Abstract:

The science of complexity is increasing its importance exponentially in the last years. A wide variety of sparse knowledge has been put together to conform a new discipline that has lead to very important conclusions in little time.
The science of complexity is not present in most of study plans in university. There are some reasons that lead to this problem, but the most important reason is that complexity it is a multidisciplinary field, and it is difficult to fit into a rigid frame of knowledge areas. Complexity feeds from advanced calculus, differential geometry, statistics, physics, but also from computation theory, electronic design and software development. The relationships between these different fields allow deriving conclusions from a very general point of view, that are accomplished in a wide variety of systems, from electronics to biology, from physics to sociology.
Maybe, two of the main achievements of the new science are chaos theory and fractals.
Chaos theory describes the behavior of dynamical systems that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.
Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions, fluid dynamics, and mechanical and magneto-mechanical devices. Observations of chaotic behavior in nature include the dynamics of satellites, population growth in ecology, the dynamics of the action potentials in neurons, and molecular vibrations. Everyday examples of chaotic systems include weather and climate.
Very related with chaos theory are fractals.
A fractal is, generally, a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus, meaning “broken” or “fractured”. A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snowflakes. However, not all self-similar objects are fractals.
Applications of fractals include classification of histopathology slides in medicine, signal and image compression, fractography and fracture mechanics, design of antennas and technical analysis of series.
In this paper, an initiative is described of including all these concepts in the Telecommunication School from the Polytechnic University of Valencia. The course is offered as a part of the Electronic Systems Design Master for postgraduate students and, with some few variations, as a free choice subject for undergraduate students.

Keywords:

chaos, fractals, complexity, cellular automata, neural networks.