Energy function for diffeomorphisms on surfaces with finite hyperbolic chain recurrent set

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.