**Contents**

*Cover**Preface and Contents*- Chapter 1: Introduction
- Chapter 2: Computability with Systems
- Chapter 3: Universality and Beyond

*Bibliography**Appendix A: Simple Samples**Appendix B: Universal Machine**Appendix C: Bomb Systems*

**Abstract**

This report studies a specific nonlinear system structure and shows that for this class of systems the theoretical undecidability issues become acute: It is possible to construct fixed systems within this class that defy all analysis attempts. Tools are introduced to utilize this framework, so that systems with special properties can be determined using a high-level description formalism.

**Keywords**: Computability, undecidability, dynamic systems, stability

**Comments**

- The compiler, written in Python, is available. The py-files are WinZipped; unpack the files in some folder, install Python, define the paths accordingly, and you are ready to run compilations. Manuals are available in the same archive.
- There is also a poster available.
- Papers where the same positive system structure has been studied from another point of view are also available; see also the introduction to complex systems.

**Errata**

- The proof on page 24 is incomplete; this version only proves that for a fixed
*A*infinite number of alternative initial vectors*s*(0) exist for which the qualitative properties of the system remain intact. To show that there exist an infinite number of systems with the same dimension but different matrices*A*having the same undecidability properties, one needs to recognize that if one defines a diagonal square matrix*D*having arbitrary positive-valued diagonal entries, one can define the vector*z = Ds*so that for a sequence of such*z*(*k*) the original behavior remains qualitatively intact if the system matrix, instead of being*A*, is*DAD*^{-1}. - In Reference [15], the author’s name has been dropped out; the author is Eduardo Sontag.

**On “Systems Cosmology” or “Systems Astronomy”**

Let us be a little poetic here.

*Some people say that there exists a star in the sky for every soul that ever lived.
It turns out that in the Universe of Systems there exists a “Star System” for every algorithm that was ever written.
These star systems can be dull and uninteresting, but they can also be full of magnificent life forms never seen before.
What is more, it turns out that there also exist “black holes” in that universe – objects that never let information be extracted from them.*