Reflection of a self-propelling rigid disk from a boundary

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