Petak, 8.05.2015. u 14:00h, sala 301f, MI SANU
Katica R. (Stevanovic) Hedrih, Matematicki institut SANU
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.