Vol.2, No 9, 1999 pp. 999-1004
OF SCIENTIFIC CONTENTS OF
ON THE BORDER OF MILLENIUMS"
dedicated to the
275th Annibersary Of the Russian Academy of Sciences
June 22?24, 1999
In the period from June 22nd to 24th, at the
University of Saint-Petersburg and at the Institute for precise mechanics
and optics - Technical University of Saint-Petersburg, an international
scientific conference called "Nonlinear sciences on the border of millenniums"
was held. It was dedicated to the 275th anniversary of the Russian Academy
of Science. The organizers of this conference were The Russian Academy
of Science, the Academy of Nonlinear sciences, Saint-Petersburg's state-owned
Institute of Precise mechanics and optics - the Technical University.
The scientific and program committee was
constituted of well-known scientists from around the world. The participants
were also from around the world.
The scientific part of the conference
was held plenary and by sections, with a certain number of invited lectures.
In the plenary part of the program lectures
had an overview character, and in the sections they were specialized.
The conference had the following sections:
Nonlinear mechanics and applications,
Nonlinear differential equations and analysis,
Nonlinear control systems theory,
Nonlinear methods and models in natural,
technical and humanitarian sciences,
Stability and nonlinear oscillations.
Among others there were the following
V.M. Matrosov: The development of the
stability theory at the Russian Academy of Science.
A review of papers in the area of the
motion stability theory development at the Russian Academy of Science published
since the foundation of RAS 275 years ago is given. The classic works of
Euler (18th century) on the matter of equilibrium stability of elastic
rods and A.M. Ljapunov in the area of setting of method of nonlinear theory
of motion stability and its applications to the problems of mechanics (19th
century) are described. After that the ideas, generalized and modified
function methods of Ljapunov in the 19th century are presented, and the
paper also pointed out to the tasks of their further development in the
21st century and that applies both to the basic methods of nonlinear dynamical
analysis of different systems, including the logical-dynamical ones, and
method of analysis and synthesis of complex systems which allow hybrid
defining and also many others.
He considered the basic applications of
the Ljapunov's function methods in application to problems of mechanics,
technology, physics, economy, in the tasks of stability preservation, controlling
and projecting and similar, and especially for aero-kinetic, electromechanical,
energy and other systems.
Important problems of science, techniques
of global development, no-spherical control, large-scale systems theories,
intellectual support to the accepting of solutions, dynamical theories
of games and many others for the research of which, the Ljapunov's function
method can be applied, are defined in the conclusion.
K.F. Chernih: Nonlinear singular theory
This lecture pointed to the popular areas
of mechanics and physics of deformable bodies through: the brittle fracture
mechanics, dislocation and disinclination in crystals and the concentrated
forces and moments, through comparation of approaches in linear and nonlinear
setting of the problem. It points to the small number of papers in the
area of nonlinear approach to this problems group and mainly through physical
and geometrical sources of non-linearity. The lecturer pointed to his own
results and to the results of his associates in the area of nonlinear singular
theory of elasticity.
V.A.Yakibovih: The square criteria of
absolute stability of nonlinear systems.
M.B. Ignyatev: Vagueness and the phenomenon
of adaptation maximum in complex nonlinear systems.
The lecture takes into consideration the
evolution of the problem of vagueness in science and technology since the
appearance of the probability theory, through the Schredinger's vagueness
principle and till the latest worked-out systems with vagueness in linguistics,
mechanics, biology, chemistry, physics, numerical systems, and in the artificial
intelligence systems. The research of nonlinear systems with vagueness
are only starting and promise many new discoveries. In the year 1963 the
phenomenon of adaptation maximum in systems which develop through degrees
of freedom increase was discovered. By using the phenomenon of adaptation
maximum we can explain many anti-entropic processes both in nature and
in the society. In the process of interaction with a changeable environment,
the appearance of random coefficients in the system of equivalent equations
allows the system to conduct changes which are of such nature that the
more vague coefficients there are the greater are the adaptation possibilities
of the system. Considerations about the possibility of controlling such
systems - the systems which make possible the keeping of such systems within
the area of adaptation maximum in the current of changes through the removal
of old boundaries and their integration into a collective system, are presented.
