Figure 1.
(A) A laboratory experiment of camphor motions. (B) A trajectory of the center of the disk governed by the moving boundary model (7) below
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March
2021, 14(3): 803-817.
doi: 10.3934/dcdss.2020229
Reflection of a self-propelling rigid disk from a boundary
| 1. | Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido 060-0810, Japan |
| 2. | Department of Mathematical Engineering, Musashino University, 3-3-3 Ariake, Koto-ku, Tokyo 135-8181, Japan |
| 3. | Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan |
A system of ordinary differential equations that describes the motion of a self-propelling rigid disk is studied. In this system, the disk moves along a straight-line and reflects from a boundary. Interestingly, numerical simulation shows that the angle of reflection is greater than that of incidence. The purpose of this study is to present a mathematical proof for this attractive phenomenon. Moreover, the reflection law is numerically investigated. Finally, existence and asymptotic stability of a square-shaped closed orbit for billiards in square table with inelastic reflection law are discussed.
Keywords:
Self-propelling particle,
billiards,
inelastic reflection,
heteroclinic orbit,
ordinary differential equations,
maps on the interval.
Mathematics Subject Classification:
Primary: 37N99; Secondary: 34C37.
Citation:
Shin-Ichiro Ei, Masayasu Mimura, Tomoyuki Miyaji. Reflection of a self-propelling rigid disk from a boundary. Discrete and Continuous Dynamical Systems - S,
2021, 14
(3)
: 803-817.
doi: 10.3934/dcdss.2020229
![]()
References:
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X. F. Chen, S.-I. Ei and M. Mimura,
Self-motion of camphor discs. Model and analysis, Netw. Heterog. Media, 4 (2009), 1-18.
doi: 10.3934/nhm.2009.4.1. |
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S.-I. Ei, K. Ikeda, M. Nagayama and A. Tomoeda,
Reduced model from a reaction-diffusion system of collective motion of camphor boats, Discrete Contin. Dyn. Syst. S, 8 (2015), 847-856.
doi: 10.3934/dcdss.2015.8.847. |
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S.-I. Ei, H. Kitahata, Y. Koyano and M. Nagayama,
Interaction of non-radially symmetric camphor particles, Phys. D, 366 (2018), 10-26.
doi: 10.1016/j.physd.2017.11.004. |
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S.-I. Ei, M. Mimura and M. Nagayama, Interacting spots in reaction diffusion systems, Discrete Contin. Dyn. Syst., 14 (2006) 31-62.
doi: 10.3934/dcds.2006.14.31. |
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E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems, Second edition. Springer Series in Computational Mathematics, 8. Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-540-78862-1. |
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J. D. Hunter,
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K. Iida, H. Kitahata and M. Nagayama,
Theoretical study on the translation and rotation of an elliptic camphor particle, Phys. D, 272 (2014), 39-50.
doi: 10.1016/j.physd.2014.01.005. |
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Y. S. Ikura, E. Heisler, A. Awazu, H. Nishimori and S. Nakata, Collective motion of symmetric camphor papers in an annular water channel, Phys. Rev. E, 88 (2013), 012911.
doi: 10.1103/PhysRevE.88.012911. |
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Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
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J. Langham and D. Barkley, Non-specular reflections in a macroscopic system with wave-particle duality: Spiral waves in bounded media, Chaos, 23 (2013), 013134, 9 pp.
doi: 10.1063/1.4793783. |
| [11] |
M. Matsumoto, Analysis of the Particle Model Describing Motions of a Camphor, Master thesis, Hiroshima University, 2002. |
| [12] |
T. Matsumoto, A Billiard Problem Under Nonlinear and Nonequilibrium Conditions, Master theses, Hiroshima University, 2003. |
| [13] |
A. S. Mikhailov and V. Calenbuhr, From Cells to Societies: Models of Complex Coherent Action, Springer Series in Synergetics. Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-05062-0. |
| [14] |
M. Mimura, T. Miyaji and I. Ohnishi,
A billiard problem in nonlinear and nonequilibrium systems, Hiroshima Math. J., 37 (2007), 343-384.
doi: 10.32917/hmj/1200529808. |
| [15] |
T. Miyaji,
Arnold tongues in a billiard problem in nonlinear and nonequilibrium systems, Phys. D, 340 (2017), 14-25.
doi: 10.1016/j.physd.2016.09.003. |
| [16] |
T. Morihara, A Nonequilibrium Billiard Problem: Simulations and Analyses, Master theses, Hiroshima University, 2004. |
| [17] |
M. Nagayama, S. Nakata, Y. Doi and Y. Hayashima,
A theoretical and experimental study on the unidirectional motion of a camphor disk, Physica D: Nonlinear Phenomena, 194 (2004), 151-165.
doi: 10.1016/j.physd.2004.02.003. |
| [18] |
S. Nakata and Y. Hayashima,
Spontaneous dancing of a camphor scraping, J. Chem. Soc., Faraday Trans., 94 (1998), 3655-3658.
