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TMF, 1998, Volume 114, Number 3, Pages 349–365 (Mi tmf845)  

This article is cited in 32 scientific papers (total in 32 papers)

$p$-Adic dynamic systems

S. A. Albeverioa, B. Tirozzib, A. Yu. Khrennikovc, S. de Smedtd

a University of Bonn, Institute for Applied Mathematics
b University of Rome "La Sapienza"
c Växjö University
d Vrije Universiteit

Abstract: Dynamic systems in non-Archimedean number fields (i. e. fields with non-Archimedean valuations) are studied. Results are obtained for the fields of $p$-adic numbers and complex $p$-adic numbers. Simple $p$-adic dynamic systems have a very rich structure–attractors, Siegel disks, cycles, and a new structure called a “fuzzy cycle”. The prime number $p$ plays the role of a parameter of the $p$-adic dynamic system. Changing $p$ radically changes the behavior of the system: attractors may become the centers of Siegel disks, and vice versa, and cycles of different lengths may appear or disappear.

DOI: https://doi.org/10.4213/tmf845

Full text: PDF file (298 kB)

English version:
Theoretical and Mathematical Physics, 1998, 114:3, 276–287

Bibliographic databases:

Received: 28.08.1997

Citation: S. A. Albeverio, B. Tirozzi, A. Yu. Khrennikov, S. de Smedt, “$p$-Adic dynamic systems”, TMF, 114:3 (1998), 349–365; Theoret. and Math. Phys., 114:3 (1998), 276–287

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. S. A. Albeverio, P. E. Kloeden, A. Yu. Khrennikov, “Human memory as a $p$-adic dynamic system”, Theoret. and Math. Phys., 117:3 (1998), 1414–1422  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Dubischar, D, “A p-adic model for the process of thinking disturbed by physiological and information noise”, Journal of Theoretical Biology, 197:4 (1999), 451  crossref  isi  scopus  scopus  scopus
    3. Khrennikov, A, “p-adic discrete dynamical systems and collective behaviour of information states in cognitive models”, Discrete Dynamics in Nature and Society, 5:1 (2000), 59  crossref  zmath  isi
    4. Khrennikov, A, “Classical and quantum dynamics on p-adic trees of ideas”, Biosystems, 56:2–3 (2000), 95  crossref  isi  scopus  scopus  scopus
    5. Gundlach, M, “On ergodic behavior of p-adic dynamical systems”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 4:4 (2001), 569  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    6. Khrennikov, A, “Small denominators in complex p-adic dynamics”, Indagationes Mathematicae-New Series, 12:2 (2001), 177  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    7. Khrennikov, A, “On the number of cycles of p-adic dynamical systems”, Journal of Number Theory, 90:2 (2001), 255  crossref  mathscinet  zmath  isi  scopus  scopus
    8. Kotovich, NV, “Representation and compression of images with the aid of m-adic coordinate systems”, Doklady Mathematics, 66:3 (2002), 330  zmath  isi
    9. Vivaldi, F, “Pseudo-randomness of round-off errors in discretized linear maps on the plane”, International Journal of Bifurcation and Chaos, 13:11 (2003), 3373  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    10. Kaneko, H, “Time-inhomogeneous stochastic processes on the p-adic number field”, Tohoku Mathematical Journal, 55:1 (2003), 65  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    11. Proc. Steklov Inst. Math., 245 (2004), 105–116  mathnet  mathscinet  zmath
    12. Proc. Steklov Inst. Math., 245 (2004), 250–257  mathnet  mathscinet  zmath
    13. Kotovich, NV, “Compression of images with the use of representations by p-adic mappings and approximations by Mahler polynomials”, Doklady Mathematics, 69:3 (2004), 373  mathscinet  zmath  isi
    14. Kaneko, H, “Capacities associated with Dirichlet space on an infinite extension of a local field”, Forum Mathematicum, 17:6 (2005), 1011  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    15. Bryk, J, “Measurable dynamics of simple p-adic polynomials”, American Mathematical Monthly, 112:3 (2005), 212  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    16. Kaneko, H, “Transition semi-groups on a local field induced by Galois group and their representation”, Journal of Theoretical Probability, 19:1 (2006), 221  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    17. Kaneko, H, “(r, p)-capacity and Hausdorff measure on a local field”, Indagationes Mathematicae-New Series, 17:2 (2006), 251  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    18. Khamraev, M, “On a class of rational p-adic dynamical systems”, Journal of Mathematical Analysis and Applications, 315:1 (2006), 76  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    19. Kaneko, H, “Sobolev space and Dirichlet space associated with symmetric Markov process on a local field”, Potential Analysis, 24:1 (2006), 87  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    20. Boeckler, F, “The structural evolution of dopamine D-3 receptor ligands: Structure-activity relationships and selected neuropharmacological aspects”, Pharmacology & Therapeutics, 112:1 (2006), 281  crossref  isi  scopus  scopus  scopus
    21. Mukhamedov, F, “On the chaotic behavior of a generalized logistic p-adic dynamical system”, Journal of Differential Equations, 243:2 (2007), 125  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    22. Dragovich, B, “Linear fractional P-ADIC and adelic dynamical systems”, Reports on Mathematical Physics, 60:1 (2007), 55  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    23. Al-Fulaij, MA, “Identification of amino acid determinants of dopamine 2 receptor synthetic agonist function”, Journal of Pharmacology and Experimental Therapeutics, 321:1 (2007), 298  crossref  isi  scopus  scopus  scopus
    24. F. M. Mukhamedov, “On the Chaotic Behavior of Cubic $p$-Adic Dynamical Systems”, Math. Notes, 83:3 (2008), 428–431  mathnet  crossref  crossref  mathscinet  zmath  isi
    25. F. M. Mukhamedov, U. A. Rozikov, “A polynomial $p$-adic dynamical system”, Theoret. and Math. Phys., 170:3 (2012), 376–383  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    26. Albeverio S., Rozikov U.A., Sattarov I.A., “P-Adic (2,1)-Rational Dynamical Systems”, J. Math. Anal. Appl., 398:2 (2013), 553–566  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    27. Khrennikov A., Yurova E., “Criteria of Measure-Preserving for P-Adic Dynamical Systems in Terms of the Van der Put Basis”, J. Number Theory, 133:2 (2013), 484–491  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    28. Mukhamedov F. Rozali Wan Nur Fairuz Alwani Wan, “On a P-Adic Cubic Generalized Logistic Dynamical System”, International Conference on Advancement in Science and Technology 2012 (Icast): Contemporary Mathematics, Mathematical Physics and their Applications, Journal of Physics Conference Series, 435, ed. Ganikhodjaev N. Mukhamedov F. Hee P., IOP Publishing Ltd, 2013  crossref  isi  scopus  scopus  scopus
    29. Anashin V., Khrennikov A., Yurova E., “Ergodicity Criteria for Non-Expanding Transformations of 2-Adic Spheres”, Discret. Contin. Dyn. Syst., 34:2 (2014), 367–377  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    30. Khrennikov A., Yurova E., “Criteria of Ergodicity For P-Adic Dynamical Systems in Terms of Coordinate Functions”, Chaos Solitons Fractals, 60 (2014), 11–30  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    31. Axelsson E.Yu., “on the Representation of the Genetic Code By the Attractors of 2-Adic Function”, Phys. Scr., T165 (2015), 014043  crossref  adsnasa  isi  scopus  scopus  scopus
    32. Rozikov U.A. Sattarov I.A., “P-Adic Dynamical Systems of (2,2)-Rational Functions With Unique Fixed Point”, Chaos Solitons Fractals, 105 (2017), 260–270  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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