The presentation pointed out that stable
development is possible only in the zone of adaptation maximum and that
is possible for biological, social-economical, physical-chemical systems
and systems of self-sustained reactions. In order to increase the harmlessness
of system they must be kept within the area of the adaptation maximum.
On the basis of the system with vagueness
a unified nonlinear theory of matter has being built, which explains the
behavior of gases, liquids, rigid bodies, plasma and living structures.
A unified theory of matter is being created on the basis of analysis of
the transfer from one point of rest to another.
Kolesnikov A.A: A Synergetic approach
to nonlinear control systems theory.
A new synergetic concept in the control
theory, based on the fundamental characteristic of self-organization of
natural disipative systems is proposed in this lecture. Invariance, self-organization,
nonlinearity, optimization and synthesis appear as basic terms of the synergetic
control theory developed in the lecture which determine its essence, innovations
At the basis of the synergetic approach
lie two fundamental principles of nature - they are, first- the invariance
principle and second - the principle of contraction (shrinking) - expansion
of phase volume in dissipate dynamical systems of random nature. Based
on the synergetic concept an essentially new invariant-group approach to
the analytical construction of nonlinear multi-measure dynamical objects
control systems with complex interconnections, based on the idea of introduction
of attracting invariant multiplicities - attractors, on which the natural
(energetic, mechanical, heat etc.) properties of object consigned in the
best way. Such attractors (synergy) form internal dynamic connections,
as a result of what coherent collective motion in the phase space of the
system appears. That permits the realization of a purposive (directed)
self-organization of the collective state in dynamic systems of different
In a developed synergetic approach the
laws of control which take into consideration the internal interactions
of concrete physical (chemical, biological) phenomena and processes are
synthesized. That approach enabled a breakthrough in solving the fundamental
applied problems of setting up physical (chemical, biological, social-economical)
theories of control as problems of searching for general objective laws
of control processes. The introduced language of invariants, as a primary
element of synergetic control theory, permits the attachment of natural-mathematical
unity to the theory and the establishment of a direct connection to preservation
laws i.e. to basic prepositions of natural properties of objects which
correspond to nature.
As the first, the new synergetic approach
permits a breakthrough in the area of synthesis of continuos, discrete,
selectively-invariant, multi-criterial, terminal and adaptiv nonlinear
dynamical objects of different physical nature control systems with complex
interconnections. Such approach found a direct application to solving of
complex problems of control of nonlinear technical objects (flying machines,
turbo-generators, robots, electric power suppliers, technological aggregates
etc.), as well as in control tasks in ecology, biotechnology etc.
O.V. Vasiljev and V.A. Srochko: On numerical
methods of optimal control problems solving.
This lecture considers the types of classes
of problems of optimal programmed control in normal dynamical systems.
A prehistory of development and advancement of numerical solving methods
which are connected to the maximum principle and gradient approximations
is given. The lecture also gives an overview of modern approaches to the
development of numerical methods which posses the properties of improving
through (the use of) effectiveness characteristics.
H. Miyagi (Okinawa): Application of fuzzy
relations to nonlinear uncertain systems.
Fuzzy relations are well known as the
tools for the description, representation and symbolic manipulation of
knowledge or system uncertainty. Computational algorithms and systematic
methodology of fuzzy relational equations and inequalities are engaged
in their practical utilization in nonlinear knowledge-based systems, information
retrieval, medical and psychological diagnosis and some business applications.
Solutions of fuzzy relation equations
presented by Sanchez have been widely applied to the fuzzy inferences,
system identifications, diagnosis problems etc. In company with relation
equation, another important formula is relation inequality which is also
used in the area of fuzzy reasoning and fuzzy problem diagnostics which
is supported by the upper or the lower boundary of normal searching for
solutions by the use of fuzzy relation inequality. Whatsoever, such a method
depends on the solution algorithm which constitutes of several rules and
that complicates everything.
In this lecture, an effective way of solving
fuzzy relation equations and inequalities by introducing fuzzy operators
is presented. Operator definitions are made capable of the simpler solving
procedures and a method suitable for application is considered. By using
the operators we get a whole set of solutions if we solve the new linear
V.A. Pavlov: The phenomena of catastrophic
change of shape of flying objects.
