American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations
  • DCDS-S Home
  • This Issue
  • Next Article
    Nonlinear lattice models for biopolymers: Dynamical coupling to a ionic cloud and application to actin filaments
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

[1]

Razvan C. Fetecau, Beril Zhang. Self-organization on Riemannian manifolds. Journal of Geometric Mechanics, 2019, 11 (3) : 397-426. doi: 10.3934/jgm.2019020
[2]

Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161
[3]

Wen Shen, Karim Shikh-Khalil. Traveling waves for a microscopic model of traffic flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2571-2589. doi: 10.3934/dcds.2018108
[4]

Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231
[5]

Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028
[6]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395
[7]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339
[8]

Scott Gordon. Nonuniformity of deformation preceding shear band formation in a two-dimensional model for Granular flow. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1361-1374. doi: 10.3934/cpaa.2008.7.1361
[9]

Michael Shearer, Nicholas Giffen. Shock formation and breaking in granular avalanches. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 693-714. doi: 10.3934/dcds.2010.27.693
[10]

Dong Li, Tong Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2011, 6 (4) : 681-694. doi: 10.3934/nhm.2011.6.681
[11]

Guillaume Bal, Olivier Pinaud. Self-averaging of kinetic models for waves in random media. Kinetic & Related Models, 2008, 1 (1) : 85-100. doi: 10.3934/krm.2008.1.85
[12]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
[13]

Yongki Lee, Hailiang Liu. Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 323-339. doi: 10.3934/dcds.2015.35.323
[14]

Wenzhang Huang. Weakly coupled traveling waves for a model of growth and competition in a flow reactor. Mathematical Biosciences & Engineering, 2006, 3 (1) : 79-87. doi: 10.3934/mbe.2006.3.79
[15]

Martial Agueh, Guillaume Carlier, Reinhard Illner. Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds. Kinetic & Related Models, 2015, 8 (2) : 201-214. doi: 10.3934/krm.2015.8.201
[16]

Wen Shen. Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition. Networks & Heterogeneous Media, 2018, 13 (3) : 449-478. doi: 10.3934/nhm.2018020
[17]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[18]

Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555
[19]

Taylan Sengul, Shouhong Wang. Pattern formation and dynamic transition for magnetohydrodynamic convection. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2609-2639. doi: 10.3934/cpaa.2014.13.2609
[20]

Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences