Three, four and five point vortices which exhibit the relaxation - PowerPoint PPT Presentation

Three, four and five point vortices which exhibit the relaxation oscillation Tatsuyuki N AKAKI nakaki@math.kyushu-u.ac.jp Faculty of Mathematics, Kyushu University, Japan Euromech448 (ESPCI, Paris, France, 6 September 2004) Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.1/19

Purpose of this talk To show the assemblies of point vortices (in 2-dim) which exhibit the relaxation oscillation Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.2/19

Purpose of this talk To show the assemblies of point vortices (in 2-dim) which exhibit the relaxation oscillation Three point vortices exhibiting the relaxation oscillation Four point vortices exhibiting the relaxation oscillation Five point vortices exhibiting the relaxation oscillation What is the relaxation oscillation? Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.2/19

✁ � ✁ ✁ ✁ ✁ � � Purpose of this talk To show the assemblies of point vortices (in 2-dim) which exhibit the relaxation oscillation Three point vortices exhibiting the relaxation oscillation Four point vortices exhibiting the relaxation oscillation Five point vortices exhibiting the relaxation oscillation What is the relaxation oscillation? It is the oscillation such that steady state rapid motion steady state rapid motion �✂✁ Click here for a numerical simulation. Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.2/19

Relaxation oscillation Mathematically, such a relaxation oscillation is induced by chain of heteroclinic orbits. Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.3/19

Relaxation oscillation Mathematically, such a relaxation oscillation is induced by chain of heteroclinic orbits. Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.3/19

Relaxation oscillation Mathematically, such a relaxation oscillation is induced by chain of heteroclinic orbits. Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.3/19

Relaxation oscillation Mathematically, such a relaxation oscillation is induced by chain of heteroclinic orbits. Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.3/19

� ✁ ✁ ✁ Relaxation oscillation (cont.) Chain of heteroclinic orbits The existence of a heteroclinic orbit (*) What we should do is to prove (*). Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.4/19

Agenda The basic equation to be solved Known results Formulation of our problem Relaxation oscillation (five point vortices) Relaxation oscillation (four point vortices) Relaxation oscillation (three point vortices) Conclusions and Future works Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.5/19

✁ ✆ ✆ ✒ ✂ ☞ ☎ ✁ ✂ ✏ ✄ ☎ ✁ ✆ ✏ ✄ ☞ ☎ ✄ ✆ � � ✁ ✂ ✄ ☎ ✁ ✓ ✝ ✞ ☛ ☞ Basic equation Classical point vortices problem (2-dim) ✟✡✠ ✂✑✄ ✌✎✍ where : complex coordinate (unknown) : circulation (given real constant) of th point vortex Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.6/19

✁ � � � ✝ ✟ � ✝ ✄ ✂ Known results Let be the number of vortices in fluid : Easy to solve. : A qualitative analysis with arbitrary strength is done by Aref (1979). : The ODE is not solved yet in general cases Chaotic behavior occurs (Aref and Pomphrey, 1982) Stationary configuration (O’neil, 1987) Morikawa and Swenson’s results (1971) Some five point vortices exhibit the relaxation oscillation (N. 1999) — another configuration shall be shown today Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.7/19

� ✁ ✂ ☎ ✞ ✆ ✂ ✟ Problem for Two parameters and ✄✆☎ ✏ ✞✝ Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.8/19

✆ ✄ ✞ ✂ ✂ ☎ ✄ ✆ ✝ ✒ ✂ ☎ ✄ ✄ ✆ ✝ ☛ ✁ ✂ ☎ ☎ ✄ ✆ ✝ ✄ ☎ ✂ � ✁ ✂ ☎ ✞ ✆ ✂ ✟ � ☎ ✄ ✆ ✝ ✒ ✂ ✁ ✝ Problem for Two parameters and ✄✆☎ ✏ ✞✝ Initial configuration Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.8/19