doi: 10.1039/a806281a. |
| [19] |
S. Nakata, Y. Iguchi, S. Ose, M. Kuboyama, T. Ishii and K. Yoshikawa,
Self-rotation of a camphor scraping on water: New insight into the old problem, Langmuir, 13 (1997), 4454-4458.
doi: 10.1021/la970196p. |
| [20] |
S. Nakata, M. Nagayama, H. Kitahata, N. J. Suematsu and T. Hasegawa,
Physicochemical design and analysis of self-propelled objects that are characteristically sensitive to environments, Phys. Chem. Chem. Phys., 17 (2015), 10326-10338.
doi: 10.1039/C5CP00541H. |
| [21] |
S. Nakata, V. Pimienta, I. Lagzi, H. Kitahata and N. J Suematsu, Self-organized Motion: Physicochemical Design based on Nonlinear Dynamics, Royal Society of Chemistry, 2018.
doi: 10.1039/9781788013499. |
| [22] |
S. Protière, A. Boudaoud and Y. Couder,
Particle-wave association on a fluid interface, J. Fluid Mech., 554 (2006), 85-108.
doi: 10.1017/S0022112006009190. |
| [23] |
R. J. Strutt,
Ⅳ. Measurements of the amount of oil necessary in order to check the motions of camphor upon water, Proc. R. Soc. Lond., 47 (1889), 286-291.
doi: 10.1098/rspl.1889.0099. |
| [24] |
S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, 30. American Mathematical Society, Providence, RI, Mathematics Advanced Study Semesters, University Park, PA, 2005.
doi: 10.1090/stml/030. |
| [25] |
S. Tanaka, Y. Sogabe and S. Nakata., Spontaneous change in trajectory patterns of a self-propelled oil droplet at the air-surfactant solution interface, Phys. Rev. E, 91 (2015), 032406.
doi: 10.1103/PhysRevE.91.032406. |
| [26] |
O. Tange, GNU parallel - The command-line power tool, Login: The USENIX Magazine, (2011), 42-47. |
| [27] |
C. Tomlinson,
Ⅱ. On the motions of camphor on the surface of water, Proc. Roy. Soc. London, 11 (1860), 575-577.
doi: 10.1098/rspl.1860.0124. |
| [28] |
T. Williams and C. Kelley, Gnuplot homepage, (2018), http://gnuplot.info/. |
show all references
References:
| [1] |
X. F. Chen, S.-I. Ei and M. Mimura,
Self-motion of camphor discs. Model and analysis, Netw. Heterog. Media, 4 (2009), 1-18.
doi: 10.3934/nhm.2009.4.1. |
| [2] |
S.-I. Ei, K. Ikeda, M. Nagayama and A. Tomoeda,
Reduced model from a reaction-diffusion system of collective motion of camphor boats, Discrete Contin. Dyn. Syst. S, 8 (2015), 847-856.
doi: 10.3934/dcdss.2015.8.847. |
| [3] |
S.-I. Ei, H. Kitahata, Y. Koyano and M. Nagayama,
Interaction of non-radially symmetric camphor particles, Phys. D, 366 (2018), 10-26.
doi: 10.1016/j.physd.2017.11.004. |
| [4] |
S.-I. Ei, M. Mimura and M. Nagayama, Interacting spots in reaction diffusion systems, Discrete Contin. Dyn. Syst., 14 (2006) 31-62.
doi: 10.3934/dcds.2006.14.31. |
| [5] |
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems, Second edition. Springer Series in Computational Mathematics, 8. Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-540-78862-1. |
| [6] |
J. D. Hunter,
Matplotlib: A 2D graphics environment, Comput. Sci. Eng., 9 (2007), 90-95.
doi: 10.1109/MCSE.2007.55. |
| [7] |
K. Iida, H. Kitahata and M. Nagayama,
Theoretical study on the translation and rotation of an elliptic camphor particle, Phys. D, 272 (2014), 39-50.
doi: 10.1016/j.physd.2014.01.005. |
| [8] |
Y. S. Ikura, E. Heisler, A. Awazu, H. Nishimori and S. Nakata, Collective motion of symmetric camphor papers in an annular water channel, Phys. Rev. E, 88 (2013), 012911.
doi: 10.1103/PhysRevE.88.012911. |
| [9] |
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
| [10] |
J. Langham and D. Barkley, Non-specular reflections in a macroscopic system with wave-particle duality: Spiral waves in bounded media, Chaos, 23 (2013), 013134, 9 pp.
doi: 10.1063/1.4793783. |
| [11] |
M. Matsumoto, Analysis of the Particle Model Describing Motions of a Camphor, Master thesis, Hiroshima University, 2002. |
| [12] |
T. Matsumoto, A Billiard Problem Under Nonlinear and Nonequilibrium Conditions, Master theses, Hiroshima University, 2003. |
| [13] |
A. S. Mikhailov and V. Calenbuhr, From Cells to Societies: Models of Complex Coherent Action, Springer Series in Synergetics. Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-05062-0. |
| [14] |
M. Mimura, T. Miyaji and I. Ohnishi,
A billiard problem in nonlinear and nonequilibrium systems, Hiroshima Math. J., 37 (2007), 343-384.
