János Karsai

Models of Impulsive Phenomena

Mathematica experiments


Typotex Publishing Company, Budapest, 2002


Table of contents





How to use the book and the electronic attachment  


I.     Modeling preliminaries                                                                                              

1.     Visualization exercises

        1.1.      Computer visualization: advantages and dangers

        1.2.      Animation

        1.3.      Visualization of the path of a motion

2.     Elements of computer-aided investigation of differential equations

        2.1.      Concepts, definitions

        2.2.      The built-in Mathematica functions

        2.3.      The ODESolve package


II.    Properties of impulsive systems

3.     Introduction

        3.1.      Some examples

        3.2.      The Dirac-δ  function

        3.3.      The Dirac-δ in differential equations

4.     Impulses at fixed instants

        4.1.      Definitions, basic properties

        4.2.      The IDESolve package

        4.3.      Modeling scheme for 1D systems

        4.4.      Modeling scheme for 2D systems

5.     State-dependent impulses

        5.1.      Impulses at variable instants 

        5.2.      Autonomous impulsive systems

        5.3.      General impulses

        5.4.      Visualization of general impulses

        5.5.      General solving program: IDERKSolve

        5.6.      The beating phenomenon

6.     Computer-aided study of  phase mapping

        6.1.      Theoretical overview

        6.2.      Phase maps of ordinary differential equations

        6.3.      Phase maps of impulsive systems

7.     Linear impulsive systems

        7.1.      General properties

        7.2.      Linear periodic systems

8.     Stability

        8.1.      Concepts, definitions

        8.2.      Stability of linear systems

        8.3.      Stability by linear approximation

        8.4.      Linear approximation in the case of variable impulse instants

9.     Liouville-formula, phase-volume method

        9.1.      Theoretical overview

        9.2.      Phase-volume experiments for differential equations

        9.3.      Phase-volume experiments for impulsive systems

10.   Stability by the Liapunov’s direct method 

        10.1.    Overview, stability theorems

        10.2.    Experiments for differential equations

        10.3.    Experiments for impulses at fixed instants

        10.4.    Experiments for impulses at variable instants

III. Some applications

11.   Drug dosage

        11.1.    Compartmental systems

        11.2.    Infusion or injection: systems of one compartment

        11.3.    Infusion or tablets: systems of two compartments

12.   Impulses in population dynamics

        12.1.    Control of the Malthusian model

        12.2.    Population growth in restricted life space

13.   Oscillators with impulsive perturbation

        13.1.    Oscillator equations of one degree of freedom

        13.2.    Harmonic oscillators with impulsive damping

        13.3.    Beating in oscillators

        13.4.    Impulsive external forces

        13.5.    Impulsive relaxation oscillations

14.   Beating, reflection, control

        14.1.    Introductory examples

        14.2.    Playing with a ball

        14.3.    Systems of impulsive control

IV.  Appendix

15.   References

16.   Mathematica functions developed for the book

17.  Notations

18. Index    

V.    Electronic attachment. Add-on chapters  

19.   Mathematica fundamentals

        19.1.    Concepts, data, variables, functions, lists

        19.2.    2D graphics

        19.3.    3D graphics

        19.4.    Solving equations

        19.5     Calculus: limit, derivative, integral, series

20.   Some useful method in modeling

        20.1.    Data: plotting and transformations

        20.2.    Linear approximation

        20.3.    Fixed points, zero points, and iterations

        20.4.    Maxima and minima

        20.4.    Curve fitting

21.   Modeling schemes for ordinary differential equations

        21.1.    Modeling scheme for 1D equations

        21.2.    Modeling scheme for 2D equations