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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


Perspectives on Multi-Level Dynamics

Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 313--339 | DOI:10.5890/DNC.2016.09.009

Fatihcan M. Atay$^{1}$, Sven Banisch$^{2}$, Philippe Blanchard$^{3}$, Bruno Cessac$^{4}$, Eckehard Olbrich$^{5}$, Dima Volchenkov$^{6}$

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

$^{2}$ Max Planck Institute for Mathematics in the Sciences, D-04103 Leipzig, Germany

$^{3}$ University of Bielefeld, Department of Physics D-33619 Bielefeld, Germany

$^{4}$ Inria Sophia Antipolis Méditerranée, Neuromathcomp project-team, Sophia Antipolis 06902, France

$^{5}$ Max Planck Institute for Mathematics in the Sciences, D-04103 Leipzig, Germany

$^{6}$ University of Bielefeld, Center of Excellence - Cognitive Interaction Technology (CITEC), D-33619 Bielefeld, Germany

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As Physics did in previous centuries, there is currently a common dream of extracting generic laws of nature in economics, sociology, neuroscience, by focalising the description of phenomena to a minimal set of variables and parameters, linked together by causal equations of evolution whose structure may reveal hidden principles. This requires a huge reduction of dimensionality (number of degrees of freedom) and a change in the level of description. Beyond the mere necessity of developing accurate techniques affording this reduction, there is the question of the correspondence between the initial system and the reduced one. In this paper, we offer a perspective towards a common framework for discussing and understanding multi-level systems exhibiting structures at various spatial and temporal levels. We propose a common foundation and illustrate it with examples from different fields. We also point out the difficulties in constructing such a general setting and its limitations.


The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 318723: Mathematics ofMulti-Level Anticipatory Complex Systems (MatheMACS). S.B. also acknowledges financial support by the Klaus Tschira Foundation. D.V. acknowledges the support from the Cluster of Excellence Cognitive Interaction Technology ’CITEC’ (EXC 277) at Bielefeld University, which is funded by the German Research Foundation (DFG).


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