Information Technology Reference
In-Depth Information
fractal offer new theories to model rough surface. This paper mainly focuses on
the applications of chaos and fractal in the field of electromagnetic scattering
basing on acquired studies.
2 Chaos and Fractal
Put forward by the meteorologist Lorentz in the 1970s, the chaos is regarded
as one of the three most important developments as well as the Relativity and
Quantum theory in 20th century. Chaos refers to a reciprocating non-periodic
motion which is sensitive to the initial condition in a determinate nonlinear
system. The motion is deterministic in short terms but unpredictable for long.
Comparing to linear systems and other nonlinear ones, chaos system has the
characteristics of ergodicity, boundedness, inner randomness, sensitive to the
initial value and long-term unpredictability. Withal, the chaos system always
converges at some certain attractors in the phase space. The referents of Chaos
are mainly the unstable divergence process in the nonlinear dynamics and the
evolution rules of complex system behavior in pace with time[4].
Fractal theory was set up by the French mathematician B. B. Mandelbrot in
1975. Fractal refers to structures of self-similarity, which also means that they
look the same under different magnifications. Fractal has the characteristics of
self-ane, scale-free and self-similarity. The fractal theory focuses on the un-
smooth and nondiferentiable geometric object generated by a nonlinear system.
Studies have shown that it is more useful and suitable to use fractal theory to
describe natural objects than beelines or smooth curves of Euclidean geometry,
such as the boundary of clouds, the profile of the mountain peak, coastline, forked
lightning and so on. It also fits for the forecasting of natural phenomenon. So
fractal is referred to as the language of describing the nature. Fractal dimension
is an important part of the fractal theory, as it is also a method to characterize
chaos system quantitatively[5].
As important parts of nonlinear system, chaos and fractal have close connex-
ions. The similarity in changing pattern of chaos system in time scale resembles
with the self-similarity of fractal geometry in space scale.Whats more, the fact
that the chaos system always converges at some certain attractors in the phase
space conforms to formation process of fractal structure greatly. Therefore, a
system with fractal structure usually has some characteristics of chaos, and with
appearance of chaos there often follows the distinction of fractal. Although with
different origins and development processes, these two disciplines interosculate
closely. Chaos mainly focuses on the unsteady divergence process in time scale,
while the referents of fractals relate to the irregular geometrical structure in
space scale. Taking the research on Logistic mapping as a example, during the
process of the system from the beginning to chaos, the phenomena of Period-
doubling bifurcation appears, and the proportion between adjacent stages is a
constant, which expresses the similarity of the fractal theory[6]. Thence, chaos
theory and fractal theory can be applied in the field of electromagnetic scatter-
ing simultaneously. Simply speaking, chaos is the fractal in the scale of time and
fractal is the chaos in the scale of space.
Search WWH ::
Custom Search