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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Operator-theoretic Identification of Closed Sub-systems of Dynamical Systems

Discontinuity, Nonlinearity, and Complexity 4(1) (2016) 91--109 | DOI:10.5890/DNC.2016.03.007

Oliver Pfante$^{1}$, Nihat Ay$^{1}$,$^{2}$,$^{3}$

$^{1}$ Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany

$^{2}$ Department of Mathematics and Computer Science, Leipzig University, PF 100920, 04009 Leipzig, Germany

$^{3}$ Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

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A central problem of dynamical systems theory is to identify a reduced description of the dynamical process one can deal easier. In this paper we present a systematic method of identifying those closed sub-systems of a given discrete time dynamical system in the frame of operator theory. It is shown that this problem is closely related to finding invariant sigma algebras of the dynamics.


The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 267087 and no. 318723.


  1. [1]  Albeverio, S. and Høegh-Krohn, R. (1978), Frobenius theory for positive maps of von neumann algebras, Communications in Mathematical Physics, 64, 83-94.
  2. [2]  Buchholz, P. (1995), Hierarchical markovian models: symmetries and reduction, Performance Evaluation, 22(1), 93- 110.
  3. [3]  Burke, C.K. and Rosenblatt, M. (1958),A markovian function of a markov chain, Annals of Statistics, 29, 1112-1122.
  4. [4]  Capra, L., Dutheillet, C., Franceschinis, G., and Ilié, J. (2001), On the use of partial symmetries for lumping markov chains, ACM SIGMETRICS Performance Evaluation Review, 28(4), 33-35.
  5. [5]  Cobb, G. and Chen, Y. (2003), An application of markov chain monte carlo to community ecology, The American Mathematical Monthly, 110(4), 265-288.
  6. [6]  Deuflhard, P., Huisinga, W., Fischer, A., and Schütte, Ch. (2000), Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains, Linear Algebra and its Applications, 315 (1-3), 39-59.
  7. [7]  Dixmier, J. (1981), Von neumann Algebras, North-Holland Publishing Company.
  8. [8]  Furstenberg, H. (1980), Recurrence in ergodic theory and combinatorial number theory, Princeton University Press.
  9. [9]  Görnerup, Olof and Jacobi, Martin Nilsson (2008), A dual digenvector dondition for strong lumpability of markov chains, SFI WORKING PAPER , 1-7.
  10. [10]  Görnerup, Olof and Jacobi,Martin Nilsson (2010), A method for finding aggregated representations of linear dynamical systems, Advances in Complex Systems, 13(2), 199-215.
  11. [11]  Görnerup, Olof and Jacobi, Martin Nilsson (2010), A model-independent approach to infer hierarchical codon substitution dynamics, BMC Bioinformatics, 11 201 (eng).
  12. [12]  Israeli, N. and Goldenfeld, N. (2006), Coarse-graining of cellular automata, emergence, and the predictability of complex systems, Physical Review E, 73, 026203.
  13. [13]  Snell, J.L. and Kemeny, J.G.(1976), Finite Markov Chains, Springer-Verlag, New York.
  14. [14]  Jacobi, M. Nilsson and Görnerup, O. (2009), A pectral method for aggregating variables in linear dynamical systems with application to cellular automata renormalization, Advances in Complex Systems, 12(2), 131-155.
  15. [15]  Jacobi, Martin Nilsson (2009), Hierarchical Dynamics, Encyclopedia of Complexity and Systems Science (Robert A. Meyers, ed.), Springer, New York, 4588-4608.
  16. [16]  Pfante, O., Olbrich, E., Bertschinger, N., Ay, N., and Jost, J. (2013), Comparison between different methods of level identification, to appear.
  17. [17]  Rowe, J.E., Vose, M., and Wright, A. (2005), State aggregation and population dynamics in linear systems, Artificial Life, 11(4), 473-492.
  18. [18]  Walters, P. (1982), An Introduction to Ergodic Theory, Springer, New York.