# American Institute of Mathematical Sciences

• Previous Article
Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction
• DCDS-S Home
• This Issue
• Next Article
Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows
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 & Continuous Dynamical Systems - S, 2021, 14 (3) : 803-817. doi: 10.3934/dcdss.2020229
##### References:

show all references

##### References:
(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
. (B) Numerical data of $F(\theta_{\mathrm{inc}}{)}$ for $m_0 = m_2 = 1, \delta = 0.25$ and fitted curve with $k = 1$ and $k = 2$">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 $k = 2$
Discrete-time model
 [1] Pedro Duarte, José Pedro GaivÃo, Mohammad Soufi. Hyperbolic billiards on polytopes with contracting reflection laws. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3079-3109. doi: 10.3934/dcds.2017132 [2] Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87 [3] Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353 [4] Rafael Sanabria. Inelastic Boltzmann equation driven by a particle thermal bath. Kinetic & Related Models, 2021, 14 (4) : 639-679. doi: 10.3934/krm.2021018 [5] Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008 [6] Tayel Dabbous. Identification for systems governed by nonlinear interval differential equations. Journal of Industrial & Management Optimization, 2012, 8 (3) : 765-780. doi: 10.3934/jimo.2012.8.765 [7] Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 [8] Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165 [9] W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209 [10] Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040 [11] Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036 [12] Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39 [13] Hongwei Lou, Weihan Wang. Optimal blowup/quenching time for controlled autonomous ordinary differential equations. Mathematical Control & Related Fields, 2015, 5 (3) : 517-527. doi: 10.3934/mcrf.2015.5.517 [14] Alex Bihlo, James Jackaman, Francis Valiquette. On the development of symmetry-preserving finite element schemes for ordinary differential equations. Journal of Computational Dynamics, 2020, 7 (2) : 339-368. doi: 10.3934/jcd.2020014 [15] Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283 [16] Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 [17] Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91 [18] David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, 2021, 29 (4) : 2791-2817. doi: 10.3934/era.2021014 [19] William Guo. The Laplace transform as an alternative general method for solving linear ordinary differential equations. STEM Education, 2021, 1 (4) : 309-329. doi: 10.3934/steme.2021020 [20] Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293

2020 Impact Factor: 2.425