The development of aviation technology
is in connection with constant battle for the increase of speed and the
reduction of weight of flying objects, and in connection with it the reduction
of relative thickness of wings and the increase in the resistance of material
which all led to the appearance of flexible and thin constructions, the
elastic movements of which during flight can not be considered small. Taking
into account the elastic movements of wings in the process of calculating
their load caused the development of linear problems of aero-elasticity.
Big movements also demand that the geometric non-linearities of the studied
system be taken into consideration. That is especially applicable on compositional
constructions such as the wing with wing fans.
As noticed by author, the forces in the
hanging supports in the middle area cause the appearance of untouchable
types of equilibrium, the crossing to which represents the catastrophic
changes of shape of resistance (wings). The catastrophic crossings into
streams can result statically or represent an oscillatory process by themselves-
the oscillations of catastrophic change of shape.
... The scientific importance of the authors
contribution is in the fact, that it brings essential changes in the concept
of composed bodies - like rods which are composed jointly in a stream of
gas, interaction and it can be formulated in the following way: The phenomenon
which was theoretically and experimentally unknown earlier - the catastrophic
change oscillations of composed bodies in a stream of gas, consist in the
fact that bodies of prolonged shape - the type of plain rods, which have
a big discrepancy between maximum and minimum bending rigidities of their
transversal cuts, composed jointly by the long sides of the plain of max
rigidity and which have small rigidly immobilized in one of the transversal
cuts, angles between those plains, when simultaneously bent by the load
of a moving stream catastrophically cross into a new equilibrium state
in the area of bending movements and curling of rods, which, when the transversal
suts attack angles are small, return to starting shape, forming that way
an oscillatory process.
S.A. Zagezda, N.G. Filipov M.P. Yushkov:
A combined task of dynamics and nonlinear non-holonom connections of high
By the use of the introduced tangential
space in multiplicity's of all the possible positions of a free mechanical
system, the equations of Lagrange of the second order are written in the
form of vector equation, which has the shape of the Newton's second law.
By the use of constraint equations up to the second order the involved
tangential spaces fall apart on a direct sum of two subspaces, in one of
which the component of the system acceleration vector is completely determined
by constraint equations as a function of time, generalized coordinates
and generalized velocities.
On the basis of introduced subspaces a
definition of ideal connection is given. That definition is also concerned
with linear connections of a high order. As such connections we refer to
movement program, given in the form of a differential equations system.
A system of differential equations is constituted in relation to the required
generalized coordinated and the required control forces, which secure the
fulfillment of the given program connections (non-holonom connections of
When the program is given in the form
of differential equation, nonlinear in relation to higher derivatives,
that equation is reduced to a linear form through differentiation in time.
In the process the concoction order is increased by one.
Among the section lectures the following
were especially interesting:
I.V. Matrosov: Modeling of global security
and stability of world development with taking into consideration the bio-mass
dynamics, control of BWP distribution, and scientific and technological
In the lecture author described a system
he proposed himself for the analysis of global security and stable development.
For evaluating global stability the following indicators of the security
of world development have been used: the prolongation of life, the spending
of nutritional and industrial products, average social spending per capita
in the world, pollution level and the bio-mass of plant inhabited part
of land which remained at the end of the 21st century as a natural resource.
If the higher values (in the case of pollution lower) are the border of
permissibility, then the global system of world developments in the state
of security, but if one of the inequalities is disturbed we talk about
the disturbance of the security conditions.
System is set up on the basis of the model
of D.Medous World 3.91 written in program language C++.
Additionally the following things are
taken into consideration in the system:
- The control of the total world bruto
product - time constants which take into consideration the time of making
certain branches are introduced into control equations.
- Dynamics of change of the plant inhabited
land bio-mass dependent on the pollution level.
- In the economic specter the science
progress equation is introduced (by using the results of S.V. Dubrovski).
- The artificial cleaning of air
- The conquest of new types and the regeneration
of non-regenerative resources.
- The political tension in the world in
dependence on the remaining non-regenerative resources, quantity of nutritional
products per capita, pollution and population density.
