p-adic dynamics

Thiran, E.; Verstegen, D.; Weyers, J.

doi:10.1007/bf01019780

Cited by 67 publications

(53 citation statements)

References 8 publications

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“…Actually, motivated by various "strange" phenomena of chaos observed on digital computers and in numerical simulations, pathologies of digital chaotic systems have been observed and extensively studied in the field of chaos theory [Arrowsmith & Vivaldi, 1994;Beck & Roepstorff, 1987;Benettin et al, 1978;Binder, 1992;Binder & Jensen, 1986;Blank, 1994Blank, , 1997Borcherds & McCauley, 1993;Bosioand & Vivaldi, 2000;Chambers, 1999;Chirkikov & Vivaldi, 1999;Diamond et al, 1994Diamond et al, , 1995Earn & Tremaine, 1992;Fryska & Zohdy, 1992;Góra & Boyarsku, 1988;Grebogi et al, 1988;Hogg & Huberman, 1985;Huberman, 1986;Kaneko, 1988;Karney, 1983;Keating, 1991;Levy, 1982;Li et al, 2001a;Lowenstein & Vivaldi, 1998;Masuda & Aihara, 2002b;McCauley & Palmore, 1986;Palmore & Herring, 1990;Palmore & McCauley, 1987;Percival & Vivaldi, 1987;Pokrovskii et al, 1999;Rannou, 1974;Thiran et al, 1989;Čermák, 1996;Vivaldi, 1994;Waelbroeck & Zertuche, 1999;Zhang & Vivaldi, 1998]. To show how such dynamical degradation occurs, assume that the discretized space has ...…”

Section: Theoretical Work: Dynamical Degradation Of Digital Chaotic Smentioning

confidence: 99%

“…Considering the significance of numerical experiments in the study of chaos theory, many efforts have been made to answer this question [Beck & Roepstorff, 1987;Binder, 1992;Binder & Jensen, 1986;Chambers, 1999;Chirkikov & Vivaldi, 1999;Earn & Tremaine, 1992;Góra & Boyarsku, 1988;Grebogi et al, 1988;Huberman, 1986;Kaneko, 1988;Karney, 1983;Levy, 1982;Rannou, 1974;Vivaldi, 1994;Zhang & Vivaldi, 1998]. Some special techniques have been developed to facilitate theoretical analysis, such as tree structures proposed in [Hogg & Huberman, 1985] and number theory based (and/or algebra based) tools developed in [Arrowsmith & Vivaldi, 1994;Bosioand & Vivaldi, 2000;Keating, 1991; Lowenstein Percival & Vivaldi, 1987;Thiran et al, 1989]. However, till now the use of these tools are limited, since they are mainly useful for chaotic systems discretized in special (Type-II) forms, such as p-adic maps and 2-D Hamilton maps.…”

Section: Long-term Dynamics: Unavoidable Periodic Pseudo-orbitsmentioning

confidence: 99%

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On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

Chen

Mou

2005

Int. J. Bifurcation Chaos

358

189

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When chaotic systems are realized with finite precisions in digital computers, their dynamical properties are often found to be entirely different from the original versions in the continuous setting. In the literature, there does not seem to be much work on quantitative analysis of such degradation of digitized chaos and how to reduce its negative influence on chaos-based digital systems. Focusing on 1D piecewise linear chaotic maps (PWLCM), this paper reports some findings on a new series of dynamical indicators, which can quantitatively reflect the degradation effects on a digital PWLCM realized with a fixed-point finite precision. On top of that, the paper introduces a new method for studying digital chaos from an algorithmic point of view. In addition, the theoretical results obtained in this paper should be very helpful for the consideration of reducing negative influence of dynamical degradation in real design of various digital chaotic systems. As typical examples, the proposed dynamical indicators are applied to the performance comparison of different remedies for improving dynamical degradation, cryptanalysis of digital chaotic ciphers based on 1D PWLCM, and the design of chaotic pseudo-random number generators with desired characteristics.

