CNN Symbolic Dynamical Systems

Cellular neural networks (CNN) , a nonlinear circuits model, was proposed by Professor L.O. Chua (University of California at Berkeley in USA) and L.Yang in 1988.CNN combines advantages of Hopfield's neural networks (HNN) with Neumann's cellular automata (CA). CNN can be realized as very-large-scale-integrated (VLSI) chips and can operate at a very high speed. Therefore, CNN has wildly applied prospect as a new information processing system. In recent decade, the theory, design and application research on CNN have become an interesting object that multitudinous scientific and technical workers pay close attention to. CNN has been extensively applied in image processing, pattern recognition, VLSI, robot, brain function simulating etc. The first analogic cellular computer based on the theory of CNN have come into being.In mathematical view, CNN is a nonlinear dynamic system, its function and importance have exceeded normal nervous systems more and more. Thereby, with theory researches going deep, CNN has become the abbreviation which include two meanings, cellular neural networks and cellular nonlinear networks. We should specially pay attention to the infinite-dimensions lattices dynamic system that we obtained when finite state quantities are changed to infinite in CNN and the dynamic behavior of the lattices dynamic system will be very complexity. In recent years, researches on this CNN have got some results, but we have met lots of difficulties. Researches on a fully standard CNN are nearly a blank at present.In this thesis, under the standpoints of symbolic dynamics, we make a new try at CNN that have infinite state qualities, which is, to avoid the complexity of dynamic behavior we regard the input of CNN as a symbolic sequence of a specific symbolic space and the output of CNN is also a symbolic sequence of the same space. Thus, there is a self-map between input and output on the space, we call it input-output map. While we add an appropriate topology to the space, the input-output map will become a topological dynamic system, and we call it CNN symbol dynamic system. In this thesis, the key contents are to propose the symbolic dynamic system and to investigate its dynamic behavior.In Chapter 1 we introduce some background and significance of producing CNN, such as how to establish CNN model, CNN's mathematical definition and a brief description of research progress of CNN etc. In Chapter 2, to illustrate the complex dynamic behavior of CNN that have infinite state quantities, we discuss the stationary solution of a CNN model which is no input, and make a map that is induced by the stationary solution and construct the map's Smale Horseshoe. In Chapter 3, we establish a new symbolic dynamic system. For 2-dimensions sequence space with k symbol, we give a appropriate measure, several shift maps and prove strictly that these maps have abundant and complex dynamic behaviors. These contents constitute basic theory of CNN symbolic dynamic. In Chapter 4, we apply the theory on 2-dimensions symbolic dynamic that mentioned in the above chapter to input-outputmap of CNN. In a saturated state and independent initial value, or CNN gene is Boolean gene, a large number of input-output maps are shift maps on 2-dimensions symbolic sequence space, and have abundant dynamic behaviors, such as Li-York chaos, Devaney chaos, having positive topological entropy and topological mixing etc. Hence, all of the above form essential contents of CNN symbolic dynamic. Lastly, in Chapter 5, we make a brief summary on this thesis, and prospect for further research on CNN symbolic dynamics.