Energy function for diffeomorphisms on surfaces with finite hyperbolic chain recurrent set
T.M. Mitryakova1, O.V. Pochinka2, A.E. Shishenkova.3
| Annotation | In the present paper the existence of energy function for diffeomorphisms of surfaces with finite hyperbolic chain recurrent set is established. The constructed function is Morse function which closely connected with dynamic of the cascade and its existence allows to prove Poincare-Hopf theorem for systems of the considered class. Construction of energy function for this dynamical system, that is a function which is not increasing along trajectories, in many cases is key to understanding of structure of artificial neural network. On the existence of such function Hebb's method of training of perseptron is based. |
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| Keywords | discrete dynamical systems, chain recurrent set, energy function, Morse function, training of perseptron. |
1Assistant Professor of Chair of Theory of Functions, Lobachevskii State University, Nizhny Novgorod; tatiana.mitryakova@yandex.ru
2Associated Professor of Chair of Theory of Functions, Lobachevskii State University, Nizhny Novgorod, olga-pochinka@yandex.ru
3Associated Professor of Chair, Agriculture Academy of Nizhniy Novgorod, Nizhnii Novgorod, vgrines@yandex.ru
Citation: T.M. Mitryakova, O.V. Pochinka, A.E. Shishenkova., "[Energy function for diffeomorphisms on surfaces with finite hyperbolic chain recurrent set]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,14:1 (2012) 98–106 (In Russian)
