Aleksić, Zoran

### Description

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...[Show more] 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.

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