Despite this book’s title, we have only focused on camphor disk motion. There are other dissipative systems that involve billiard-like motion with nonspecular reflection. One of the classical examples is a droplet walker [6, 69]. Under certain conditions, a liquid droplet moves in a straight line and reflects from a tank wall at a nonspecular angle. Shirokoff has demonstrated by a mathematical model that the trajectory of the droplet approaches either non-periodic dense curves or a quasiperiodic orbit in a large domain [73]. Altmann et al. [3] have studied some optical systems in which the site of reflection shifts and the angle of reflection is greater than that of incidence. In [3], the authors have employed a discrete-time model to study a billiard problem in an annular domain, where reflections at the outer boundary are specular, and reflections at the inner boundary are non-specular. Certain microorganisms exhibit similar behavior. Kantsler et al. have demonstrated that Chlamydomonas approaching a flat wall moves away from the wall at a fixed angle, independent of the angle of incidence [33]. Similarly to Altmann et al.’s system, the site of reflection shifts when it reflects from a boundary wall. Spagnolie et al. [74] have studied a billiard problem in regular polygons using a discrete-time model which exhibits a fixed angle of reflection and shift of reflection sites. They have demonstrated stable cycles and chaotic orbits in regular polygons and trajectories in the presence of obstacles. Uspal et al. [80] have studied the motion of Janus particles, which are self-driven by interaction from hydrodynamic and chemical fields induced by their motion. They have shown that three types of motion near a boundary can occur depending on parameters: reflection, sliding and hovering. Some mathematical models of a cavity soliton formulated in terms of a nonlinear Schrödinger equation and ODEs also exhibit billiard-like motion with nonspecular reflection, but this has not been realized in laboratory experiments [67]. In addition, cores of spiral patterns in some reaction-diffusion systems also exhibit similar motions [44].

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Conclusion

  • Tomoyuki Miyaji,
  • Shin-Ichiro Ei,
  • Masayasu Mimura

摘要

Despite this book’s title, we have only focused on camphor disk motion. There are other dissipative systems that involve billiard-like motion with nonspecular reflection. One of the classical examples is a droplet walker [6, 69]. Under certain conditions, a liquid droplet moves in a straight line and reflects from a tank wall at a nonspecular angle. Shirokoff has demonstrated by a mathematical model that the trajectory of the droplet approaches either non-periodic dense curves or a quasiperiodic orbit in a large domain [73]. Altmann et al. [3] have studied some optical systems in which the site of reflection shifts and the angle of reflection is greater than that of incidence. In [3], the authors have employed a discrete-time model to study a billiard problem in an annular domain, where reflections at the outer boundary are specular, and reflections at the inner boundary are non-specular. Certain microorganisms exhibit similar behavior. Kantsler et al. have demonstrated that Chlamydomonas approaching a flat wall moves away from the wall at a fixed angle, independent of the angle of incidence [33]. Similarly to Altmann et al.’s system, the site of reflection shifts when it reflects from a boundary wall. Spagnolie et al. [74] have studied a billiard problem in regular polygons using a discrete-time model which exhibits a fixed angle of reflection and shift of reflection sites. They have demonstrated stable cycles and chaotic orbits in regular polygons and trajectories in the presence of obstacles. Uspal et al. [80] have studied the motion of Janus particles, which are self-driven by interaction from hydrodynamic and chemical fields induced by their motion. They have shown that three types of motion near a boundary can occur depending on parameters: reflection, sliding and hovering. Some mathematical models of a cavity soliton formulated in terms of a nonlinear Schrödinger equation and ODEs also exhibit billiard-like motion with nonspecular reflection, but this has not been realized in laboratory experiments [67]. In addition, cores of spiral patterns in some reaction-diffusion systems also exhibit similar motions [44].