Discontinuity, Nonlinearity, and Complexity
Automatic Recognition and Tagging of Topologically Different Regimes in Dynamical Systems
Discontinuity, Nonlinearity, and Complexity 3(4) (2014) 413426  DOI:10.5890/DNC.2014.12.004
Jesse J. Berwald; Marian Gidea; Mikael VejdemoJohansson
$^{1}$ Institute for Mathematics and its Applications, University of Minnesota Minneapolis, Minnesota, USA
$^{2}$ Yeshiva University New York City, New York, USA
$^{3}$ AI Laboratory, Jožef Stefan Institute, Ljubljana, Slovenia, Computer Vision and Active Perception Laboratory KTH Royal Institute of Technology, Stockholm, Sweden
Download Full Text PDF
Abstract
Complex systems are commonly modeled using nonlinear dynamical systems. These models are often highdimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the system undergoes changes in qualitative behavior. A detailed description of the dynamics can be difficult or impossible to obtain for highdimensional and chaotic systems. Therefore, a more sensible goal is to recognize and mark transitions of a system between qualitatively different regimes of behavior. In practice, one is interested in developing techniques for detection of such transitions from sparse observations, possibly contaminated by noise. In this paper we develop a framework to accurately tag different regimes of complex systems based on topological features. In particular, our framework works with a high degree of success in picking out a cyclically orbiting regime from a stationary equilibrium regime in highdimensional stochastic dynamical systems.
Acknowledgments
JJB would like to thank Dr. Richard McGehee for providing the Vostok ice core data.
The first author was partially supported by theMathematics and Climate Research Network under grant NSF DMS0940363. The work described in this article is a result of a collaboration made possible while the author was a postdoctoral fellow at the Institute for Mathematics and its Applications during the IMA’s annual program on Scientific and Engineering Applications of Algebraic Topology.
The research of the second author was partially supported by NSF grants: DMS1201357 and DMS0940363 and by the Mathematics and Climate Research Network under grant NSF DMS0940363.
The third author was supported by Toposys grant FP7ICT318493STREP. The author gratefully acknowledges the support and hospitality provided by the IMA during his visit which took place during the annual program on Scientific and Engineering Applications of Algebraic Topology.
References

[1]  Scheffer, M., Bascompte, J., Brock, W.A., Brovkin, V., Carpenter, S.R., Dakos, V., Held, H., Van Nes, E.H., Rietkerk, M., and Sugihara, G. (2009), Earlywarning signals for critical transitions, Nature, 461(7260), 5359. 

[2]  P. F. Hoffman. A Neoproterozoic Snowball Earth. Science, 281(5381):13421346,August 1998. 

[3]  Vellinga,M. andWood, R.A. (2002), Global Climatic Impacts of a Collapse of the Atlantic Thermohaline Circulation, Climate Change, 54, 251267. 

[4]  Keigwin, L.D. (1996), The Little Ice Age and Medieval Warm Period in the Sargasso Sea, Science (New York, N.Y.), 274(5292), 15031508. 

[5]  Scheffer, M., Carpenter, S., Foley, J.A. A., Folke, C., andWalker, B. (2001), Catastrophic shifts in ecosystems, Nature, 413(6856), 591596. 

[6]  Chen, X.P. Duan, J.Q., and Fu, X.C. (2010), A sufficient condition for bifurcation in random dynamical systems, Proceedings of the American Mathematical Society, 965973. 

[7]  Dakos, V., Scheffer, M., van Nes, Egbert H, Brovkin, V., Petoukhov, V., and Held, H. (2008), Slowing down as an early warning signal for abrupt climate change. Proceedings of the National Academy of Sciences of the United States of America, 105(38), 1430814312. 

[8]  Ditlevsen, P.D. and Johnsen, S.J. (2010), Tipping points: Early warning and wishful thinking. Geophysical Research Letters,37(19), 25. 

[9]  Lenton, T.M., Held, H., Kriegler, E., Hall, J.W., Lucht, W., Rahmstorf, S., and Schellnhuber, H.J. (2008), Tipping elements in the Earth's climate system, Proceedings of the National Academy of Sciences of the United States of America, 105(6), 17861793. 

[10]  Livina, V. and Lenton, T. (2007), A modified method for detecting incipient bifurcations in a dynamical system, Geophysical Research Letters, 34. 

[11]  CohenSteiner, D. and Edelsbrunner, H. (2007), Stability of persistence diagrams. Discrete and Computational Geometry, 37(1), 103120. 

[12]  Arnold, L. (1998), Random Dynamical Systems, Springer, Berlin. 

[13]  Evans, L.C. (2013), An Introduction to Stochastic Differential Equations, American Mathematical Society. 

[14]  Takens, F. (1981), Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, Warwick, 1980, 366381. 

[15]  Sauer, T., Yorke, J.A., and Casdagli, M. (1991), Embedology, Journal of Statistical Physics, 65(3), 579616. 

[16]  Edelsbrunner, H., Letscher, D., and Zomorodian, A. (2002), Topological persistence and simplification, Discrete & Computational Geometry , 28(40), 511533. 

[17]  Hatcher, A. (2002), Algebraic Topology, Cambridge University Press, Cambridge. 

[18]  Barcodes, R.G. (2008), The persistent topology of data, Bulletin of the American Mathematical Society, 6175. 

[19]  Carlsson, G. (2009), Topology and data, Bulletin (New Series) of the American Mathematical Society, 46 (2), 255308. 

[20]  Edelsbrunner, H. and Harer, J. (2009), Computational Topology: an Introduction, AMS Press. 

[21]  Zomorodian, A. and Carlsson, G. (2005), Computing persistent homology, Discrete & Computational Geometry, 33, 249274. 

[22]  Kaczynski, T., Mischaikow, K., and Mrozek, M. (2004), Computational Homology, Applied Mathematical Sciences, 157, SpringerVerlag. 

[23]  Bishop, ChristopherM. (2007), Pattern Recognition and Machine Learning, Springer. 

[24]  Doedel, E.J, Krauskopf, B., and Osinga, H.M. (2006), Global bifurcations of the Lorenz manifold. Nonlinearity, 19(12), 29472972. 

[25]  Monnin, E., Indermuhle, A., Dällenbach, A., Flückiger, J., Stauffer, B., Stocker, T.F., Raynaud, D., and Barnola, J.M. (2001), Atmospheric co2 concentrations over the last glacial termination, Science, 291, 112114. 

[26]  Higham, D.J. (2001), An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, 3(22), 48634869. 

[27]  Berwald, J. and Gidea, M. (2014), Critical transitions in a model of a genetic regulatory system, Mathematical Biosciences and Engineering (MBE), 11(4), 723 740. 

[28]  Lum, P.Y., Singh, G., Lehman, A., Ishkanov, T., VejdemoJohansson, M., Alagappan, M., Carlsson, J., and Carlsson, G.(2013), Extracting insights from the shape of complex data using topology. Scientific Reports, 3, 2013. 