×

Chaotic synchronization in a type of coupled lattice maps. (English) Zbl 1475.37047

Summary: In this paper we explore the dynamics of a class of maps defined on a coupled lattice. We make a special emphasis on whether the system synchronizes and when such synchronization is produced in a chaotic way. We use circulant matrices for computing the tangential and normal Lyapunov exponents on an invariant set of the system.

MSC:

37E25 Dynamical systems involving maps of trees and graphs
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37B40 Topological entropy
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adler, R. L.; Konheim, A. G.; McAndrew, M. H., Topological entropy, Trans Am Math Soc, 114, 309-319 (1965) · Zbl 0127.13102
[2] Alsedà, L.; Llibre, J.; Misiurewicz, M., Combinatorial dynamics and entropy in dimension one, Advances series in nonlinear dynamics, vol. 5 (1993), World Scientific Publishing Co. Inc: World Scientific Publishing Co. Inc River Edge, NJ · Zbl 0843.58034
[3] Ashwin, P.; Buescu, J.; Stewart, I., From attractor to chaotic saddle: a tale of transverse instability, Nonlinearity, 9, 703-737 (1996) · Zbl 0887.58034
[4] Balibrea, F.; Cánovas, J. S.; Linero, A., Minimal sets of antitriangular maps, Int J Bifur Chaos Appl Sci Eng, 13, 1733-1741 (2003) · Zbl 1056.37005
[5] Balibrea, F.; Cánovas, J. S.; Linero, A., \(ω\)-limit sets of antitriangular maps, Topol Appl, 137, 13-20 (2004) · Zbl 1042.54026
[6] Balibrea, F.; López, V. J., The measure of scrambled sets: a survey, Acta Univ M Belii Ser Math, 7, 3-11 (1999) · Zbl 0967.37021
[7] Block, L. S.; Coppel, W. A., Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513 (1992), Springer-Verlag: Springer-Verlag Berlin · Zbl 0746.58007
[8] Block, L.; Keesling, J.; Li, S. H.; Peterson, K., An improved algorithm for computing topological entropy, J Stat Phys, 55, 929-939 (1989) · Zbl 0714.54018
[9] Davis, P. J., Circulant matrices (1979), John Wiley & Sons: John Wiley & Sons New York · Zbl 0418.15017
[10] Guirao, J. L.G.; Lampart, M., Positive entropy of a coupled lattice system related with Belusov-Zhabotinskii reaction, J Math Chem, 48, 66-71 (2010) · Zbl 1196.92068
[11] Guirao, J. L.G.; Lampart, M., Chaos of a coupled lattice system related with Belusov-Zhabotinskii reaction, J Math Chem, 48, 159-164 (2010) · Zbl 1196.92019
[12] Graczyck, J.; Śviatek, G., Smooth unimodal maps in the 1990s, Ergodic Theory Dyn Syst, 19, 263-287 (1999) · Zbl 0941.37024
[13] Guckhenheimer, J., Sensitive dependence to initial conditions for one-dimensional maps, Comm Math Phys, 70, 133-160 (1979) · Zbl 0429.58012
[14] Kaneko, K., Period-doubling of kink-antikink patterns, quasiperiodicity and antiferro-like structures and spatial intermittency in coupled logistic lattice, Progr Theor Phys, 72, 3, 480-486 (1984) · Zbl 1074.37521
[15] Kaneko, K., Globally coupled chaos violates law of large numbers, but not the central-limit theorem, Phys Rev Lett, 65, 1391-1394 (1990) · Zbl 1050.82550
[16] Kuznetsov, Y. A., Elements of applied bifurcation theory, Applied Mathematical Sciences, vol. 112 (2004), Springer-Verlag: Springer-Verlag New York · Zbl 1082.37002
[17] Li, T. Y.; Yorke, J. A., Period three implies chaos, Amer Math Mon, 82, 985-992 (1975) · Zbl 0351.92021
[18] Li, S., \(ω\)-chaos and topological entropy, Trans Amer Math Soc, 339, 243-249 (1993) · Zbl 0812.54046
[19] Li, R.; Wang, J.; Lu, T.; Jiang, R., Remark on topological entropy and \(P\)-chaos of a coupled lattice system with non-zero coupling constant related with Belusov-Zhabotinskii reaction, J Math Chem, 54, 1110-1116 (2016) · Zbl 1345.37012
[20] Liu, J.; Lu, T.; Li, R., Topological entropy and \(P\)-chaos of a coupled lattice system with non-zero coupling constant related with Belousov-Zhabotinsky reaction, J Math Chem, 53, 1220-1226 (2015) · Zbl 1318.54019
[21] Martens, M.; de Melo, W.; van Strien, S., Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math, 168, 273-318 (1992) · Zbl 0761.58007
[22] Milnor, J., On the concept of attractor, Commun Math Phys, 99, 177-195 (1985) · Zbl 0595.58028
[23] Oseledets, V. I., A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans Moscow Math Soc, 19, 197-231 (1968) · Zbl 0236.93034
[24] Schweizer, B.; Smítal, J., Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans Am Math Soc, 344, 737-754 (1994) · Zbl 0812.58062
[25] Singer, D., Stable orbits and bifurcation of maps of the interval, SIAM J App Math, 35, 260-267 (1978) · Zbl 0391.58014
[26] Thunberg, H., Periodicity versus chaos in one-dimensional dynamics, SIAM Rev, 43, 3-30 (2001) · Zbl 1049.37027
[27] Wu, X.; Zhu, P., Li-Yorke chaos in a coupled lattice system related with Belusov-Zhabotinskii reaction, J Math Chem, 50, 1304-1308 (2012) · Zbl 1401.92236
[28] Wu, X.; Zhu, P., The principal measure and distributional (p, q)-chaos of a coupled lattice system related with Belusov-Zhabotinskii reaction, J Math Chem, 50, 2439-2445 (2012) · Zbl 1307.37018
[29] Yamada, T.; Fujisaka, H., Stability theory of synchronized motion in coupled-oscillator systems. II The mapping approach, Progr Theor Phys, 70, 5, 1240-1248 (1983) · Zbl 1171.70307
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.