doi: 10.32917/hmj/1200529808. |
| [15] |
T. Miyaji,
Arnold tongues in a billiard problem in nonlinear and nonequilibrium systems, Phys. D, 340 (2017), 14-25.
doi: 10.1016/j.physd.2016.09.003. |
| [16] |
T. Morihara, A Nonequilibrium Billiard Problem: Simulations and Analyses, Master theses, Hiroshima University, 2004. |
| [17] |
M. Nagayama, S. Nakata, Y. Doi and Y. Hayashima,
A theoretical and experimental study on the unidirectional motion of a camphor disk, Physica D: Nonlinear Phenomena, 194 (2004), 151-165.
doi: 10.1016/j.physd.2004.02.003. |
| [18] |
S. Nakata and Y. Hayashima,
Spontaneous dancing of a camphor scraping, J. Chem. Soc., Faraday Trans., 94 (1998), 3655-3658.
doi: 10.1039/a806281a. |
| [19] |
S. Nakata, Y. Iguchi, S. Ose, M. Kuboyama, T. Ishii and K. Yoshikawa,
Self-rotation of a camphor scraping on water: New insight into the old problem, Langmuir, 13 (1997), 4454-4458.
doi: 10.1021/la970196p. |
| [20] |
S. Nakata, M. Nagayama, H. Kitahata, N. J. Suematsu and T. Hasegawa,
Physicochemical design and analysis of self-propelled objects that are characteristically sensitive to environments, Phys. Chem. Chem. Phys., 17 (2015), 10326-10338.
doi: 10.1039/C5CP00541H. |
| [21] |
S. Nakata, V. Pimienta, I. Lagzi, H. Kitahata and N. J Suematsu, Self-organized Motion: Physicochemical Design based on Nonlinear Dynamics, Royal Society of Chemistry, 2018.
doi: 10.1039/9781788013499. |
| [22] |
S. Protière, A. Boudaoud and Y. Couder,
Particle-wave association on a fluid interface, J. Fluid Mech., 554 (2006), 85-108.
doi: 10.1017/S0022112006009190. |
| [23] |
R. J. Strutt,
Ⅳ. Measurements of the amount of oil necessary in order to check the motions of camphor upon water, Proc. R. Soc. Lond., 47 (1889), 286-291.
doi: 10.1098/rspl.1889.0099. |
| [24] |
S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, 30. American Mathematical Society, Providence, RI, Mathematics Advanced Study Semesters, University Park, PA, 2005.
doi: 10.1090/stml/030. |
| [25] |
S. Tanaka, Y. Sogabe and S. Nakata., Spontaneous change in trajectory patterns of a self-propelled oil droplet at the air-surfactant solution interface, Phys. Rev. E, 91 (2015), 032406.
doi: 10.1103/PhysRevE.91.032406. |
| [26] |
O. Tange, GNU parallel - The command-line power tool, Login: The USENIX Magazine, (2011), 42-47. |
| [27] |
C. Tomlinson,
Ⅱ. On the motions of camphor on the surface of water, Proc. Roy. Soc. London, 11 (1860), 575-577.
doi: 10.1098/rspl.1860.0124. |
| [28] |
T. Williams and C. Kelley, Gnuplot homepage, (2018), http://gnuplot.info/. |
Figure 2.
Reflection from the boundary $ \{ x = 0 \} $ . (A) Three orbits of (1) with $ m_0 = m_2 = 1, \delta = 0.06 $ starting at $ (x, y) = (20, 0) $ . (B) Illustration of definition of angles
Figure 3.
A trajectory of the center of the disk governed by the moving boundary model (7). Parameters are $ r_0 = 1.0, D = 0.13, k = 1.0, \beta = 1.0, \gamma_0 = 1.0, a = 2.0 $ , and $ F(u) \equiv 2.0 $
Figure 5.
Graphs of $ F(\theta_{\mathrm{inc}}) $ and its derivative for (1) with $ \delta = 0.05 $ and $ m_2 = 1 $ . Red, blue, and green curves are results for $ m_0 = 0, 0.5, 1.0 $ , respectively
Figure 6.
The results of linear regression for (1) with $ m_2 = 1 $ . Only the results satisfying $ E < 0.001 $ are plotted. (A) The estimated values of $ (c_0, c_1) $ for various $ \delta $ and $ m_0 $ . (B) The dependence of $ \sqrt{c_0^2 + c_1^2} $ on $ \delta $ . Red, blue, and green circles are results for $ m_0 = 0, 0.5, 1.0 $ , respectively
Figure 8.
Discrete-time model
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