Correlation integral and determinism for a family of 2^\infty maps

Previous Article
Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions
DCDS Home
This Issue
Next Article
Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement

September 2016, 36(9): 5067-5096. doi: 10.3934/dcds.2016020

Correlation integral and determinism for a family of $2^\infty$ maps

1.	Slovanet a.s., Záhradnícka 151, 821 08 Bratislava, Slovak Republic

Received June 2015 Revised March 2016 Published May 2016

Abstract
Related Papers

The correlation integral and determinism are quantitative characteristics of a dynamical system based on the recurrence of orbits. For strongly non-chaotic interval maps, the determinism equals $1$ for every small enough threshold. This means that trajectories of such systems are perfectly predictable in the infinite horizon. In this paper we study the correlation integral and determinism for the family of $2^\infty$ non-chaotic maps, first considered by Delahaye in 1980. The determinism in a finite horizon equals $1$. However, the behaviour of the determinism in the infinite horizon is counter-intuitive. Sharp bounds on the determinism are provided.

Keywords: odometer, Correlation integral, predictability, distance matrix., asymptotic determinism.

Mathematics Subject Classification: Primary: 37E05, 37E15; Secondary: 37M2.

Citation: Jana Majerová. Correlation integral and determinism for a family of $2^\infty$ maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5067-5096. doi: 10.3934/dcds.2016020

References:

[1]	L. S. Block and W. A. Coppel, Dynamics in One Dimension,, Springer-Verlag, (1992). Google Scholar
[2]	L. S. Block and J. Keesling, A characterization of adding machine maps,, Topology Appl., 140 (2004), 151. doi: 10.1016/j.topol.2003.07.006. Google Scholar
[3]	J. P. Boroński and P. Oprocha, On indecomposability in chaotic attractors,, Proc. Amer. Math. Soc., 143 (2015), 3659. doi: 10.1090/S0002-9939-2015-12526-9. Google Scholar
[4]	P. Collas and D. Klein, An ergodic adding machine on the Cantor set,, Enseign. Math. (2), 40 (1994), 249. Google Scholar
[5]	J.-P. Delahaye, Fonctions admettant des cycles d'ordre n'importe quelle puissance de $2$ et aucun autre cycle,, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980). Google Scholar
[6]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, $2^{nd}$ edition, (1989). Google Scholar
[7]	M. Grendár, J. Majerová and V. Špitalský, Strong laws for recurrence quantification analysis,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013). doi: 10.1142/S0218127413501472. Google Scholar
[8]	P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors,, Phys. D, 9 (1983), 189. doi: 10.1016/0167-2789(83)90298-1. Google Scholar
[9]	R. Hric, Topological sequence entropy for maps of the interval,, Proc. Amer. Math. Soc., 127 (1999), 2045. doi: 10.1090/S0002-9939-99-04799-1. Google Scholar
[10]	H. Kantz and T. Schreiber, Nonlinear Time Series Analysis,, $2^{nd}$ edition, (2004). Google Scholar
[11]	M. Misiurewicz, Invariant measures for continuous transformations of $[0,1]$ with zero topological entropy,, in Ergodic theory (Proc. Conf., 729* (1979), 144. Google Scholar*
[12]	A. Manning and K. Simon, A short existence proof for correlation dimension,, J. Statist. Phys., 90 (1998), 1047. doi: 10.1023/A:1023253709865. Google Scholar
[13]	Ya. B. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions,, J. Statist. Phys., 71 (1993), 529. doi: 10.1007/BF01058436. Google Scholar
[14]	Ya. B. Pesin and A. Tempelman, Correlation dimension of measures invariant under group actions,, Random Comput. Dynam., 3 (1995), 137. Google Scholar
[15]	S. Ruette, Chaos for continuous interval maps,, 2003. Available from: , (). Google Scholar
[16]	J. Smítal, Chaotic functions with zero topological entropy,, Trans. Amer. Math. Soc., 297 (1986), 269. doi: 10.1090/S0002-9947-1986-0849479-9. Google Scholar
[17]	J. P. Zbilut and C. L. Webber Jr., Embeddings and delays as derived from quantification of recurrence plots,, Physics Letters A, 171 (1992), 199. doi: 10.1016/0375-9601(92)90426-M. Google Scholar

show all references

References:

