American Institute of Mathematical Sciences

Advanced
August  2002, 2(3): 415-431. doi: 10.3934/dcdsb.2002.2.415

Computational information for the logistic map at the chaos threshold

C. Bonanno 1, and  G. Menconi 2,

1. 

Department of Mathematics, University of Pisa, via Buonarroti, 2/a, 56127 Pisa, Italy
2. 

Centro Interdisciplinare per lo Studio dei Sistemi Complessi, University of Pisa, via Bonanno, 25/b, 56126 Pisa, Italy

Received  August 2001 Revised  November 2001 Published  May 2002

We study the logistic map $f(x)=\lambda x (1-x)$ on the unit square at the chaos threshold. By using the methods of symbolic dynamics, the information content of an orbit of a dynamical system is defined as the Algorithmic Information Content (AIC) of a symbolic sequence. We give results for the behaviour of the AIC for the logistic map. Since the AIC is not a computable function we use, as approximation of the AIC, a notion of information content given by the length of the string after it has been compressed by a compression algorithm, and in particular we introduce a new compression algorithm called CASToRe. The information content is then used to characterise the chaotic behaviour.
Keywords: compression algorithms., logistic map, Algorithmic Information Content.
Mathematics Subject Classification: 28D20, 37E05, 37M25, 60J1.
Citation: C. Bonanno, G. Menconi. Computational information for the logistic map at the chaos threshold. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 415-431. doi: 10.3934/dcdsb.2002.2.415
[1]

C. Bonanno. The algorithmic information content for randomly perturbed systems. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 921-934. doi: 10.3934/dcdsb.2004.4.921
[2]

Mahendra Piraveenan, Mikhail Prokopenko, Albert Y. Zomaya. On congruity of nodes and assortative information content in complex networks. Networks and Heterogeneous Media, 2012, 7 (3) : 441-461. doi: 10.3934/nhm.2012.7.441
[3]

Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927
[4]

Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075
[5]

H. T. Banks, John E. Banks, R. A. Everett, John D. Stark. An adaptive feedback methodology for determining information content in stable population studies. Mathematical Biosciences & Engineering, 2016, 13 (4) : 653-671. doi: 10.3934/mbe.2016013
[6]

Qingqing Ye. Algorithmic computation of MAP/PH/1 queue with finite system capacity and two-stage vacations. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2459-2477. doi: 10.3934/jimo.2019063
[7]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[8]

Lassi Roininen, Markku S. Lehtinen. Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions. Inverse Problems and Imaging, 2013, 7 (2) : 649-661. doi: 10.3934/ipi.2013.7.649
[9]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Period doubling and reducibility in the quasi-periodically forced logistic map. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1507-1535. doi: 10.3934/dcdsb.2012.17.1507
[10]

Santiago Moral, Victor Chapela, Regino Criado, Ángel Pérez, Miguel Romance. Efficient algorithms for estimating loss of information in a complex network: Applications to intentional risk analysis. Networks and Heterogeneous Media, 2015, 10 (1) : 195-208. doi: 10.3934/nhm.2015.10.195
[11]

Stefano Galatolo. Orbit complexity and data compression. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477
[12]

R. Estrada. Boundary layers and spectral content asymptotics. Conference Publications, 1998, 1998 (Special) : 242-252. doi: 10.3934/proc.1998.1998.242
[13]

Rafail Krichevskii and Vladimir Potapov. Compression and restoration of square integrable functions. Electronic Research Announcements, 1996, 2: 42-49.

[14]

Matthias Ngwa, Ephraim Agyingi. A mathematical model of the compression of a spinal disc. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1061-1083. doi: 10.3934/mbe.2011.8.1061
[15]

Philip N. J. Eagle, Steven D. Galbraith, John B. Ong. Point compression for Koblitz elliptic curves. Advances in Mathematics of Communications, 2011, 5 (1) : 1-10. doi: 10.3934/amc.2011.5.1
[16]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1673-1692. doi: 10.3934/dcdss.2020449
[17]

Ouayl Chadli, Hicham Mahdioui, Jen-Chih Yao. Bilevel mixed equilibrium problems in Banach spaces : existence and algorithmic aspects. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 549-561. doi: 10.3934/naco.2011.1.549
[18]

Lassi Roininen, Markku S. Lehtinen, Petteri Piiroinen, Ilkka I. Virtanen. Perfect radar pulse compression via unimodular fourier multipliers. Inverse Problems and Imaging, 2014, 8 (3) : 831-844. doi: 10.3934/ipi.2014.8.831
[19]

Markku Lehtinen, Baylie Damtie, Petteri Piiroinen, Mikko Orispää. Perfect and almost perfect pulse compression codes for range spread radar targets. Inverse Problems and Imaging, 2009, 3 (3) : 465-486. doi: 10.3934/ipi.2009.3.465
[20]

Richard Archibald, Hoang Tran. A dictionary learning algorithm for compression and reconstruction of streaming data in preset order. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 655-668. doi: 10.3934/dcdss.2021102

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy