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:


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                                                                                                


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                                                        


Chapter 3. Stability of dynamical systems                            

    Stability of dynamical systems. Lyapunov stability theory        

    Linearization and stability.                                                                                


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

    State machines                                                                                                  

    Event-driven systems                                                                                         


    States and transitions                                                                                        

    Internal transitions                                                                                            


    Hybrid time                                                                                                      

    Discrete  variables                                                                                            


    Broadcast signals                                                                                              

    State activity                                                                                                    

    Activity composition                                                                                       

    Orthogonal activity                                                                                           

    Hidden hybrid models. Conditional equations.                                                 

    Bad types of hybrid behavior                                                                            

    Zeno’ behavior                                                                                                  



Chapter 5. Introduction in theory of oscillations               



   Limit cycles.                                                                                                    

   Poincaré cross-section                                                                                    



 Chapter 6.  Bifurcation                                                     

   Bifurcation in continuous and discrete systems.

   Bifurcation diagrams                .                                                                   

   Lamerey diagram.                                                                                          

   Strange attractors.                                                                                          


 Chapter 7. Markov chains.                                   

    Continuous and discrete chains.                                                                    

    Markov equations.                                                                                         


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                                                                                                      




University of Bremen






The European Commission support for the production of this publication does not constitute an endorsement of the contents which reflects the views only of the authors, and the Commission cannot be held responsi­ble for any use which may be made of the information contained therein

Project news

As part of on-going monitoring of CBHE projects, an advisory field monitoring has taken place on 11 September 2018 at the premises of Novosibirsk State Technical University. It was carried out by the National Erasmus+ Office Russia.

The  project meeting of InMotion steering group  was hold in Novosibirsk  State Technical Universityin th  faculty of Automation and Computer Engineering. The members of the project steering group get together to report about the results of the work during the first two project years and plan the activities for the final year.

You can read more information about these and other events in our Newsletter Oct 2018

25-27.03.119 Project Meeting in UniKL (Kuala Lumpur)



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