Complex spatiotemporal behavior in a chain of one-way nonlinearly coupled elements

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October 2011, 4(5): 1167-1179. doi: 10.3934/dcdss.2011.4.1167

Complex spatiotemporal behavior in a chain of one-way nonlinearly coupled elements

Yuri B. Gaididei ^1, , Rainer Berkemer ^2, , Carlos Gorria ^3, , Peter L. Christiansen ^4, , Atsushi Kawamoto ^5, , Takahiro Shiga ^5, , Mads P. Sørensen ^6, and Jens Starke ^2,

1.	Bogolyubov Institute for Theoretical Physics, Metrologichna str. 14 B, 01413, Kiev, Ukraine
2.	Department of Mathematics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark, Denmark
3.	Department of Applied Mathematics and Statistics, University of the Basque Country, E-48080 Bilbao, Spain
4.	Department of Informatics and Mathematical Modeling & Department of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
5.	Toyota Central R&D Labs, Inc., Nagakute, 480-1192 Aichi, Japan, Japan
6.	Department of Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

Received September 2009 Revised January 2010 Published December 2010

Abstract
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The dynamics of asymmetrically coupled nonlinear elements is considered. It is shown that there are two distinctive regimes of oscillatory behavior of one-way nonlinearly coupled elements depending on the relaxation time and the strength of the coupling. In the subcritical regime when the relaxation time is shorter than a critical one a spatially uniform stationary state is stable. In the supercritical regime due to a Hopf bifurcation traveling waves spontaneously create and propagate along the system. Our analytical approach is in good agreement with numerical simulations of the fully nonlinear model.

Keywords: granular media, Self-organization, pattern formation. traveling waves, traffic flow..

Mathematics Subject Classification: Primary: 34K18, 35C07, 35B36; Secondary: 37C7.

Citation: Yuri B. Gaididei, Rainer Berkemer, Carlos Gorria, Peter L. Christiansen, Atsushi Kawamoto, Takahiro Shiga, Mads P. Sørensen, Jens Starke. Complex spatiotemporal behavior in a chain of one-way nonlinearly coupled elements. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1167-1179. doi: 10.3934/dcdss.2011.4.1167

References:

[1]	M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions," Dover Publications, Inc., New York, 1972.
[2]	M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042. doi: 10.1103/PhysRevE.51.1035.
[3]	A. R. Bulsara, V. In, A. Kho, A. Palacios, P. Longhini, J. D. Neff, G. Anderson, C. Obra, S. Baglio and B. Ando, Exploiting nonlinear dynamics in a coupled-core fluxgate magnetometer, Meas. Sci. Technol., 19 (2008), 075203-075221. doi: 10.1088/0957-0233/19/7/075203.
[4]	A. H. Cohen, P. J. Holmes and R. H. Rand, The nature of the coupling between segmental oscillators and the lamprey spinal generator for locomotion: A mathematical model,, J. Math. Biol., 13 (): 345. doi: 10.1007/BF00276069.
[5]	Yu. B. Gaididei, R. Berkemer, J. G. Caputo, P. L. Christiansen, A. Kawamoto, T. Shiga, M. P. Sørensen and J. Starke, Analytical solutions of jam pattern formation on a ring for a class of optimal velocity traffic models, New Journal of Phys., 11 (2009), 073012-073030. doi: 10.1088/1367-2630/11/7/073012.
[6]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," Springer-Verlag, New York, Heidelberg, Berlin, 1997.
[7]	D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.
[8]	V. In, A. Kho, J. D. Neff, A. Palacios, P. Longhini and B. K. Meadows, Experimental observation of multifrequency patterns in arrays of coupled nonlinear oscillators, Phys. Rev. Lett., 91 (2003), 244101-244104. doi: 10.1103/PhysRevLett.91.244101.
[9]	V. In, A. Palacios, P. Longhini, A. Kho, J. D. Neff, S. Baglio and B. Ando, Complex behavior in driven unidirectionally coupled overdamped Duffing oscillators, Phys. Rev. E, 73 (2006), 066121-066129. doi: 10.1103/PhysRevE.73.066121.
[10]	B. S. Kerner, "The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory," Springer, Heidelberg, 2004.
[11]	B. S. Kerner, "Introduction to Modern Traffic Flow Theory and Control. The Long Road to Three-Phase Traffic Theory," Springer, Berlin, 2009. doi: 10.1007/978-3-642-02605-8.
[12]	T. Nagatani, The physics of traffic jams, Rep. Prog. Phys., 65 (2002), 1331-1386. doi: 10.1088/0034-4885/65/9/203.
[13]	C. M. A. Pinto and M. Golubitsky, Central pattern generators for bipedal locomotion, J. Math. Biol., 53 (2006), 474-489. doi: 10.1007/s00285-006-0021-2.
[14]	Yu. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, Traffic jams without bottlenecks-experimental evidence for the physical mechanism of the formation of a jam, New Journal of Phys., 10 (2008), 033001-033007. doi: 10.1088/1367-2630/10/3/033001.
[15]	A. Takamatsu, R. Tanaka, T. Nakagaki, T. Fujii, and I. Endo, Spatiotemporal symmetry in rings of coupled biological oscillators of Physarum plasmoidal slime mold, Phys. Rev. Lett., 87 (2001), 078102-078105. doi: 10.1103/PhysRevLett.87.078102.
[16]	A. Takamatsu, R. Tanaka and T. Fujii, Hidden symmetry in chains of biological coupled oscillators, Phys. Rev. Lett., 92 (2004), 228102-228105. doi: 10.1103/PhysRevLett.92.228102.
[17]	K. van der Weele, G. Kannelopulos, C. Tsiavos and D. van der Meer, Transient granular shock waves and upstream motion on a staircase, Phys. Rev. E, 80 (2009), 011305 (16 pages).
[18]	S. Wiggins, "Introduction to Applied Dynamical Systems and Chaos," Springer-Verlag, New York, Heidelberg, Berlin, 1990.
[19]	D. E. Wolf, M. Schreckenberg and A. Bachem (ed), "Traffic and Granular Flow," World Scientific, Singapore, 1996.
[20]	M. Yamamoto, Y. Nomura and Y. Sugiyama, Dissipative system with asymmetric interaction and Hopf bifurcation, Phys. Rev. E, 80 (2009), 026203-026209. doi: 10.1103/PhysRevE.80.026203.

