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describes well the dynamics of the nuclei in the molecule. The effect of the
asymmetry of the potential energy function is relatively small. It can be taken
into account if the expression for the force acting between the nuclei is consecu-
tively supplemented by quadratic, cubic, etc. terms, i.e., moving from a linear
dependence of force on the displacement of nuclei to the nonlinear one.
In this simple example, the main features of the physics of the last century
become apparent:
Understanding of high complexity of physical phenomena
Desire to use a linear model of the phenomenon, if possible
The belief that taking into account nonlinearity only makes the conclusions of
the linear model more precise
Nevertheless, the development of the physical understanding of natural phe-
nomena resulted in the second half of the last century in a gradual understanding of
the much more complex role and the possibilities of manifestation of nonlinear
processes. Let us begin treating the features of this situation with a few simple and
obvious examples encountered in everyday life.
When buying apples on the market, nobody thinks about the linearity of the
process, when upon having decided to buy 4 kg instead of 2, the buyer assumes that
there will be twice as many of apples. At the same time, few take into account that
increasing the speed of the car by a factor of 2, for example, from 50 to 100 km/h,
the driver dramatically exacerbates the consequences of a possible accident,
because the kinetic energy of the car is proportional to the square of its velocity.
These examples show that everyday life is characterized by both linear and
nonlinear phenomena, the essence of which one generally ignores.
Examples of nonlinear phenomena are easy to find among biological objects.
Leafed trees are characterized by a large number of branched branches, which
sharply increases the amount of foliage. This is called fractal structure. A fractal is
defined as a mathematical object, in which upon a given transformation, the number
of its characteristic details increases nonlinearly. As one possible example consider
a geometric structure, the initial state of which includes three segments (see
Fig.
6.10
). We define as the transformation of the object the drawing of a perpen-
dicular line through the middle of each segment. As a result of each successive
transformation, a segment is turned into an x-shaped structure with equal half lines.
It is easy to see that as a result of each conversion both the number of crossings and
the number of segments increase nonlinearly. A large number of fractal structures
are known that differ both in their initial state and in the rules of its transformation.
Trees, including leafed trees, can also be considered fractal structures. For them, the
fractal structure was, apparently, the decisive factor for survival during the evolu-
tion of plants.
The first plants appeared about 500 million years ago in the Paleozoic era. It all
started with Rhyniophyta that originated from green algae and were the first to
populate the land. During the carboniferous period giant lycopsids, calamites, and
horsetails grew on earth with a small degree of fractality of trunks, stems, and
leaves assimilating solar energy. In the course of evolution, plants have been
