**Mathematical Colloquim
**

**PROGRAM**

ODELJENJE ZA MATEMATIKUMATEMATIČKOG INSTITUTA SANU |

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PROGRAM ZA MAJ 2015.
**

NAPOMENA: Predavanja ce se odrzavati u Sali 301f na trecem spratu Matematickog instituta SANU, Knez-Mihailova 36 (zgrada preko puta SANU).

__Petak, 8.05.2015. u 14:00h, sala 301f, MI SANU__

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Katica R. (Stevanovic) Hedrih, Matematicki institut
SANU
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PETROVIC'S ELEMENTS OF MATHEMATICAL PHENOMENOLOGY AND
PHENOMENOLOGICAL MAPPINS: THEORY AND APPLICATIONS

Abstract: Lecture starts with short description of Element of
Mathematical Phenomenology and Phenomenological Mappings published in
Petrovic's theory. The biographical data of Mihailo Petrovic (1868-1943)
is presented. Petrovic was a famous Serbian mathematician, one of three
Henrei Poincare's doctoral students. Next it is a description of
abstraction of real system to the physical, chemical or biological and
mathematical model.

Some of basic elements of mathematical phenomenology are elements of
non-linear-functional transformations of coordinates from one to other
functional curvilinear coordinate system. Some of these elements, as it is
basic vectors of tangent space of kinetic point vector position and their
changes (velocity of their magnitude extensions and component angular
velocities of rotations), are presented in different functional coordinate
systems.

Mihailo Petrovic's theory contains two types of analogies: mathematical
and qualitative, and in this lecture third type - structural analogy is
described. Taking into account large possibility for applications of all
three types of analogies, numerous original examples are presented using,
between other, fractional system dynamics with one degree of freedom,
finite number of degrees of freedom as well as multi-body discrete
continuum hybrid fractional order system dynamics.

Mathematical analogies between vector models in local area of stress
state, strain stare of the point in stressed and deformed deformable body
as well as with vector model of the mass inertia moment state at point of
rigid body, used mass inertia moment vectors coupled for pole and axis,
are presented, also.

Using discrete continuum method, fractional order mode analysis in hybrid
system dynamics is presented. For a class of fractional order system
dynamics with finite number of degrees of freedom, independent eigen main
fractional order modes are determined with corresponding eigen main
coordinates of the system and presented by Tables. A number of theorems of
energy fractional order dissipation presented in corresponding Tables,
also. It is shown that applications of qualitative, structural and
mathematical analogies in analysis of fractional order modes appear in
analogous mechanical, electrical and biological fractional order chains,
and that is very power, suitable and useful tools to reduce research
models to corresponding minimal numbers, and, in same time, develop power of
analysis use phenomenological mappings between local and global phenomena
and properties.

An analogy between kinetic parameters of collision of two rigid body in
translator motions and collision of two rolling billiards' balls is
presented and corresponding new theorems are defined.

Phenomenological approximate mappings on nonlinear phenomena, in local
area around stationary points or stationary states, are presented.
Corresponding kinetic parameters of model of nonlinear dynamics of real
system behavior are presented, also. For obtaining approximate differential
equations and approximate solutions in local area around singular points,
linear and non-liner approximations are used. Method of local analysis
based on phenomenological approximate mappings between local linear as
well as nonlinear phenomena is power to obtain information of all local
nonlinear phenomena in the nonlinear dynamics of the system for completing
kinetic elements for global analysis of the system nonlinear dynamics and
stability and to use different analogies.

Stevan Pilipovic, Odeljenje za matematiku Matematickog instituta SANU