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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

Shin-Ichiro Ei 1, , Masayasu Mimura 2,†, and  Tomoyuki Miyaji 3,‡,,

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

* Corresponding author: Tomoyuki Miyaji

† Present address: Program of Mathematical and Life Sciences, Graduate School of Integrated Sciences for Life, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
‡ Present address: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan.

Received  January 2019 Revised  September 2019 Published  December 2019

Fund Project: This work was supported by JST CREST Grants No. JPMJCR14D3 and JSPS KAKENHI Grant Number 16K17649, 19K03626, and JP26310212

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 & Continuous Dynamical Systems - S, 2021, 14 (3) : 803-817. doi: 10.3934/dcdss.2020229
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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T. Morihara, A Nonequilibrium Billiard Problem: Simulations and Analyses, Master theses, Hiroshima University, 2004. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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O. Tange, GNU parallel - The command-line power tool, Login: The USENIX Magazine, (2011), 42-47. Google Scholar

[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.  Google Scholar

[28]

T. Williams and C. Kelley, Gnuplot homepage, (2018), http://gnuplot.info/. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. D. Hunter, Matplotlib: A 2D graphics environment, Comput. Sci. Eng., 9 (2007), 90-95.  doi: 10.1109/MCSE.2007.55.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Matsumoto, Analysis of the Particle Model Describing Motions of a Camphor, Master thesis, Hiroshima University, 2002. Google Scholar

[12]

T. Matsumoto, A Billiard Problem Under Nonlinear and Nonequilibrium Conditions, Master theses, Hiroshima University, 2003. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

T. Morihara, A Nonequilibrium Billiard Problem: Simulations and Analyses, Master theses, Hiroshima University, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

O. Tange, GNU parallel - The command-line power tool, Login: The USENIX Magazine, (2011), 42-47. Google Scholar

[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.  Google Scholar

[28]

T. Williams and C. Kelley, Gnuplot homepage, (2018), http://gnuplot.info/. Google Scholar

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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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
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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 $
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Figure 4.  Direction field of $ (r, z) $ and the curve $ \{ \dot{r} = 0 \} $ at a fixed $ \theta $
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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
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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
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Figure 7.  (A) $ \delta $ versus $ \log_{10}E $. The parameters are same as Fig. 6. (B) Numerical data of $ F(\theta_{\mathrm{inc}}{)} $ for $ m_0 = m_2 = 1, \delta = 0.25 $ and fitted curve with $ k = 1 $ and $ k = 2 $
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Figure 8.  Discrete-time model
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