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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

* 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  March 2021 Early access  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.

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:
 [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/.

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/.
(A) A laboratory experiment of camphor motions. (B) A trajectory of the center of the disk governed by the moving boundary model (7) below
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
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 $F(u) \equiv 2.0$
Direction field of $(r, z)$ and the curve $\{ \dot{r} = 0 \}$ at a fixed $\theta$
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
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
(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 $k = 2$
Discrete-time model
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