[1]	L. S. Block and W. A. Coppel, Dynamics in One Dimension,, Springer-Verlag, (1992). Google Scholar
[2]	L. S. Block and J. Keesling, A characterization of adding machine maps,, Topology Appl., 140 (2004), 151. doi: 10.1016/j.topol.2003.07.006. Google Scholar
[3]	J. P. Boroński and P. Oprocha, On indecomposability in chaotic attractors,, Proc. Amer. Math. Soc., 143 (2015), 3659. doi: 10.1090/S0002-9939-2015-12526-9. Google Scholar
[4]	P. Collas and D. Klein, An ergodic adding machine on the Cantor set,, Enseign. Math. (2), 40 (1994), 249. Google Scholar
[5]	J.-P. Delahaye, Fonctions admettant des cycles d'ordre n'importe quelle puissance de $2$ et aucun autre cycle,, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980). Google Scholar
[6]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, $2^{nd}$ edition, (1989). Google Scholar
[7]	M. Grendár, J. Majerová and V. Špitalský, Strong laws for recurrence quantification analysis,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013). doi: 10.1142/S0218127413501472. Google Scholar
[8]	P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors,, Phys. D, 9 (1983), 189. doi: 10.1016/0167-2789(83)90298-1. Google Scholar
[9]	R. Hric, Topological sequence entropy for maps of the interval,, Proc. Amer. Math. Soc., 127 (1999), 2045. doi: 10.1090/S0002-9939-99-04799-1. Google Scholar
[10]	H. Kantz and T. Schreiber, Nonlinear Time Series Analysis,, $2^{nd}$ edition, (2004). Google Scholar
[11]	M. Misiurewicz, Invariant measures for continuous transformations of $[0,1]$ with zero topological entropy,, in Ergodic theory (Proc. Conf., 729* (1979), 144. Google Scholar*
[12]	A. Manning and K. Simon, A short existence proof for correlation dimension,, J. Statist. Phys., 90 (1998), 1047. doi: 10.1023/A:1023253709865. Google Scholar
[13]	Ya. B. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions,, J. Statist. Phys., 71 (1993), 529. doi: 10.1007/BF01058436. Google Scholar
[14]	Ya. B. Pesin and A. Tempelman, Correlation dimension of measures invariant under group actions,, Random Comput. Dynam., 3 (1995), 137. Google Scholar
[15]	S. Ruette, Chaos for continuous interval maps,, 2003. Available from: , (). Google Scholar
[16]	J. Smítal, Chaotic functions with zero topological entropy,, Trans. Amer. Math. Soc., 297 (1986), 269. doi: 10.1090/S0002-9947-1986-0849479-9. Google Scholar
[17]	J. P. Zbilut and C. L. Webber Jr., Embeddings and delays as derived from quantification of recurrence plots,, Physics Letters A, 171 (1992), 199. doi: 10.1016/0375-9601(92)90426-M. Google Scholar

[1]	Houduo Qi, ZHonghang Xia, Guangming Xing. An application of the nearest correlation matrix on web document classification. Journal of Industrial & Management Optimization, 2007, 3 (4) : 701-713. doi: 10.3934/jimo.2007.3.701
[2]	Joshua Du, Jun Ji. An integral representation of the determinant of a matrix and its applications. Conference Publications, 2005, 2005 (Special) : 225-232. doi: 10.3934/proc.2005.2005.225
[3]	Yutian Lei, Chao Ma. Asymptotic behavior for solutions of some integral equations. Communications on Pure & Applied Analysis, 2011, 10 (1) : 193-207. doi: 10.3934/cpaa.2011.10.193
[4]	Josef Diblík, Zdeněk Svoboda. Asymptotic properties of delayed matrix exponential functions via Lambert function. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 123-144. doi: 10.3934/dcdsb.2018008
[5]	Jiongxuan Zheng, Joseph D. Skufca, Erik M. Bollt. Heart rate variability as determinism with jump stochastic parameters. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1253-1264. doi: 10.3934/mbe.2013.10.1253
[6]	Lassi Roininen, Markku S. Lehtinen, Sari Lasanen, Mikko Orispää, Markku Markkanen. Correlation priors. Inverse Problems & Imaging, 2011, 5 (1) : 167-184. doi: 10.3934/ipi.2011.5.167
[7]	Jiyoung Han, Seonhee Lim, Keivan Mallahi-Karai. Asymptotic distribution of values of isotropic here quadratic forms at S-integral points. Journal of Modern Dynamics, 2017, 11: 501-550. doi: 10.3934/jmd.2017020
[8]	José M. Arrieta, Esperanza Santamaría. Estimates on the distance of inertial manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 3921-3944. doi: 10.3934/dcds.2014.34.3921
[9]	Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565
[10]	Ming Su, Arne Winterhof. Hamming correlation of higher order. Advances in Mathematics of Communications, 2018, 12 (3) : 505-513. doi: 10.3934/amc.2018029
[11]	Vladimír Špitalský. Local correlation entropy. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5711-5733. doi: 10.3934/dcds.2018249
[12]	Nian Li, Xiaohu Tang, Tor Helleseth. A class of quaternary sequences with low correlation. Advances in Mathematics of Communications, 2015, 9 (2) : 199-210. doi: 10.3934/amc.2015.9.199
[13]	Xin-Guo Liu, Kun Wang. A multigrid method for the maximal correlation problem. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 785-796. doi: 10.3934/naco.2012.2.785
[14]	Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems & Imaging, 2018, 12 (3) : 697-731. doi: 10.3934/ipi.2018030
[15]	Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009
[16]	Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014
[17]	Meijuan Shang, Yanan Liu, Lingchen Kong, Xianchao Xiu, Ying Yang. Nonconvex mixed matrix minimization. Mathematical Foundations of Computing, 2019, 2 (2) : 107-126. doi: 10.3934/mfc.2019009
[18]	Konstantinos Drakakis, Roderick Gow, Scott Rickard. Common distance vectors between Costas arrays. Advances in Mathematics of Communications, 2009, 3 (1) : 35-52. doi: 10.3934/amc.2009.3.35
[19]	Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139
[20]	Yujuan Li, Guizhen Zhu. On the error distance of extended Reed-Solomon codes. Advances in Mathematics of Communications, 2016, 10 (2) : 413-427. doi: 10.3934/amc.2016015

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences