Abstract
Three quantitative measures of the spatiotemporal behavior of the coupled map lattices: reduced density matrix, reduced wave function, and an analog of particle number, have been introduced. Making extensive use of two computer algebra systems (Maxima and Mathematica) various properties of the above mentioned parameters have been thoroughly studied. Their behavior suggests that the logistic coupled-map lattices approach the states which resemble the condensed states of systems of Bose particles. In addition, pattern formation in two-dimensional coupled map lattices based on the logistic mapping has been investigated with respect to the non-linear parameter, the diffusion constant and initial as well as boundary conditions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bianconi, G., Barabasi, A.-L.: Bose–Einstein condensation in complex networks. Phys. Rev. Lett. 86, 5632–5635 (2001)
Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999)
Chazottes, J.R., Fernandez, B.: Dynamics of Coupled Map Lattices and Related Spatially Extended Systems. Springer, New York (2005)
Ghoshdastidar, P.S., Chakraborty, I.: A coupled map lattice model of flow boiling in a horizontal tube. J. Heat Transfer 129, 1737–1741 (2007)
Góral, K., Gajda, M., Rzążewski, K.: Multi-mode description of an interacting Bose–Einstein condensate. Opt. Express 8, 92–98 (2001)
Góral, K., Gajda, M., Rzążewski, K.: Thermodynamics of an interacting trapped Bose–Einstein gas in the classical field approximation. Phys. Rev. A 66, 051602(R) (4 pages) (2002)
Hale, J., Koçak, H.: Dynamics and Bifurcations. Springer, Berlin (1991)
Ilachinski, A.: Cellular Automata. A Discrete Universe. World Scientific, Singapore (2001)
Janosi, I.M., Flepp, L., Tel, T.: Exploring transient chaos in an NMR-laser experiment. Phys. Rev. Lett. 73, 529–532 (1994)
Janowicz, M., Orłowski, A.: Coherence properties of coupled chaotic map lattices. Acta Phys. Polon. A 120, A-114–A-118 (2011)
Kadio, D., Gajda, M., Rzążewski, K.: Phase fluctuations of a Bose–Einstein condensate in low-dimensional geometry. Phys. Rev. A 72, 013607 (9 pages) (2005)
Kaneko, K.: Period-doubling of kink-antikink patterns, quasi-periodicity in antiferro-like structures and spatial intermittency in coupled map lattices – Toward a prelude to a “Field Theory of Chaos”. Prog. Theor. Phys. 72, 480–486 (1984)
Kaneko, K.: Pattern dynamics in spatiotemporal chaos. Physica D 34, 1–41 (1989)
Kaneko, K.: Simulating physics with coupled map lattices – Pattern dynamics, information flow, and thermodynamics of spatiotemporal chaos. In: Kawasaki, K., Onuki, A., Suzuki, M. (eds.) Pattern Dynamics, Information Flow, and Thermodynamics of Spatiotemporal Chaos, pp. 1–52. World Scientific, Singapore (1990)
Kapral, R.: Pattern formation in two-dimensional arrays of coupled, discrete-time Oscillators. Phys. Rev. A 31, 3868–3879 (1985)
Leggett, A.: Bose–Einstein condensation in the alkali gases: some fundamental concepts. Rev. Mod. Phys. 73, 307–356 (2001)
Muruganandam, P., Francisco, F., de Menezes, M., Ferreira, F.F.: Low dimensional behavior in three-dimensional coupled map lattices. Chaos, Solitons and Fractals 41, 997–1004 (2009)
Penrose, O., Onsager, L.: Bose–Einstein condensation and liquid helium. Phys. Rev. 104, 576–584 (1956)
Petrov, D.S., Shlyapnikov, G.V., Walraven, J.T.M.: Regimes of quantum degeneracy in trapped 1D gases. Phys. Rev. Lett. 85, 3745–3749 (2000)
Petrov, D.S., Shlyapnikov, G.V., Walraven, J.T.M.: Phase-fluctuating 3D Bose–Einstein condensates in elongated traps. Phys. Rev. Lett. 87, 050404 (4 pages) (2001)
Reka, A., Barabasi, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)
Schmidt, H., Góral, K., Floegel, F., Gajda, M., Rzążewski, K.: Probing the classical field approximation – thermodynamics and decaying vortices. J. Opt. B: Quantum Semiclassical Opt. 5, S96 (2003)
Sinha, S.: Transient 1/f Noise. Phys. Rev. E 53, 4509–4513 (1996)
Waller, I., Kapral, R.: Spatial and temporal structure in systems of coupled nonlinear oscillators. Phys. Rev. A 30, 2047–2055 (1984)
Yanagita, T.: Coupled map lattice model for boiling. Phys. Lett. A 165, 405–408 (1992)
Yanagita, T., Kaneko, K.: Rayleigh–Benard convection: Pattern, chaos, spatiotemporal chaos and turbulence. Physica D 82, 288–313 (1995)
Yanagita, T., Kaneko, K.: Modeling and characterization of cloud dynamics. Phys. Rev. Lett. 78, 4297–4300 (1997)
Yang, C.N.: Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors. Rev. Mod. Phys. 34, 694–704 (1962)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Janowicz, M., Orłowski, A. (2014). Coherence and Large-Scale Pattern Formation in Coupled Logistic-Map Lattices via Computer Algebra Systems. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-10515-4_17
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10514-7
Online ISBN: 978-3-319-10515-4
eBook Packages: Computer ScienceComputer Science (R0)