Geodesic
Application of a center manifold theory to a reaction-diffusion system of collective motion of camphor disks and boats
Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 363-371

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MR Zbl
Unidirectional motion along an annular water channel can be observed in an experiment even with only one camphor disk or boat. Moreover, the collective motion of camphor disks or boats in the water channel exhibits a homogeneous and an inhomogeneous state, depending on the number of disks or boats, which looks like a kind of bifurcation phenomena. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Hence it suffices to investigate a linearized eigenvalue problem in order to prove the destabilization of a traveling wave solution. However, the eigenvalue problem is too difficult to analyze even if the number of camphor disks or boats is 2. Hence we need to make a reduction on the model. In the present paper, we apply the center manifold theory and reduce the model to an ordinary differential system.
Unidirectional motion along an annular water channel can be observed in an experiment even with only one camphor disk or boat. Moreover, the collective motion of camphor disks or boats in the water channel exhibits a homogeneous and an inhomogeneous state, depending on the number of disks or boats, which looks like a kind of bifurcation phenomena. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Hence it suffices to investigate a linearized eigenvalue problem in order to prove the destabilization of a traveling wave solution. However, the eigenvalue problem is too difficult to analyze even if the number of camphor disks or boats is 2. Hence we need to make a reduction on the model. In the present paper, we apply the center manifold theory and reduce the model to an ordinary differential system.
DOI : 10.21136/MB.2014.143861
Classification : 34K18, 35C07, 35K57, 37L10, 70K50, 76Z10
Keywords: center manifold theory; bifurcation; traveling wave solution; collective motion 
Ei, Shin-Ichiro; Ikeda, Kota; Nagayama, Masaharu; Tomoeda, Akiyasu. Application of a center manifold theory to a reaction-diffusion system of collective motion of camphor disks and boats. Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 363-371. doi: 10.21136/MB.2014.143861
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     journal = {Mathematica Bohemica},
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[1] Chaudhury, M. K., Whitesides, G. M.: How to make water run uphill. Science 256 (1992), 1539-1541. | DOI

[2] Ei, S.-I.: The motion of weakly interacting pulses in reaction-diffusion systems. J. Dyn. Differ. Equations 14 (2002), 85-137. | DOI | MR | Zbl

[3] Ei, S.-I., Mimura, M., Nagayama, M.: Pulse-pulse interaction in reaction-diffusion systems. Physica D 165 (2002), 176-198. | DOI | MR | Zbl

[4] Helbing, D., Farkas, I., Vicsek, T.: Simulating dynamical features of escape panic. Nature 407 (2000), 487-490. | DOI

[5] Inaba, M., Yamanaka, H., Kondo, S.: Pigment pattern formation by contact-dependent depolarization. Science 335 (2012), 677. | DOI

[6] Miura, T., Tanaka, R.: In vitro vasculogenesis models revisited---measurement of VEGF diffusion in Matrigel. Math. Model. Nat. Phenom. 4 (2009), 118-130. | DOI | MR | Zbl

[7] Nagayama, M., Nakata, S., Doi, Y., Hayashima, Y.: A theoretical and experimental study on the unidirectional motion of a camphor disk. Physica D 194 (2004), 151-165. | DOI | Zbl

[8] Nakata, S., Iguchi, Y., Ose, S., Kuboyama, M., Ishii, T., Yoshikawa, K.: Self-rotation of a camphor scraping on water: new insight into the old problem. Langmuir 13 (1997), 4454-4458. | DOI

[9] Suematsu, N. J., Nakata, S., Awazu, A., Nishimori, H.: Collective behavior of inanimate boats. Physical Review E 81 (2010), Article ID 056210, 5 pages DOI 10.1103/PhysRevE.81.056210. | DOI

[10] Tomoeda, A., Chowdhury, K. Nishinari. D., Schadschneider, A.: An information-based traffic control in a public conveyance system: Reduced clustering and enhanced efficiency. Physica A: Statistical Mechanics and Its Applications 384 (2007), 600-612.

[11] Tomoeda, A., Yanagisawa, D., Imamura, T., Nishinari, K.: Propagation speed of a starting wave in a queue of pedestrians. Physical Review E 86 (2012), Article ID 036113, 8 pages DOI 10.1103/PhysRevE.86.036113. | DOI

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