**Mathematical modeling of complex dynamical systems**

Yu.B. Kolesov, Yu.B.Senichenkov

** **

Mathematical modeling is commonly used in up-to-date industry and it calls for well-placed acknowledged experts which are able to

- design complex models using tools for modeling and simulation,
- analyse their properties carring out computational experiments,
- employ them for solving real-wold problems.

Computer models of complex dynamical systems are abundant and called-for types of models. Models based on ordinary differential and difference equations are used not only for modeling and simution real-wold physical phenomenona, but for design new technical systems too. Not long ago mathematical modeling and simulation was prerogative of mathematicans, but now, when tools for visual modeling are easy for using and they have intitive and friendly user interface, modern engineers are able and have to design and use computer models by themselfs.

Engineering education has changed a lot of late fiаty years and now it is inconceivable without such basic and former traditional mathematical disciplines as informatics, theory of algorithms, numerical analysis. The adaptation of these disciplines for engineers is demanding task and «mathematical modeling» gives a good example of incipient problems. There are fine textbooks for mathematican students and there is not enouth good books for engineers. Similar situation was in the middle of last century seventieth. There were books on numerical methods for linear algebra, ordinary and partial differential equations for mathematicans, and the were no books for engineers. The books for engineers written by mathematicans appeared only after appearance of application packages (Linlack, Eispack, OdePack). It would be adequate to list just books: George Elmer Forsythe, Michael A. Malcolm, Cleve B. Moler «Computer Methods for Mathematical Computations», John R. Rice «Matrix Computations and Mathematical Software» and others, translated in Russian later. These books based on theoretical results are teaching not to design of applicational packages but to use engineered software for solving practical problems. This is the main feature of these books and they are demanded till now.

The main goal of this book is to tell engineers about special type of mathematical models named dynamical systems, discuss their properties and their using in engineering, illustrate possibility their designing and analyzing with the help of tools for visual modeling and simulation.

It is possible to use tools for modelling and simulation intuitively, but backgroung knowledge of modeling theory will help to do it better.

Contents of the textbook:

**Introduction**.

**Chapter 1. Using Mathematical Modeling for cognition and design **

Mathematical models.

Models based on differential and difference equations

Models based on partial differential equations

Simulation modeling

Building of mathematical models

Computational experiments

Model adequacy

Model analyzing

References

**Chapter 2. Dynamical systems **

Contituous dynamical systems ane thier discrete approximation

Discrete dynamical systems

One-dimensional and two-dimensional dynamical systems

Dynamical systems on line

Linear dynamical systems on plane

Non-Linear dynamical systems on plane

References

**Chapter 3. Stability of dynamical systems **

Stability of dynamical systems. *Lyapunov stability* theory

Linearization and stability.

References

**Chapter 4. Event-driven dynamical systems (hybrid). **

State machines

Event-driven systems

B-Charts

States and transitions

Internal transitions

Activity

Hybrid time

Discrete variables

Signals

Broadcast signals

State activity

Activity composition

Orthogonal activity

Hidden hybrid models. Conditional equations.

Bad types of hybrid behavior

Zeno’ behavior

Sliding

References

**Chapter 5. Introduction in theory of oscillations **

Oscillators

Self-oscillations

Limit cycles.

Poincaré cross-section

Examples

References

** Chapter 6. Bifurcation **

Bifurcation in continuous and discrete systems.

Bifurcation diagrams .

Lamerey diagram.

Strange attractors.

References

** Chapter 7. Markov chains. **

Continuous and discrete chains.

Markov equations.

References

**Chapter 8. Computational experiments **

Computational experiments in Rand Model Designer.

Examle. Chemical kinetics

Hierarchy of models. Inheritance

DAE models

Possible forms of model specification

Equations in Matrix form

Discrete activity: functions, procedures

Date types

Final example

**Сonclusion **