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Relating Dynamical Systems to

Software Engineering

“Some of the most exciting interactions between mathematics and engineering are occurring in the area of analysis and control of uncertain, multivariable, and nonlinear systems. While changing technology has made control and dynamical systems theory increasingly relevant to a much broader class of problems, the interdisciplinary nature of this area means that they no longer have a natural home exclusively or even primarily within any one of the traditional engineering disciplines.”

From CalTech’s DS&C Website

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Dynamical Systems Basics

• The change of state in a system over time• Autonomous & control systems• Continuous & discrete systems• Characterized by some level of complexity

– Degrees of freedom => # of dimensions– Incompleteness – Godelian

Dynamical Systems Basics

• Possible meta-states of outcomes– Deterministic– Vascillation– Chaotic

• Some combination of the above

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SE Basics

• Best practices• Formalized processes & procedures• Phases and iterations• Possible meta-state outcomes

– Deterministic (hopefully)– Vascillation– Chaotic

Decisions, decisions

• Is it important to be precise?– π = 3.14…– π = 3.141592653589792384626433832795…

(6.4 Billion* digits and counting)

• Is it important to be correct?– π = the constant ratio of the circumference of

a circle to the diameter*American

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Heisenberg Uncertaintyin the Software Engineering

SystemThe more precisely you know

where you are (your state),the less precisely you know where

you’re going (your transition)

Lorenz AttractorPhase shifting between statespace and transition space

Tools from MOF / UML World

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A Change of State …

Understanding

Confusion

Analysis ?

Over time

Requirement n

Epiphany*

*A sudden manifestation of the essence or meaning of something.

State

Transition

Start

Event

End

Outside Observer

…Can Become Complex

Understanding

Confusion

Epiphany*

Mealy

Moore

Time

*A sudden manifestation of the essence or meaning of something.

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Space(The Final Frontier?)

State SpaceTransition SpacePhase Space

Model SpacePaper Space

Defining Time & Space

• Iteration => Discrete time (ordered)• A State Space• A Transitional (Phase) Space

– In the ‘state’ of transition?– Transitional vector between

• State spaces• State space and a Phase Space• Phase Spaces => meta-Phase Space

Discrete

Discrete

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A State Space

Blue => confusionWhite => understanding

Blue => DocumentaryWhite => AnalysisYellow => NormalizationRed => SynthesisAqua => Realization

Embedding

Meaning of orderIn space

A Transition Space

State 1Confusion

State 2Understanding

PossiblePseudo-States?

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A Phase Space

AutomataRules for Transitions

• Rules can be expressed in mathematical, textual and graphical dimensions

• The iteration of rules can show behavior and meta-state evolution.

• Rules may be unidirectional or bi-directional => rules may be Hamiltonian or dissipative.

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Possible Conclusions and Derivative concepts

• Methods may be formalized using concepts of space, state, transition and phase to predict meta-state behavior in software engineering systems.

• Replicate the space concepts discussed to include a model space of all known informational dimensions – not just an abstraction.

• Paper space may be a projection of 2 dimensions of the model onto a traditional medium.

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Contact Information

Kenneth A. Lloyd, Jr.PresidentWatt Systems Technologies, Inc.3444 Briar PlaceCarmel, IN 46033 USA

Phone: 317-846-4088Email: kalloyd@wattsys.com

Web: http://www.wattsys.com

The companies and individuals listed above hereby grant a royalty-free license to the Object ManagementGroup, Inc. (OMG) for worldwide distribution of this document or any derivative works thereof within OMG andto OMG members for evaluation purposes, so long as the OMG reproduces the copyright notices and the belowparagraphs on all distributed copies.

The information contained in this document is subject to change with notice.

UML is a trademark of the Object Management Group.

Copyright © 2002, Watt Systems Technologies, Inc.Copyright © 2002. Kenneth A. Lloyd, Jr.

All rights reserved.