{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T11:13:14Z","timestamp":1720609994840},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2011,8]]},"DOI":"10.1007\/s00332-011-9094-1","type":"journal-article","created":{"date-parts":[[2011,3,16]],"date-time":"2011-03-16T12:52:42Z","timestamp":1300279962000},"page":"579-593","source":"Crossref","is-referenced-by-count":8,"title":["Dynamical Properties of a Perceptron Learning Process: Structural Stability Under Numerics and\u00a0Shadowing"],"prefix":"10.1007","volume":"21","author":[{"given":"Andrzej","family":"Bielecki","sequence":"first","affiliation":[]},{"given":"Jerzy","family":"Ombach","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2011,3,17]]},"reference":[{"key":"9094_CR1","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/S0362-546X(00)00105-X","volume":"2","author":"A. Bielecki","year":"2001","unstructured":"Bielecki, A.: Dynamical properties of learning process of weakly nonlinear and nonlinear neurons. Nonlinear Anal., Real World Appl. 2, 249\u2013258 (2001)","journal-title":"Nonlinear Anal., Real World Appl."},{"key":"9094_CR2","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1002\/cnm.516","volume":"18","author":"A. Bielecki","year":"2002","unstructured":"Bielecki, A.: Estimation of the Euler method error on a Riemannian manifold. Commun. Numer. Methods Eng. 18, 757\u2013763 (2002)","journal-title":"Commun. Numer. Methods Eng."},{"key":"9094_CR3","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1016\/S0362-546X(01)00896-3","volume":"51","author":"A. Bielecki","year":"2002","unstructured":"Bielecki, A.: Topological conjugacy of discrete-time map and Euler discrete dynamical systems generated by gradient flow on a two-dimensional compact manifold. Nonlinear Anal., Theory Methods Appl. 51, 1293\u20131317 (2002)","journal-title":"Nonlinear Anal., Theory Methods Appl."},{"key":"9094_CR4","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0377-0427(03)00486-2","volume":"164\u2013165","author":"A. Bielecki","year":"2004","unstructured":"Bielecki, A., Ombach, J.: Shadowing property in analysis of neural networks dynamics. J. Comput. Appl. Math. 164\u2013165, 107\u2013115 (2004)","journal-title":"J. Comput. Appl. Math."},{"key":"9094_CR5","first-page":"25","volume":"63","author":"B. Garay","year":"1994","unstructured":"Garay, B.: Discretization and Morse\u2013Smale dynamical systems on planar discs. Acta Math. Univ. Comen. 63, 25\u201338 (1994)","journal-title":"Acta Math. Univ. Comen."},{"key":"9094_CR6","volume-title":"Introduction to the Theory of Neural Computation","author":"J. Hertz","year":"1991","unstructured":"Hertz, J., Krogh, A., Palmer, R.G.: Introduction to the Theory of Neural Computation. Addison-Wesley, Redwood City (1991)"},{"key":"9094_CR7","volume-title":"Theoretical Neurocybernetic","author":"G. Hess","year":"2009","unstructured":"Hess, G.: Synaptic transmission and synaptic plasticity. In: Tadeusiewicz, R. (ed.) Theoretical Neurocybernetic. Warsaw University Press, Warsaw (2009) (in Polish)"},{"key":"9094_CR8","doi-asserted-by":"crossref","first-page":"171","DOI":"10.4064\/ap-65-2-171-177","volume":"65","author":"P.E. Kloeden","year":"1997","unstructured":"Kloeden, P.E., Ombach, J.: Hyperbolic homeomorphisms are bishadowing. Ann. Pol. Math. 65, 171\u2013177 (1997)","journal-title":"Ann. Pol. Math."},{"key":"9094_CR9","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1137\/S003614109529238X","volume":"28","author":"M.C. Li","year":"1997","unstructured":"Li, M.C.: Structural stability of Morse\u2013Smale gradient-like flows under discretization. SIAM J. Math. Anal. 28, 381\u2013388 (1997)","journal-title":"SIAM J. Math. Anal."},{"key":"9094_CR10","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1137\/S0036141097322558","volume":"30","author":"M.C. Li","year":"1999","unstructured":"Li, M.C.: Structural stability of the Euler method. SIAM J. Math. Anal. 30, 747\u2013755 (1999)","journal-title":"SIAM J. Math. Anal."},{"key":"9094_CR11","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-21752-9","volume-title":"Introduction to Smooth Manifolds","author":"J.M. Lee","year":"2003","unstructured":"Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2003)"},{"key":"9094_CR12","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-5703-5","volume-title":"Geometric Theory of Dynamical Systems","author":"J. Palis","year":"1982","unstructured":"Palis, J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, New York (1982)"},{"key":"9094_CR13","series-title":"Lecture Notes in Mathematics","volume-title":"Shadowing in Dynamical Systems","author":"S.Yu. Pilyugin","year":"1999","unstructured":"Pilyugin, S.Yu.: Shadowing in Dynamical Systems. Lecture Notes in Mathematics, vol.\u00a01706. Springer, Berlin (1999)"},{"key":"9094_CR14","doi-asserted-by":"crossref","DOI":"10.21236\/AD0256582","volume-title":"Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms","author":"F. Rosenblatt","year":"1961","unstructured":"Rosenblatt, F.: Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms. Spartan Books, Washington (1961)"},{"key":"9094_CR15","volume-title":"Problems of Biocybernetics","author":"R. Tadeusiewicz","year":"1994","unstructured":"Tadeusiewicz, R.: Problems of Biocybernetics. PWN, Warsaw (1994) (in Polish)"},{"key":"9094_CR16","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/S0377-0427(01)00571-4","volume":"144","author":"W. Wu","year":"2002","unstructured":"Wu, W., Xu, Y.: Deterministic convergence of an online gradient method for neural networks. J. Comput. Appl. Math. 144, 335\u2013347 (2002)","journal-title":"J. Comput. Appl. Math."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.springerlink.com\/index\/pdf\/10.1007\/s00332-011-9094-1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,9]],"date-time":"2019-06-09T04:37:39Z","timestamp":1560055059000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-011-9094-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,3,17]]},"references-count":16,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2011,8]]}},"alternative-id":["9094"],"URL":"https:\/\/doi.org\/10.1007\/s00332-011-9094-1","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,3,17]]}}}