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Section: Theoretical Work: Dynamical Degradation Of Digital Chaotic Smentioning

confidence: 99%

Section: Long-term Dynamics: Unavoidable Periodic Pseudo-orbitsmentioning

confidence: 99%

On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

Chen

Mou

2005

Int. J. Bifurcation Chaos

358

189

View full text Add to dashboard Cite

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“…For somewhat related work on extending notions of classical mechanics to more general algebraic settings see, for example [7,10]. For related work in discrete dynamical systems see [2,12] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

An algebraic approach to discrete mechanics

Baez

Gilliam

1994

Lett Math Phys

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Using basic ideas from algebraic geometry, we extend the methods of Lagrangian and symplectic mechanics to treat a large class of discrete mechanical systems, that is, systems such as cellular automata in which time proceeds in integer steps and the configuration space is discrete. In particular, we derive an analog of the Euler-Lagrange equation from a variational principle, and prove an analog of Noether's theorem. We also construct a symplectic structure on the analog of the phase space, and prove that it is preserved by time evolution.

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“…These investigations show that the study of perturbated dynamical systems is important (see [16]). Note that for a quadratic function f (x) = x 2 + c, c ∈ Q p its chaotic behavior is complicated (see [41,4,40,46]). In [40,14] the Fatou and Julia sets of logistic p-adic dynamical system, i.e.…”

Section: Introductionmentioning

confidence: 99%

On ap-adic Cubic Generalized Logistic Dynamical System

Mukhamedov

Rozali

2013

J. Phys.: Conf. Ser.

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Abstract. Applications of p-adic numbers mathematical physics, quantum mechanics stimulated increasing interest in the study of p-adic dynamical system. One of the interesting investigations is p-adic logistics map. In this paper, we consider a new generalization, namely we study a dynamical system of the form fa(x) = ax(1 − x 2 ). The paper is devoted to the investigation of a trajectory of the given system. We investigate the generalized logistic dynamical system with respect to parameter a and we restrict ourselves for the investigation of the case |a|p < 1. We study the existence of the fixed points and their behavior. Moreover, we describe their size of attractors and Siegel discs since the structure of the orbits of the system is related to the geometry of the p-adic Siegel discs. IntroductionOver the last century, p-adic numbers and p-adic analysis have come to play major role in the number theory. P -adic numbers were first introduced by K. Hensel. During a century after their discovery they were considered mainly objects of pure mathematics. Starting from 1980's various models described in the language of p-adic analysis have been actively studied. P -adic numbers have been widely used in many applications mostly in p-adic mathematical physics, quantum mechanics and others which roses the interest in the study of p-adic dynamical systems ( see for example, [6,18,42,44,45]).On the other hand, the study of p-adic dynamical systems arises in Diophantine geometry in the constructions of canonical heights, used for counting rational points on algebraic vertices over a number field, as in [12]. In [24] the p-adic field have arisen in physics in the theory of superstrings, promoting questions about their dynamics. Also some applications of p-adic dynamical systems to some biological, physical systems were proposed in [8,3,4,15,25]. In [9],[27] dynamical systems (not only monomial) over finite field extensions of the p-adic numbers were considered. Other studies of non-Archimedean dynamics in the neighborhood of a periodic and of the counting of periodic points over global fields using local fields appeared in [26,20,29,28,36]. Certain rational p-adic dynamical systems were investigated in [22], [31],[33], which appear from problems of p-adic Gibbs measures [23,32,34,35]. Note that in [37,10,11] a general theory of p-adic rational dynamical systems over complex p-adic filed C p has been developed.The most studied discrete p-adic dynamical systems (iterations of maps) are so-called monomial systems. In [5], [25] the behavior of a p-adic dynamical system f (x) = x n in the fields of p-adic numbers Q p and C p was investigated. In [25] perturbated monomial dynamical systems defined by functions f q (x) = x n + q(x), where the perturbation q(x) is a polynomial whose coefficients have small p-adic absolute value, have been studied. It was investigated the connection between monomial and perturbated monomial systems. These investigations show that the study of perturbated dynamical systems is important (see [16]). No...

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p-adic dynamics

Cited by 67 publications

References 8 publications

On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

An algebraic approach to discrete mechanics

On ap-adic Cubic Generalized Logistic Dynamical System

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