American Institute of Mathematical Sciences

Advanced
November  2003, 9(6): 1607-1624. doi: 10.3934/dcds.2003.9.1607

Global and local complexity in weakly chaotic dynamical systems

Stefano Galatolo 1,

1. 

Dipartimento di Matematica Applicata, Università di Pisa, Via Bonanno Pisano, Italy

Received  July 2002 Revised  June 2003 Published  September 2003

The generalized complexity of an orbit of a dynamical system is defined by the asymptotic behavior of the information that is necessary to describe $n$ steps of the orbit as $n$ increases. This local complexity indicator is also invariant up to topological conjugation and is suited for the study of $0$-entropy dynamical systems. First, we state a criterion to find systems with "non trivial" orbit complexity. Then, we consider also a global indicator of the complexity of the system. This global indicator generalizes the topological entropy, having non trivial values for systems were the number of essentially different orbits increases less than exponentially. Then we prove that if the system is constructive ( if the map can be defined up to any given accuracy by some algorithm) the orbit complexity is everywhere less or equal than the generalized topological entropy. Conversely there are compact non constructive examples where the inequality is reversed, suggesting that the notion of constructive map comes out naturally in this kind of complexity questions.
Keywords: generalized topological entropy, computable analisys., Weak chaos, generalized orbit complexity.
Mathematics Subject Classification: 37B99, 37B40, 37A3.
Citation: Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607
[1]

Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2020, 14 (3) : 413-422. doi: 10.3934/amc.2020063
[2]

Luis Barreira, Liviu Horia Popescu, Claudia Valls. Generalized exponential behavior and topological equivalence. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3023-3042. doi: 10.3934/dcdsb.2017161
[3]

Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210
[4]

Stefano Galatolo. Orbit complexity and data compression. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477
[5]

Alina Ostafe, Igor E. Shparlinski, Arne Winterhof. On the generalized joint linear complexity profile of a class of nonlinear pseudorandom multisequences. Advances in Mathematics of Communications, 2010, 4 (3) : 369-379. doi: 10.3934/amc.2010.4.369
[6]

Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure and Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465
[7]

Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127
[8]

Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451
[9]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[10]

Youngae Lee. Non-topological solutions in a generalized Chern-Simons model on torus. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1315-1330. doi: 10.3934/cpaa.2017064
[11]

Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani. Entropy and chaos in the Kac model. Kinetic and Related Models, 2010, 3 (1) : 85-122. doi: 10.3934/krm.2010.3.85
[12]

Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353
[13]

Lin Yi, Xiangyong Zeng, Zhimin Sun, Shasha Zhang. On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021019
[14]

Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221
[15]

Valentin Afraimovich, Maurice Courbage, Lev Glebsky. Directional complexity and entropy for lift mappings. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3385-3401. doi: 10.3934/dcdsb.2015.20.3385
[16]

Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial and Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621
[17]

Mahmut Çalik, Marcel Oliver. Weak solutions for generalized large-scale semigeostrophic equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 939-955. doi: 10.3934/cpaa.2013.12.939
[18]

Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure and Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45
[19]

Chien-Hong Cho, Marcus Wunsch. Global weak solutions to the generalized Proudman-Johnson equation. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1387-1396. doi: 10.3934/cpaa.2012.11.1387
[20]

Do Sang Kim, Nguyen Ngoc Hai, Bui Van Dinh. Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 63-78. doi: 10.3934/naco.2021051

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (112)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy