Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Chaotic Behavior of Cellular Automata

  • Julien Cervelle
  • Alberto Dennunzio
  • Enrico Formenti
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_65-4

Glossary

Equicontinuity

All points are equicontinuity points (in compact settings).

Equicontinuity point

A point for which the orbits of nearby points remain close.

Expansivity

From two distinct points, orbits eventually separate.

Injectivity

The next state function is injective.

Linear CA

A CA with additive local rule.

Regularity

The set of periodic points is dense.

Sensitivity to initial conditions

For any point x there exist arbitrary close points whose orbits eventually separate from the orbit of x.

Strong transitivity

There always exist points which eventually move from any arbitrary neighborhood to any point.

Surjectivity

The next state function is surjective.

Topological mixing

There always exist points which definitely move from any arbitrary neighborhood to any other.

Transitivity

There always exist points which eventually move from any arbitrary neighborhood to any other.

Definition of the Subject

A discrete time dynamical system (DTDS) is a pair 〈X, F〉 where Xis a set...

This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

This work has been supported by the Interlink/MIUR project “Cellular Automata:

Topological Properties, Chaos and Associated Formal Languages”, by the ANR Blanc Project “Sycomore” and by the PRIN/MIUR project “Formal Languages and Automata: Mathematical and Applicative Aspects”.

Bibliography

Primary Literature

  1. Acerbi L, Dennunzio A, Formenti E (2007) Shifting and lifting of cellular automata. In: Third conference on computability in Europe, CiE 2007, Siena, Italy, 18–23 June 2007. Lecture notes in computer science, vol 4497. Springer, Berlin, pp 1–10Google Scholar
  2. Adler R, Konheim A, McAndrew J (1965) Topological entropy. Trans Am Math Soc 114:309–319MathSciNetCrossRefzbMATHGoogle Scholar
  3. Akin E, Auslander E, Berg K (1996) When is a transitive map chaotic? In: Bergelson V, March P, Rosenblatt J (eds) Convergence in ergodic theory and probability. de Gruyter, Berlin, pp 25–40Google Scholar
  4. Amoroso S, Patt YN (1972) Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. J Comput Syst Sci 6:448–464MathSciNetCrossRefzbMATHGoogle Scholar
  5. Assaf D IV, Gadbois S (1992) Definition of chaos. Am Math Mon 99:865MathSciNetGoogle Scholar
  6. Auslander J, Yorke JA (1980) Interval maps, factors of maps and chaos. Tohoku Math J 32:177–188MathSciNetCrossRefzbMATHGoogle Scholar
  7. Banks J, Brooks J, Cairns G, Davis G, Stacey P (1992) On Devaney’s definition of chaos. Am Math Mon 99:332–334MathSciNetCrossRefzbMATHGoogle Scholar
  8. Blanchard F, Cervelle J, Formenti E (2005) Some results about chaotic behavior of cellular automata. Theor Comp Sci 349:318–336MathSciNetCrossRefzbMATHGoogle Scholar
  9. Blanchard F, Formenti E, Kurka K (1998) Cellular automata in the Cantor, Besicovitch and Weyl topological spaces. Compl Syst 11:107–123MathSciNetzbMATHGoogle Scholar
  10. Blanchard F, Glasner E, Kolyada S, Maass A (2002) On Li-Yorke pairs. J Reine Angew Math 547:51–68MathSciNetzbMATHGoogle Scholar
  11. Blanchard F, Kurka P, Maass A (1997) Topological and measure-theoretic properties of one-dimensional cellular automata. Phys D 103:86–99MathSciNetCrossRefzbMATHGoogle Scholar
  12. Blanchard F, Maass A (1997) Dynamical properties of expansive one- sided cellular automata. Israel J Math 99:149–174MathSciNetCrossRefzbMATHGoogle Scholar
  13. Blanchard F, Tisseur P (2000) Some properties of cellular automata with equicontinuity points. Ann Inst Henri Poincaré Probab Stat 36:569–582ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. Boyle M, Kitchens B (1999) Periodic points for cellular automata. Indag Math 10:483–493MathSciNetCrossRefzbMATHGoogle Scholar
  15. Boyle M, Maass A (2000) Expansive invertible one-sided cellular automata. J Math Soc Jpn 54(4):725–740CrossRefzbMATHGoogle Scholar
  16. Cattaneo G, Dennunzio A, Margara L (2002) Chaotic subshifts and related languages applications to one-dimensional cellular automata. Fundam Inform 52:39–80MathSciNetzbMATHGoogle Scholar
  17. Cattaneo G, Dennunzio A, Margara L (2004) Solution of some conjectures about topological properties of linear cellular automata. Theor Comp Sci 325:249–271MathSciNetCrossRefzbMATHGoogle Scholar
  18. Cattaneo G, Finelli M, Margara L (2000) Investigating topological chaos by elementary cellular automata dynamics. Theor Comp Sci 244:219–241MathSciNetCrossRefzbMATHGoogle Scholar
  19. Cattaneo G, Formenti E, Manzini G, Margara L (2000) Ergodicity, transitivity, and regularity for linear cellular automata. Theor Comp Sci 233:147–164. A preliminary version of this paper has been presented to the Symposium of Theoretical Computer Science (STACS’97). LNCS, vol 1200MathSciNetCrossRefzbMATHGoogle Scholar
  20. Cattaneo G, Formenti E, Margara L, Mazoyer J (1997) A shift-invariant metric on S Z inducing a non-trivial topology. In: Mathematical Foundations of Computer Science 1997. Lecture notes in computer science, vol 1295. Springer, Berlin, pp 179–188Google Scholar
  21. D’Amico M, Manzini G, Margara L (2003) On computing the entropy of cellular automata. Theor Comp Sci 290:1629–1646MathSciNetCrossRefzbMATHGoogle Scholar
  22. Denker M, Grillenberger C, Sigmund K (1976) Ergodic theory on compact spaces. Lecture notes in mathematics, vol 527. Springer, BerlinGoogle Scholar
  23. Devaney RL (1989) An Introduction to chaotic dynamical systems, 2nd edn. Addison-Wesley, ReadingzbMATHGoogle Scholar
  24. Di Lena P (2006) Decidable properties for regular cellular automata. In: Navarro G, Bertolossi L, Koliayakawa Y (eds) Proceedings of fourth IFIP international conference on theoretical computer science. Springer, Santiago de Chile, pp 185–196Google Scholar
  25. Durand B, Formenti E, Varouchas G (2003) On undecidability of equicontinuity classification for cellular automata. Discrete Math Theor Comp Sci AB:117–128MathSciNetzbMATHGoogle Scholar
  26. Edgar GA (1990) Measure, topology and fractal geometry. Undergraduate texts in Mathematics. Springer, New YorkCrossRefGoogle Scholar
  27. Formenti E (2003) On the sensitivity of additive cellular automata in Besicovitch topologies. Theor Comp Sci 301(1–3):341–354MathSciNetCrossRefzbMATHGoogle Scholar
  28. Formenti E, Grange A (2003) Number conserving cellular automata II: dynamics. Theor Comp Sci 304(1–3):269–290MathSciNetCrossRefzbMATHGoogle Scholar
  29. Furstenberg H (1967) Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation. Math Syst Theor Theor Comp Syst 1(1):1–49MathSciNetCrossRefzbMATHGoogle Scholar
  30. Glasner E, Weiss B (1993) Sensitive dependence on initial condition. Nonlinearity 6:1067–1075ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. Guckenheimer J (1979) Sensitive dependence to initial condition for one-dimensional maps. Commun Math Phys 70:133–160ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. Haeseler FV, Peitgen HO, Skordev G (1992) Linear cellular automata, substitutions, hierarchical iterated system. In: Fractal geometry and computer graphics. Springer, BerlinGoogle Scholar
  33. Haeseler FV, Peitgen HO, Skordev G (1993) Multifractal decompositions of rescaled evolution sets of equivariant cellular automata: selected examples. Technical report, Institut für Dynamische Systeme, Universität BremenGoogle Scholar
  34. Haeseler FV, Peitgen HO, Skordev G (1995) Global analysis of self-similarity features of cellular automata: selected examples. Phys D 86:64–80MathSciNetCrossRefzbMATHGoogle Scholar
  35. Hedlund GA (1969) Endomorphism and automorphism of the shift dynamical system. Math Sy Theor 3:320–375MathSciNetCrossRefzbMATHGoogle Scholar
  36. Hurd LP, Kari J, Culik K (1992) The topological entropy of cellular automata is uncomputable. Ergodic. Theor Dyn Sy 12:255–265zbMATHGoogle Scholar
  37. Hurley M (1990) Ergodic aspects of cellular automata. Ergod Theor Dyn Sy 10:671–685MathSciNetzbMATHGoogle Scholar
  38. Ito M, Osato N, Nasu M (1983) Linear cellular automata over z m. J Comp Sy Sci 27:127–140MathSciNetzbMATHGoogle Scholar
  39. Kannan V, Nagar A (2002) Topological transitivity for discrete dynamical systems. In: Misra JC (ed) Applicable mathematics in golden age. Narosa Pub, New DelhiGoogle Scholar
  40. Kari J (1994a) Reversibility and surjectivity problems of cellular automata. J Comp Sy Sci 48:149–182MathSciNetCrossRefzbMATHGoogle Scholar
  41. Kari J (1994b) Rice’s theorem for the limit, set of cellular automata. Theor Comp Sci 127(2):229–254MathSciNetCrossRefzbMATHGoogle Scholar
  42. Knudsen C (1994) Chaos without nonperiodicity. Am Math Mon 101:563–565MathSciNetCrossRefzbMATHGoogle Scholar
  43. Kolyada S, Snoha L (1997) Some aspect of topological transitivity – a survey. Grazer Math Ber 334:3–35MathSciNetzbMATHGoogle Scholar
  44. Kurka P (1997) Languages, equicontinuity and attractors in cellular automata. Ergo Theor Dyn Sy 17:417–433MathSciNetCrossRefzbMATHGoogle Scholar
  45. Kurka P (2004) Topological and symbolic dynamics. Cours Spécialisés, vol 11. Société Mathématique de France, ParisGoogle Scholar
  46. Li TY, Yorke JA (1975) Period three implies chaos. Am Math Mon 82:985–992MathSciNetCrossRefzbMATHGoogle Scholar
  47. Manzini G, Margara L (1999) A complete and efficiently computable topological classification of D-dimensional linear cellular automata over Z m. Theor Comp Sci 221(1–2):157–177MathSciNetCrossRefzbMATHGoogle Scholar
  48. Margara L (1999) On some topological properties of linear cellular automata. Kutylowski M, Pacholski L, Wierzbicki T Mathematical foundations of computer science 1999 (MFCS99). Lecture notes in computer science, vol 1672. Springer, Berlin, pp 209–219Google Scholar
  49. Moothathu TKS (2005) Homogenity of surjective cellular automata. Discret Contin Dyn Syst 13:195202CrossRefGoogle Scholar
  50. Morris G, Ward T (1998) Entropy bounds for endomorphisms commuting with k actions. Israel J Math 106:1–12MathSciNetCrossRefzbMATHGoogle Scholar
  51. Nasu M (1995) Textile systems for endomorphisms and automorphisms of the shift. Memoires of the American Mathematical Society, vol 114. American Mathematical Society, ProvidenceGoogle Scholar
  52. Pesin YK (1997) Dimension theory in dynamical systems. Chicago lectures in Mathematics. The University of Chicago Press, ChicagoCrossRefGoogle Scholar
  53. Shereshevsky MA (1993) Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indag Math 4:203–210MathSciNetCrossRefzbMATHGoogle Scholar
  54. Shereshevsky MA, Afraimovich VS (1993) Bipermutative cellular automata are topologically conjugate to the one-sided Bernoulli shift. Random Comput Dynam 1(1):91–98MathSciNetzbMATHGoogle Scholar
  55. Sutner K (1999) Linear cellular automata and de Bruijn automata. In: Delorme M, Mazoyer J (eds) Cellular automata, a parallel model, number 460 in mathematics and its applications. Kluwer, DordrechtGoogle Scholar
  56. Takahashi S (1992) Self-similarity of linear cellular automata. J Comput Syst Sci 44:114–140MathSciNetCrossRefzbMATHGoogle Scholar
  57. Vellekoop M, Berglund R (1994) On intervals, transitivity = chaos. Am Math Mon 101:353–355MathSciNetCrossRefzbMATHGoogle Scholar
  58. Walters P (1982) An introduction to ergodic theory. Springer, BerlinCrossRefzbMATHGoogle Scholar
  59. Weiss B (1971) Topological transitivity and ergodic measures. Math Syst Theor 5:71–75ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. Willson S (1984) Growth rates and fractional dimensions in cellular automata. Phys D 10:69–74MathSciNetCrossRefzbMATHGoogle Scholar
  61. Willson S (1987a) Computing fractal dimensions for additive cellular automata. Phys D 24:190–206ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. Willson S (1987b) The equality of fractional dimensions for certain cellular automata. Phys D 24:179–189ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. Wolfram S (1986) Theory and applications of cellular automata. World Scientific, Singapore, SingaporezbMATHGoogle Scholar

Books and Reviews

  1. Akin E (1993) The general topology of dynamical systems. Graduate studies in mathematics, vol 1. American Mathematical Society, ProvidenceGoogle Scholar
  2. Akin E, Kolyada S (2003) Li-Yorke sensitivity. Nonlinearity 16:1421–1433ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. Block LS, Coppel WA (1992) Dynamics in one dimension. Springer, BerlinCrossRefzbMATHGoogle Scholar
  4. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  5. Kitchens PB (1997) Symbolic dynamics: one-sided, two-sided and countable state Markov shifts. Universitext Springer, BerlinzbMATHGoogle Scholar
  6. Kolyada SF (2004) Li-Yorke sensitivity and other concepts of chaos. Ukr Math J 56(8):1242–1257MathSciNetCrossRefzbMATHGoogle Scholar
  7. Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambidge University Press, CambidgeCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Julien Cervelle
    • 1
  • Alberto Dennunzio
    • 2
  • Enrico Formenti
    • 3
  1. 1.Laboratoire d’Informatique de l’Institut Gaspard–MongeUniversité Paris-EstMarne- la-ValléeFrance
  2. 2.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi di Milano-BicoccaMilanItaly
  3. 3.Laboratoire I3SUniversité de Nice-Sophia AntipolisSophia AntipolisFrance