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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


Transient Dynamics in Complex Systems: Heteroclinic Sequences with Multidimensional Unstable Manifolds

Discontinuity, Nonlinearity, and Complexity 2(1) (2012) 21--41 | DOI:10.5890/DNC.2012.11.001

Valentin Afraimovich$^{1}$; Irma Tristan$^{2}$; Pablo Varona$^{3}$; Mikhail Rabinovich$^{2}$

$^{1}$ Instituto de Investigacion en Comunicacion Optica, Universidad Autonoma de San Luis Potosi, Karakorum 1470, Lomas 4a 78210, San Luis Potosi, S.L.P., Mexico

$^{2}$ BioCircuits Institute University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0402, USA

$^{3}$ Grupo de Neurocomputación Biológica, Depto. de Ingeniería Informática, Escuela Politécnica Superior, Universidad Autónoma de Madrid, 28049, Madrid, Spain

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We formulate the basic principles of multi-agents complex system dynamics following the lessons from experimental neuro- and cognitive science: 1) the cognitive dynamics in a changing environment is transient and can be considered as a temporal sequence of metastable states; 2) the available resources for the information processing are limited; 3) the transient dynamics is robust against noise and at the same time sensitive to information signals. We suggest the basic dynamical models that describe the evolution of cooperative modes. We focus on two limit cases: a) the unstable manifold of metastable states has one leading direction and many others that are characterized by small positive eigenvalues (system on the edge of instability), and b) the unstable manifold is characterized by small number of positive eigenvalues having the same range (integration of different flows - binding).


V.A. was partially supported by PROMEP, UASLP-CA21. I.T. acknowledges support from UC MEXUSCONACYT Fellowship. P.V. was supported by MICINNBFU2009-08473 and M.I.R. acknowledges support from ONR grant N00014-07-1-074.


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