A billiard problem in nonlinear and nonequilibrium systems

In this study, the interaction between two non-radially symmetric camphor particles is theoretically investigated and the equation describing the motion is derived as an ordinary differential system for the locations and the rotations. In particular, slightly modified non-radially symmetric cases from radial symmetry are extensively investigated and explicit motions are obtained. For example, it is theoretically shown that elliptically deformed camphor particles interact so as to be parallel with major axes. Such predicted motions are also checked by real experiments and numerical simulations.

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“…Note that A m 0 is invertible in X R by the invertibility of L 0 in L 2 (R 2 ). Then the equation of 20) and…”

Section: By Proposition 24 We Havementioning

confidence: 99%

“…In fact, there has been a lot of research related to multi-camphor particles (e.g. [19,20,21,22,23,24,25,26,27,28]) such as analysis of the jam of camphor particles on a circle (e.g. [28]) although almost all of them are related to the interaction in one-dimensional space.…”

Section: Introductionmentioning

confidence: 99%

Interaction of non-radially symmetric camphor particles

Kitahata

Koyano

et al. 2018

Physica D: Nonlinear Phenomena

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“…In other words, a camphor particle releases its repellants (camphor molecules) and moves to the region with less repellent concentration. The camphor-water system is so simple that various geometries can be realized [21][22][23][24][25][26][27][28][29][30][31][32][33]. For example, the size of the water chamber can be one of the parameters which can affect the motion of the camphor disk [21,22].…”

Section: Introductionmentioning

confidence: 99%

Rotational motion of a camphor disk in a circular region

2019

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In a two-dimensional axisymmetric system, the system symmetry allows rotational or oscillatory motion as stable stationary motion for a symmetric self-propelled particle. In the present paper, we studied the motion of a camphor disk confined in a two-dimensional circular region. By reducing the mathematical model describing the dynamics of the motion of a camphor disk and the concentration field of camphor molecules on a water surface, we analyzed the reduced equations around a bifurcation point where the rest state at the center of the system becomes unstable. As a result, we found that rotational motion is stably realized through the double-Hopf bifurcation from the rest state. The theoretical results were confirmed by numerical calculation and well corresponded to the experimental results.

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“…These properties appear in the reflection of a selfpropelling particle [22,25]. As a consequence of these properties, motions in square or rectangular domains are quite different from the one in mathematical billiards [14,15]. For example, numerical simulations suggest that a typical trajectory in square approaches to a square-shaped periodic orbit such that the disk visits each edge of the square in turn ( Fig.…”

mentioning

confidence: 99%

“…We briefly explain the derivation of this model in the next section. In this model equation 1, a very attractive phenomenon in which the angle of reflection against the boundary {x = 0} is greater than that of incidence was numerically observed [11,14,15]. See Fig.…”

mentioning

confidence: 99%

Reflection of a self-propelling rigid disk from a boundary

Ei¹,

Mimura²,

Miyaji³

2021

Discrete &Amp; Continuous Dynamical Systems - S

Self Cite

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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.

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A billiard problem in nonlinear and nonequilibrium systems

Cited by 8 publications

References 2 publications

Interaction of non-radially symmetric camphor particles

Interaction of non-radially symmetric camphor particles

Rotational motion of a camphor disk in a circular region

Reflection of a self-propelling rigid disk from a boundary

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