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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

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 & 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, (1972). Google Scholar

[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. doi: 10.1103/PhysRevE.51.1035. Google Scholar

[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. doi: 10.1088/0957-0233/19/7/075203. Google Scholar

[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. Google Scholar

[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. doi: 10.1088/1367-2630/11/7/073012. Google Scholar

[6]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,", Springer-Verlag, (1997). Google Scholar

[7]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys., 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067. Google Scholar

[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. doi: 10.1103/PhysRevLett.91.244101. Google Scholar

[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. doi: 10.1103/PhysRevE.73.066121. Google Scholar

[10]

B. S. Kerner, "The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory,", Springer, (2004). Google Scholar

[11]

B. S. Kerner, "Introduction to Modern Traffic Flow Theory and Control. The Long Road to Three-Phase Traffic Theory,", Springer, (2009). doi: 10.1007/978-3-642-02605-8. Google Scholar

[12]

T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331. doi: 10.1088/0034-4885/65/9/203. Google Scholar

[13]

C. M. A. Pinto and M. Golubitsky, Central pattern generators for bipedal locomotion,, J. Math. Biol., 53 (2006), 474. doi: 10.1007/s00285-006-0021-2. Google Scholar

[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. doi: 10.1088/1367-2630/10/3/033001. Google Scholar

[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. doi: 10.1103/PhysRevLett.87.078102. Google Scholar

[16]

A. Takamatsu, R. Tanaka and T. Fujii, Hidden symmetry in chains of biological coupled oscillators,, Phys. Rev. Lett., 92 (2004), 228102. doi: 10.1103/PhysRevLett.92.228102. Google Scholar

[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). Google Scholar

[18]

S. Wiggins, "Introduction to Applied Dynamical Systems and Chaos,", Springer-Verlag, (1990). Google Scholar

[19]

D. E. Wolf, M. Schreckenberg and A. Bachem (ed), "Traffic and Granular Flow,", World Scientific, (1996). Google Scholar

[20]

M. Yamamoto, Y. Nomura and Y. Sugiyama, Dissipative system with asymmetric interaction and Hopf bifurcation,, Phys. Rev. E, 80 (2009), 026203. doi: 10.1103/PhysRevE.80.026203. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions,", Dover Publications, (1972). Google Scholar

[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. doi: 10.1103/PhysRevE.51.1035. Google Scholar

[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. doi: 10.1088/0957-0233/19/7/075203. Google Scholar

[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. Google Scholar

[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. doi: 10.1088/1367-2630/11/7/073012. Google Scholar

[6]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,", Springer-Verlag, (1997). Google Scholar

[7]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys., 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067. Google Scholar

[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. doi: 10.1103/PhysRevLett.91.244101. Google Scholar

[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. doi: 10.1103/PhysRevE.73.066121. Google Scholar

[10]

B. S. Kerner, "The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory,", Springer, (2004). Google Scholar

[11]

B. S. Kerner, "Introduction to Modern Traffic Flow Theory and Control. The Long Road to Three-Phase Traffic Theory,", Springer, (2009). doi: 10.1007/978-3-642-02605-8. Google Scholar

[12]

T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331. doi: 10.1088/0034-4885/65/9/203. Google Scholar

[13]

C. M. A. Pinto and M. Golubitsky, Central pattern generators for bipedal locomotion,, J. Math. Biol., 53 (2006), 474. doi: 10.1007/s00285-006-0021-2. Google Scholar

[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. doi: 10.1088/1367-2630/10/3/033001. Google Scholar

[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. doi: 10.1103/PhysRevLett.87.078102. Google Scholar

[16]

A. Takamatsu, R. Tanaka and T. Fujii, Hidden symmetry in chains of biological coupled oscillators,, Phys. Rev. Lett., 92 (2004), 228102. doi: 10.1103/PhysRevLett.92.228102. Google Scholar

[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). Google Scholar

[18]

S. Wiggins, "Introduction to Applied Dynamical Systems and Chaos,", Springer-Verlag, (1990). Google Scholar

[19]

D. E. Wolf, M. Schreckenberg and A. Bachem (ed), "Traffic and Granular Flow,", World Scientific, (1996). Google Scholar

[20]

M. Yamamoto, Y. Nomura and Y. Sugiyama, Dissipative system with asymmetric interaction and Hopf bifurcation,, Phys. Rev. E, 80 (2009), 026203. doi: 10.1103/PhysRevE.80.026203. Google Scholar

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