Basins of Attraction of Cellular Automata and Discrete Dynamical Networks
Glossary
 Attractor, basin of attraction, subtree

The terms “attractor” and “basin of attraction” are borrowed from continuous dynamical systems. In this context the attractor signifies the repetitive cycle of states into which the system will settle. The basin of attraction in convergent (injective) dynamics includes the transient states that flow to an attractor as well as the attractor itself, where each state has one successor but possibly zero or more predecessors (preimages). Convergent dynamics implies a topology of trees rooted on the attractor cycle, though the cycle can have a period of just one, a point attractor. Part of a tree is a subtree defined by its root and number of levels. These mathematical objects may be referred to in general as “attractor basins.”
 Basin of attraction field

One or more basins of attraction comprising all of statespace.
 Cellular automata, CA

Although CA are often treated as having infinite size, we are dealing here with finite CA, which usually...
References
 Note: Most references by A. Wuensche are available online at http://www.uncomp.ac.uk/wuensche/publications.html
 Ashby WR (1956) An introduction to cybernetics. Chapman & Hall, LondonCrossRefzbMATHGoogle Scholar
 Conway JH (1982) What is life? In: Berlekamp E, Conway JH, Guy R (eds) Winning ways for your mathematical plays, chapter 25, vol Vol. 2. Academic Press, New YorkGoogle Scholar
 Cook M (2004) Universality in elementary cellular automata. Complex Syst 15:1–40zbMATHMathSciNetGoogle Scholar
 Domain C, Gutowitz H (1997) The topological skeleton of cellular automata dynamics. Pysica D 103(1–4):155–168ADSCrossRefzbMATHGoogle Scholar
 GomezSoto JM, Wuensche A (2015) The Xrule: universal computation in a nonisotropic Lifelike Cellular Automaton. JCA 10(34):261–294. preprint: http://arxiv.org/abs/1504.01434/ zbMATHGoogle Scholar
 GomezSoto JM, Wuensche A (2016) Xrule’s precursor is also logically universal. To appear in JCA. Preprint: https://arxiv.org/abs/1611.08829/
 Harris SE, Sawhill BK, Wuensche A, Kauffman SA (2002) A model of transcriptional regulatory networks based on biases in the observed regulation rules. Complexity 7(4):23–40CrossRefGoogle Scholar
 Hopield JJ (1982) Neural networks and physical systems with emergent collective abilities, proceeding of the national. Acad Sci 79:2554–2558CrossRefGoogle Scholar
 Kauffman SA (1969) Metabolic stability and Epigenesis in randomly constructed genetic nets. Theor Biol 22(3):439–467CrossRefMathSciNetGoogle Scholar
 Kauffman SA (1993) The origins of order. Oxford University Press, New York/OxfordGoogle Scholar
 Kauffman SA (2000) Investigations. Oxford University Press, New YorkGoogle Scholar
 Langton CG (1990) Computation at the edge of chaos: phase transitions and emergent computation. Physica D 42:12–37ADSCrossRefMathSciNetGoogle Scholar
 Somogyi R, Sniegoski CA (1996) Modeling the complexity of genetic networks: understanding multigene and pleiotropic regulation. Complexity 1:45–63CrossRefGoogle Scholar
 Walker CC, Ashby WR (1966) On the temporal characteristics of behavior in certain complex systems. Kybernetick 3(2):100–108CrossRefGoogle Scholar
 Wuensche A (1993–2017) Discrete Dynamics Lab (DDLab). http://www.ddlab.org/
 Wuensche A (1994a) Complexity in 1D cellular automata; Gliders, basins of attraction and the Z parameter. Santa Fe Institute working paper 9404025Google Scholar
 Wuensche A (1994b) The ghost in the machine: basin of attraction fields of random Boolean networks. In: Langton CG (ed) Artificial Life III. AddisonWesley, Reading, pp 496–501Google Scholar
 Wuensche A (1996) The emergence of memory: categorisation far from equilibrium. In: Hameroff SR, Kaszniak AW, Scott AC (eds) Towards a science of consciousness: the first Tucson discussions and debates. MIT Press, Cambridge, pp 383–392Google Scholar
 Wuensche A (1997) Attractor basins of discrete networks: Implications on selforganisation and memory. Cognitive science research paper 461, DPhil thesis, University of SussexGoogle Scholar
 Wuensche A (1998) Genomic regulation modeled as a network with basins of attraction. Proceedings of the 1998 pacific symposium on Biocomputing. World Scientific, SingaporeGoogle Scholar
 Wuensche A (1999) Classifying cellular automata automatically; finding gliders, filtering, and relating spacetime patterns, attractor basins, and the Z parameter. Complexity 4(3):47–66CrossRefMathSciNetGoogle Scholar
 Wuensche A (2004) Basins of attraction in network dynamics: a conceptual framework for biomolecular networks. In: Schlosser G, Wagner GP (eds) Modularity in development and Evolution,chapter 13. Chicago University Press, Chicago, pp 288–311Google Scholar
 Wuensche A (2009) Cellular automata encryption: the reverse algorithm, Zparameter and chain rules. Parallel Proc Lett 19(2):283–297CrossRefMathSciNetGoogle Scholar
 Wuensche A (2010) Complex and chaotic dynamics, basins of attraction, and memory in discrete networks. Acta Phys Pol, B 3(2):463–478Google Scholar
 Wuensche A (2016) Exploring discrete dynamics, 2nd edn. Luniver Press, FromezbMATHGoogle Scholar
 Wuensche A, Adamatzky A (2006) On spiral gliderguns in hexagonal cellular automata: activatorinhibitor paradigm. Int J Mod Phys C 17(7):1009–1026ADSCrossRefzbMATHGoogle Scholar
 Wuensche A, Lesser MJ (1992) The global dynamics of cellular automata; an atlas of basin of attraction fields of onedimensional cellular automata, Santa Fe institute studies in the sciences of complexity. AddisonWesley, ReadingzbMATHGoogle Scholar