Scholar

[33]

P. Koltai, Efficient Approximation Methods for the Global Long-Term Behavior of Dynamical Systems - Theory, Algorithms and Examples,, PhD thesis, (2010).   Google Scholar

[34]

B. O. Koopman, Hamiltonian systems and transformation in Hilbert space,, Proceedings of the National Academy of Sciences of the United States of America, 17 (1931), 315.  doi: 10.1073/pnas.17.5.315.  Google Scholar

[35]

U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics,, Walter de Gruyter & Co., (1985).  doi: 10.1515/9783110844641.  Google Scholar

[36]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, vol. 97 of Applied Mathematical Sciences,, 2nd edition, (1994).  doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[37]

T.-Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture,, Journal of Approximation Theory, 17 (1976), 177.  doi: 10.1016/0021-9045(76)90037-X.  Google Scholar

[38]

J. C. Mattingly and A. M. Stuart, Geometric ergodicity of some hypo-elliptic diffusions for particle motions,, Markov Process. Related Fields, 8 (2002), 199.   Google Scholar

[39]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability,, Springer Science & Business Media, (2012).   Google Scholar

[40]

R. Murray, Discrete Approximation of Invariant Densities,, PhD thesis, (1997).   Google Scholar

[41]

R. Murray, Optimal partition choice for invariant measure approximation for one-dimensional maps,, Nonlinearity, 17 (2004), 1623.  doi: 10.1088/0951-7715/17/5/004.  Google Scholar

[42]

F. Noé and F. Nüske, A variational approach to modeling slow processes in stochastic dynamical systems,, Multiscale Modeling & Simulation, 11 (2013), 635.  doi: 10.1137/110858616.  Google Scholar

[43]

F. Nüske, B. G. Keller, G. Pérez-Hernández, A. S. J. S. Mey and F. Noé, Variational approach to molecular kinetics,, Journal of Chemical Theory and Computation, 10 (2014), 1739.   Google Scholar

[44]

F. Nüske, R. Schneider, F. Vitalini and F. Noé, Variational tensor approach for approximating the rare-event kinetics of macromolecular systems,, The Journal of Chemical Physics, 144 ().   Google Scholar

[45]

S. Ober-Blöbaum and K. Padberg-Gehle, Multiobjective optimal control of fluid mixing,, PAMM, 15 (2015), 639.   Google Scholar

[46]

D. Ornstein, Bernoulli shifts with the same entropy are isomorphic,, Advances in Mathematics, 4 (1970), 337.  doi: 10.1016/0001-8708(70)90029-0.  Google Scholar

[47]

R. Preis, M. Dellnitz, M. Hessel, C. Schütte and E. Meerbach, Dominant Paths Between Almost Invariant Sets of Dynamical Systems,, DFG Schwerpunktprogramm 1095, (1095).   Google Scholar

[48]

P. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data,, in 61st Annual Meeting of the APS Division of Fluid Dynamics, (2008).   Google Scholar

[49]

Schrödinger, LLC, The PyMOL molecular graphics system,, Version 1.7.4, (2014).   Google Scholar

[50]

C. Schütte, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules, 1999,, Habilitation Thesis., ().   Google Scholar

[51]

C. Schütte and M. Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches,, no. 24 in Courant Lecture Notes, (2013).   Google Scholar

[52]

Y. G. Sinai, On the notion of entropy of dynamical systems,, in Doklady Akademii Nauk, 124 (1959), 768.   Google Scholar

[53]

A. Tantet, V. Lucarini, F. Lunkeit and H. A. Dijkstra, Crisis of the chaotic attractor of a climate model: A transfer operator approach,, Preprint, ().   Google Scholar

[54]

A. Tantet, F. R. van der Burgt and H. A. Dijkstra, An early warning indicator for atmospheric blocking events using transfer operators,, Chaos, 25 (2015).  doi: 10.1063/1.4908174.  Google Scholar

[55]

J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and J. N. Kutz, On dynamic mode decomposition: Theory and applications,, ArXiv e-prints., ().   Google Scholar

[56]

S. M. Ulam, A Collection of Mathematical Problems,, Interscience Publisher NY, (1960).   Google Scholar

[57]

U. Vaidya, P. G. Mehta and U. V. Shanbhag, Nonlinear stabilization via control Lyapunov measure,, IEEE Transactions on Automatic Control, 55 (2010), 1314.  doi: 10.1109/TAC.2010.2042226.  Google Scholar

[58]

M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition,, J. Nonlinear Sci., 25 (2015), 1307.  doi: 10.1007/s00332-015-9258-5.  Google Scholar

[59]

M. O. Williams, C. W. Rowley and I. G. Kevrekidis, A kernel-based approach to data-driven Koopman spectral analysis,, J. Comput. Dyn., 2 (2015), 247.  doi: 10.3934/jcd.2015005.  Google Scholar

[60]

M. O. Williams, I. I. Rypina and C. W. Rowley, Identifying finite-time coherent sets from limited quantities of Lagrangian data,, Chaos, 25 (2015).  doi: 10.1063/1.4927424.  Google Scholar

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