Operator-theoretic Identification of Closed Sub-systems of Dynamical Systems
DOI:
https://doi.org/10.5890/DNC.2015.03.007Abstract
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.References
[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] Buchholz, P. (1995), Hierarchical markovian models: symmetries and reduction, Performance Evaluation, 22(1), 93- 110.
[3] Burke, C.K. and Rosenblatt, M. (1958),A markovian function of a markov chain, Annals of Statistics, 29, 1112-1122.
[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] 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] 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] Dixmier, J. (1981), Von neumann Algebras, North-Holland Publishing Company.
[8] Furstenberg, H. (1980), Recurrence in ergodic theory and combinatorial number theory, Princeton University Press.
[9] Görnerup, Olof and Jacobi, Martin Nilsson (2008), A dual digenvector dondition for strong lumpability of markov chains, SFI WORKING PAPER , 1-7.
[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] Görnerup, Olof and Jacobi, Martin Nilsson (2010), A model-independent approach to infer hierarchical codon substitution dynamics, BMC Bioinformatics, 11 201 (eng).
[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] Snell, J.L. and Kemeny, J.G.(1976), Finite Markov Chains, Springer-Verlag, New York.
[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] Jacobi, Martin Nilsson (2009), Hierarchical Dynamics, Encyclopedia of Complexity and Systems Science (Robert A. Meyers, ed.), Springer, New York, 4588-4608.
[16] Pfante, O., Olbrich, E., Bertschinger, N., Ay, N., and Jost, J. (2013), Comparison between different methods of level identification, to appear.
[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] Walters, P. (1982), An Introduction to Ergodic Theory, Springer, New York.
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