Abstract
AbstractTheoretical analysis using mathematical models is often used to understand a mechanism of collective motion in a self-propelled system. In the experimental system using camphor disks, several kinds of characteristic motions have been observed due to the interaction of two camphor disks. In this paper, we understand the emergence mechanism of the motions caused by the interaction of two self-propelled bodies by analyzing the global bifurcation structure using the numerical bifurcation method for a mathematical model. Finally, it is also shown that the irregular motion, which is one of the characteristic motions, is chaotic motion and that it arises from periodic bifurcation phenomena and quasi-periodic motions due to torus bifurcation.
Highlights
IntroductionIn order to theoretically understand the mechanism of self-propelled material motion, much computer-aided and formal analysis using mathematical models have been investigated [32,33,34,35,36]
Vicsek et al many theoretical analyses using particle motion models have been conduced to understand the collective motion of biological species [1]
In order to theoretically understand the mechanism of self-propelled material motion, much computer-aided and formal analysis using mathematical models have been investigated [32,33,34,35,36]
Summary
In order to theoretically understand the mechanism of self-propelled material motion, much computer-aided and formal analysis using mathematical models have been investigated [32,33,34,35,36]. To understand the collective motion of camphor disks in the annular water channel [40], in 2015 Nishi et al analyzed the interaction of two camphor disks using the following dimensionless mathematical model without the inertia term, which assumes that camphor disks are very light [41]: dxci = (u( L(xci + r), t)) − (u( L(xci − r), t)) , dt. Corresponding to an billiard like motion of camphor disks Both solutions are obtained by the numerical computation for (3). When N = 2 , from the computer-aided analysis of (3) under the periodic boundary condition and the additional experimental results, the model system (3) was shown to qualitatively reproduce well the motion of two camphor disks [41]. We summarize the entire study and discuss future work
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