ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
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

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

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

Automatic Recognition and Tagging of Topologically Different Regimes in Dynamical Systems

Discontinuity, Nonlinearity, and Complexity 3(4) (2014) 413--426 | DOI:10.5890/DNC.2014.12.004

Jesse J. Berwald; Marian Gidea; Mikael Vejdemo-Johansson

$^{1}$ 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

Abstract

Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional 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 high-dimensional 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 high-dimensional 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 DMS-0940363. 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: DMS-1201357 and DMS-0940363 and by the Mathematics and Climate Research Network under grant NSF DMS-0940363. The third author was supported by Toposys grant FP7-ICT-318493-STREP. 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.

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