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# Reflection of a self-propelling rigid disk from a boundary

• * 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.

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.

Mathematics Subject Classification: Primary: 37N99; Secondary: 34C37.

 Citation: • • 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

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 4.  Direction field of $(r, z)$ and the curve $\{ \dot{r} = 0 \}$ at a fixed $\theta$

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 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 $k = 2$

Figure 8.  Discrete-time model

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