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References:

[1]	M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions," Dover Publications, Inc., New York, 1972.
[2]	M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042. doi: 10.1103/PhysRevE.51.1035.
[3]	A. R. Bulsara, V. In, A. Kho, A. Palacios, P. Longhini, J. D. Neff, G. Anderson, C. Obra, S. Baglio and B. Ando, Exploiting nonlinear dynamics in a coupled-core fluxgate magnetometer, Meas. Sci. Technol., 19 (2008), 075203-075221. doi: 10.1088/0957-0233/19/7/075203.
[4]	A. H. Cohen, P. J. Holmes and R. H. Rand, The nature of the coupling between segmental oscillators and the lamprey spinal generator for locomotion: A mathematical model,, J. Math. Biol., 13 (): 345. doi: 10.1007/BF00276069.
[5]	Yu. B. Gaididei, R. Berkemer, J. G. Caputo, P. L. Christiansen, A. Kawamoto, T. Shiga, M. P. Sørensen and J. Starke, Analytical solutions of jam pattern formation on a ring for a class of optimal velocity traffic models, New Journal of Phys., 11 (2009), 073012-073030. doi: 10.1088/1367-2630/11/7/073012.
[6]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," Springer-Verlag, New York, Heidelberg, Berlin, 1997.
[7]	D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.
[8]	V. In, A. Kho, J. D. Neff, A. Palacios, P. Longhini and B. K. Meadows, Experimental observation of multifrequency patterns in arrays of coupled nonlinear oscillators, Phys. Rev. Lett., 91 (2003), 244101-244104. doi: 10.1103/PhysRevLett.91.244101.
[9]	V. In, A. Palacios, P. Longhini, A. Kho, J. D. Neff, S. Baglio and B. Ando, Complex behavior in driven unidirectionally coupled overdamped Duffing oscillators, Phys. Rev. E, 73 (2006), 066121-066129. doi: 10.1103/PhysRevE.73.066121.
[10]	B. S. Kerner, "The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory," Springer, Heidelberg, 2004.
[11]	B. S. Kerner, "Introduction to Modern Traffic Flow Theory and Control. The Long Road to Three-Phase Traffic Theory," Springer, Berlin, 2009. doi: 10.1007/978-3-642-02605-8.
[12]	T. Nagatani, The physics of traffic jams, Rep. Prog. Phys., 65 (2002), 1331-1386. doi: 10.1088/0034-4885/65/9/203.
[13]	C. M. A. Pinto and M. Golubitsky, Central pattern generators for bipedal locomotion, J. Math. Biol., 53 (2006), 474-489. doi: 10.1007/s00285-006-0021-2.
[14]	Yu. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, Traffic jams without bottlenecks-experimental evidence for the physical mechanism of the formation of a jam, New Journal of Phys., 10 (2008), 033001-033007. doi: 10.1088/1367-2630/10/3/033001.
[15]	A. Takamatsu, R. Tanaka, T. Nakagaki, T. Fujii, and I. Endo, Spatiotemporal symmetry in rings of coupled biological oscillators of Physarum plasmoidal slime mold, Phys. Rev. Lett., 87 (2001), 078102-078105. doi: 10.1103/PhysRevLett.87.078102.
[16]	A. Takamatsu, R. Tanaka and T. Fujii, Hidden symmetry in chains of biological coupled oscillators, Phys. Rev. Lett., 92 (2004), 228102-228105. doi: 10.1103/PhysRevLett.92.228102.
[17]	K. van der Weele, G. Kannelopulos, C. Tsiavos and D. van der Meer, Transient granular shock waves and upstream motion on a staircase, Phys. Rev. E, 80 (2009), 011305 (16 pages).
[18]	S. Wiggins, "Introduction to Applied Dynamical Systems and Chaos," Springer-Verlag, New York, Heidelberg, Berlin, 1990.
[19]	D. E. Wolf, M. Schreckenberg and A. Bachem (ed), "Traffic and Granular Flow," World Scientific, Singapore, 1996.
[20]	M. Yamamoto, Y. Nomura and Y. Sugiyama, Dissipative system with asymmetric interaction and Hopf bifurcation, Phys. Rev. E, 80 (2009), 026203-026209. doi: 10.1103/PhysRevE.80.026203.

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