- Model contains around 300 mathematical
dependence (differential equations and algebra relations) some from the
published literature, and some worked out by workers
By the use of the worked out system a
number of scenarios of global development has been studied including both
the known (depleted resource crises, ecological and demographically) and
the qualitatively new. Author has been specially noticed that the scenario
of Meduzov D. "A stable society" can lead to a global collapse, if the
listed additional factors are included.
Within the boundaries of the proposed
models a scenario called "A stable development" was found, and that scenarios
property is that it is not destroyed when small disturbances of the system
parameters appear. The basic thing in that scenario is the use (20 to 40%)
of the bruto world product for control. The work was carried out on a Pentium
II computer, with C++ Builder.
Harold SZU (University of George Washington:
Non-surveying learning sensors in nonlinear sciences.
Starting from biological systems and the
basis of sensory learning, the author makes conclusions which he uses to
make artificial systems which learn by using sensor and on the basis of
informational data and introduces Ljapunov's function for the study of
such systems. The lecture caused a very vivid discussion and many doubts
about the reexamination of the statements and introduced prepositions of
the presented theory.
Al.A. Kolesnikov: Directed self-organization
and controlling of objects with chaotic attractors.
V.V. Baranov: The equilibrium method in
problems of dynamic solution accepting in vagueness.
C.N. Vasiljev: From classic problems of
regulation to intelligent control.
P.E. Tovstik and V.Ju. Anisimov: On one
model of crack growth.
V.A. Shamina: On asymptotic method of
setting up nonlinear models of continual environments mechanics.
A.O. Bochkarev: MGE of Geometrically Nonlinear
Problems of Fracture Mechanics.
A.V. Bushmanov: The Analysis of Tension-deformational
State of Biomechanical Systems.
A.A. Tikhonov: On Nonlinear Oscillations
of Gravity-oriented Rigid Bodies.
F. Wegmann and F. Pfeiffer (Germany):
The Textile Threads Dynamics.
Katica (Stevanović) Hedrih (Yugoslavia):
Differential Equations of Two Material Particles Dynamics Constrained with
a Hereditary Element
and a 20 minute long, improvised speech
On Some Research Directions and the Integration of Scientific Knowledge
in the are of Nonlinear dynamical Systems.
At the opening, while presenting the list
of countries the participants come from, and among them Japan, France,
USA and others, academician Meljnikov said "The Heroic Yugoslavia". With
one keyword he expressed the feelings of organizers towards Yugoslavia
and her people, as well as the reputation Yugoslavia and her people got
by the magnificent and proud resistance to NATO aggression.
In the concluding part of the closing
session of this international scientific conference a scientific discussion
developed on perspectives of development of sciences that deal with nonlinear
dynamical processes, a especially connected to nonlinear processes and
stability in social systems, in the course of which the perspective of
civilization on Earth as a nonlinear system was analyzed - "noosphere"
with all the properties of intellect and other resources of that systems
dynamics. On that occasion the support was given to "The Heroic Yugoslavia
and Her People" in resistance which is a contribution to ways of better
and more humane system of values of civilization on Earth. I was given
the floor and in my speech I pointed to the dangers of monopoly control
and the system of deciding "by the power of wealth" and about the need
for the limitation of those powers by the "force of the power of conciseness,
moral and ethics", as well as that we by our heroic resistance gave the
UN again the chance to make decisions, which must not be missed by the
more powerful in numbers and militarily stronger. All these speeches at
the closing ceremony were held in the frames of scientific discussions
and elements of nonlinear dynamical systems, bat there were also present
open sympathies for the Yugoslav people and its proud behavior and resistance.
The president of the scientific committee
academician Matrosov held a scientific speech on perspectives scientific
research on global level in the area of nonlinear sciences from the aspect
of modeling of social processes, Earth's resources, and global nonlinear
dynamical, intelligent system on Earth - "noosphere", and which will also
be in the content of his speech at the, at that time -future, World Conference
on Science in the Third Millennium, held in Budapest one week after the
end of our conference (from June 29th to July 2nd 1999), and in the work
of which he participated as a scientist - a member of the Russian state
Katica (Stevanović) Hedrih