Abstract
The classical dynamics concepts of recurrence and attractor are analysed in the basic mathematical setting of state transition systems, where both time and space are discrete, and no structure is assumed on the state space besides a binary transition relation. This framework proves useful to the dynamical analysis of computations and biomolecular processes. Here a relational formulation of this framework is presented, where the concepts of attractor and recurrence surface in two variants, respectively relating to the two fundamental modalities. A strong link between recurrence and both existence and extent of attractors, in either variant, is established by a novel characterization theorem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ashby, W.R.: An Introduction to Cybernetics. Chapman and Hall, Boca Raton (1956)
Backhouse, R., van der Woude, J.: Demonic Operators and Monotype Factors. Mathematical Structures in Computer Science 3(4), 417–433 (1993)
Bonanno, C., Manca, V.: Discrete dynamics in biological models. Romanian Journal of Information Science and Technology 1-2(5), 45–67 (2002)
Bianco, L., Fontana, F., Franco, G., Manca, V.: P Systems for Biological Dynamics. In: Ciobanu, G., Păun, Gh., Perez-Jimenez, M.J. (eds.) Applications of Membrane Computing. Natural Computing Series, pp. 81–126. Springer, Heidelberg (2006)
Devaney, R.L.: Introduction to chaotic dynamical systems. Addison-Wesley, Reading (1989)
Franco, G.: Biomolecular Computing – Combinatorial Algorithms and Laboratory Experiments, PhD thesis, University of Verona, Italy (2006)
Kauffman, S.: Investigations. Oxford University Press, Oxford (2000)
Kůrka, P.: Topological and Symbolic Dynamics, Cours Spécialisés. Société Mathématique de France 11 (2003)
Manca, V., Franco, G., Scollo, G.: State transition dynamics: basic concepts and molecular computing perspectives. In: Gheorghe, M. (ed.) Molecular Computational Models: Unconventional Approaches, Idea Group, Hershey, PA, USA, pp. 32–55 (2005)
Manca, V., Bianco, L.: Biological Networks in Metabolic P Systems (submitted, 2006)
Manca, V., Bianco, L., Fontana, F.: Evolution and oscillation in P systems: Applications to biological phenomena. In: Mauri, G., Păun, Gh., Jesús Pérez-JÃmenez, M., Rozenberg, G., Salomaa, A. (eds.) WMC 2004. LNCS, vol. 3365, pp. 63–84. Springer, Heidelberg (2005)
Păun, Gh.: Computing with membranes. J. Comput. System Sci. 61(1), 108–143 (2000)
Păun, Gh.: Membrane Computing. An Introduction. Springer, Heidelberg (2002)
Schmidt, G., Ströhlein, T.: Relations and Graphs. Springer, Heidelberg (1993)
Tarski, A.: On the calculus of relations. Journal of Symbolic Logic 6, 73–89 (1941)
Wolfram, S.: Theory and Application of Cellular Automata. Addison-Wesley, Reading (1986)
Wuensche, A.: Basins of Attraction in Network Dynamics: A Conceptual Framework for Biomolecular Networks. In: Schlosser, G., Wagner, G.P. (eds.) Modularity in Development and Evolution, Chicago University Press (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Scollo, G., Franco, G., Manca, V. (2006). A Relational View of Recurrence and Attractors in State Transition Dynamics. In: Schmidt, R.A. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2006. Lecture Notes in Computer Science, vol 4136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11828563_24
Download citation
DOI: https://doi.org/10.1007/11828563_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37873-0
Online ISBN: 978-3-540-37874-7
eBook Packages: Computer ScienceComputer Science (R0)