Applications of nonlinear dynamics to information processing

Abstract

The reported results are direct applications of nonlinear dynamics to information processing or are relevant for the applications. In the second chapter we describe a simple method for estimating the embedding dimension that can be used as a first step in constructing nonlinear models. The method for the reduction of measurement noise in chaotic systems that is presented in the third chapter is attractive in the cases where high accuracy is necessary. Next we propose how to overcome some problems encountered in constructing models of complex nonlinear systems. Finally, the behaviour of one-dimensional cellular automata useful for the detection of velocities of patterns is shown and explained in the last chapter. The method of estimating the embedding dimension is based on the idea that when the observed dynamical system is deterministic and smooth and the embedding dimension is correctly chosen, the relationship between the successive reconstructed state vectors should be described as a continuous mapping. To check if the given embedding dimension is a good one we search for pairs of state vectors whose distance is smaller than some number. For each pair we compute the distance between the successors of the elements of pairs and represent this distance graphically. When the embedding dimension is equal or larger than the minimum correct dimension, all distances are small in comparison to distances for incorrect dimensions. The method for noise reduction is developed assuming that the map of the system is known and the noise is bounded. The closer the initial condition is to the true state of the system, the longer the computed trajectory follows the observed trajectory. To reduce the uncertainty in knowing the given state we recursively search for the state for which the computed trajectory follows the observed trajectory as long as possible. The method is demonstrated on several twodimensional invertible and noninvertible chaotic maps. When the map is known exactly an arbitrary level of noise reduction can be achieved. With the increase of the complexity of a nonlinear system it is harder to construct its model. We propose to discover first how to construct a model of a similar but simple system. Discovered heuristics can be useful in modeling more complex systems. We demonstrate the approach by constructing a deterministic feed-forward neural network that can extract velocities of onedimensional patterns. Analysing simpler models we discovered how to estimate the necessary numbers of neurons; what are the useful ranges of the parameters of the network and what are the potential functional dependencies between the parameters. The class of one-dimensional cellular automata whose state is a function of both the previous state and a time-dependant input is described. As inputs we considered the sequences of binary strings that represent black-and-white objects moving in front of a white background. As outputs we considered the trajectory of the automaton. For some rules the automaton will evolve to the zero state for all velocities of the object except for the velocities in specific narrow range. The phenomenon is persistent even when a strong noise is present in input patterns but unreliable units of the automaton or having a more complex input break it down. Show more