| Abstract: |
| A single self-driven particle exhibits various types of periodic motions like oscillatory, rotational and quasi-periodic motions through interaction with external forces. An oil droplet exhibits various periodic motions in sodium dodecyl sulfate aqueous solution by the Marangoni flow induced by the chemical reaction as described by Tanaka et al. (2015), where a mathematical model was proposed to analyze such phenomena theoretically. Such mathematical models generate periodic solutions via Hopf bifurcation. In those cases we can deduce a canonical system of an ODE form. Actually, under a suitable condition, there are quasi-periodic solutions which connect oscillatory and rotational solutions in a bifurcation diagram. Generally speaking, it is difficult to prove the existence and stability of quasi-periodic solutions in a rigorous manner. Thus we rewrite the canonical system into a lower dimensional system, which we name the RVF system. In this talk, we verify the existence and stability of a periodic solution in the RVF system. Moreover, we will state that the periodic solution in the RVF system may correspond to a quasi-periodic solution in the canonical system. |
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