Dipartimento di Scienze Economiche, Matematiche e Statistiche
Università degli Studi di Foggia
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Modeling Algorithms for Dynamical Systems
Crescenzio Gallo
Quaderno n. 01/2009
“Esemplare fuori commercio per il deposito legale agli effetti della legge 15 aprile 2004 n. 106”
Quaderno riprodotto dal Dipartimento di Scienze Economiche, Matematiche e Statistiche
nel mese di febbraio 2009 e depositato ai sensi di legge.
Authors only are responsible for the content of this reprint.
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Dipartimento di Scienze Economiche, Matematiche e Statistiche, Largo Papa Giovanni Paolo II, 1,
71100 Foggia (Italy), Phone +39 0881-75.37.30, Fax +39 0881-77.56.16
Modeling Algorithms for Dynamical Systems
Crescenzio Gallo
[email protected]
Dipartimento di Scienze Economiche, Matematiche e Statistiche
Università di Foggia – Largo Papa Giovanni Paolo II n.1, 71100 Foggia (Italy)
Abstract
human capacity for abstraction.
The management and control of complex systems
Technological development and explosion of com- behaviour are increasingly demanding challenges,
puting tools allow today to make best use of and the creation of models seems to be a fundamodeling methodologies that in the past were not mental aspect of human thought.
able to perform the calculations necessary for the
mathematical representation of a complex system.
This paper illustrates some directions in literature 1.2 Modeling dynamical systems
about the modeling of Dynamical Systems, targetDynamical systems are frequently modeled by difing the algorithmic search of numerical methods
ference and differential equations (ordinary or not).
with an exact or limited error approximation.
Certainly they are not the only type of formal model
for dynamical systems, but have a long tradition
Keywords: modeling; dynamical systems; aland have been used in different ways to describe a
gorithm.
wide range of real-world systems.
Mathematics Subject Classification (2000)
This work is focused on the construction of these
93A30; 37-02, 34A34.
models, and to this end, the “mathematical object” is conceived as consisting of several elements:
the framework, the structure and the parameters,
1 Introduction
and a methodology is considered to assemble these
components, illustrating the procedures to focus di1.1 Topics
rectly on the system in question and its model. In
The expansion of new frontiers of technology and such circumstances the computing tool may be usescience meets a strong resistance due to increasing ful in two ways: firstly, it can provide the means
complexity of the problems to be solved. The so- for resolving analytically intractable equations; seclution to this difficulty is partly from modeling, for ondly, it can provide a sufficient computing power
his ability to simplify problems by eliminating un- to use otherwise unworkable techniques.
necessary details.
Another type of model is also used for the deFor a long time modeling methodologies have scription of dynamical systems, based on partial
been plagued by a lack of an instrument being able differential equations (PDE): it is a more complex
to cover with precision and speed the enormous model, whose creation methodology is not yet at
amount of calculations necessary for the mathemat- the same level reached by the model for differential
ical representation of a complex system.
equations, but it is often used in many fields of sciNowadays, computers can tackle problems other- ence and engineering. The analytical techniques for
wise intractable, and their usefulness, however, does resolving PDE are limited and numerical solutions
not end in simple computing capabilities. The de- are so costly in terms of computation to overcome
velopment of software for modeling allows for catch- at times the computing capacity of most powerful
ing links and meanings not normally detectable be- workstations. Even in building models the computcause hidden by the vast amount of data that is ing needs are much more onerous than for ODE;
to be collected during the observation of a system. consequently, the most efficient method for buildIn this sense, then, the machine is an extension of ing PDE models is based on traditional deductive