✄ ✏ ☎ ✄ ✆ ✝ ☛ ✁ ✂ ☎ ☎ ✄ ✆ ✝ ✄ � ✝ ✝ ✏ ✁ ✝ ✞ ✏ ✂ ✝ ✏ ✄ ✝ ✏ ☎ ✝ ✒ ✂ ✆ ✆ � ✁ ✂ ☎ ✞ ✆ ✂ ✟ ✂ � ✄ ✄ ☎ ✝ ✆ ☎ ✂ ✂ ✒ ✝ ✞ ✄ ☎ ✁ ✂ Problem for Two parameters and ✄✆☎ ✏ ✞✝ Initial configuration Strength is ✏ ✞✝ ✏ ✁� where is ..... ✏ ✁� Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.8/19

� Problem for (cont.) ....., where is determined so that the five point vortices is ✏ ✁� in the relative equilibrium . Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.9/19

� ✂ ✂ ✟ ✞ ✁ ✆ ✁ ☎ ✄ ✂ ✝ ✆ ☎ ✄ ✟ ✁ � ✆ ✁ ☎ ✄ ✂ � Problem for (cont.) ....., where is determined so that the five point vortices is ✏ ✁� in the relative equilibrium . Definition. The solution is a relative equilibrium if and only if is an equilibrium for . Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.9/19

☎ � ✟ ✟ ✂ ✟ ✞ ✁ ✆ ✁ ☎ ✄ ✂ ✝ ✆ ✂ ✄ ✁ ✆ � ✁ ☎ ✄ ✂ � Problem for (cont.) ....., where is determined so that the five point vortices is ✏ ✁� in the relative equilibrium . Definition. The solution is a relative equilibrium if and only if is an equilibrium for . The vortices corotate around the origin with uniform angular velocity . Throughout this talk (including all numerical simulations), we observe the vortices in the rotating coordinates. Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.9/19

� Problem for (cont.) The motion of point vortices on this problem has rich structure: Relaxation oscillation Stability of the configuration Unstable in the linearized sense Weakly unstable Stable in the Lyapunov sense Today’s talk: relaxation oscillation Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.10/19

✝ ✆ ✝ ✄ ✝ ✒ ✟ ✁ ☎ ☎ ✒ ✞ ✁ ✆ ✒ ✏ ✄ ✝ ✝ ✒ ✞ ☎ ✄ ✞ ✏ ✝ ✝ ✟ ☎ ✆ ✝ ✁ ✏ ✝ ✞ � ✄ � ✁ � ✞ ✒ ✁ ✄ ✂ � � ✝ ✄ ✁ ✏ � ✞ � ✝ Problem for (cont.) Conjecture. For and , ✏ ✞✝ there exists a heteroclinic orbit. Theorem. For and , there exists a heteroclinic orbit. Numerical simulations when and ✄ ✂✁ and (Click here) ✄ ✂✁ and (Click here) ✄ ✂✁ ✄ ✂✁ ✄ ✂✁ and (Click here) ✄ ✂✁ ✄ ✂✁ ✄ ✂✁ Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.11/19

� ✝ ✁ � ✄ ✂ � ✁ ✂ � ✁ ✄ ✁ ✒ ✝ ✆ � ✄ ✝ � ✝ ✏ Problem for In the previous problem ( ), let us consider the case when (i.e., ). ✏ ✁� The center vortex is not a vortex anymore; it is a particle. For some , we observe the relaxation oscillation. (Click here for a numerical simulation) Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.12/19

� ☎ ✞ � ✁ ✝ ✆ ☎ ✁ ✁ ✁ ✁ ✏ ✝ � ✄ � ✄ ✝ ✁ ✟ ✒ ✞ ✏ ✝ ✝ ✒ ✞ � Problem for (cont.) Theorem. For (i.e., ), there exists a heteroclinic orbit. Conjecture. For (i.e., ), ✄ ✂✁ there exists a heteroclinic orbit. Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.13/19

Three point vortices The qualitative analysis is already done by Aref (1979). Faculty of Mathematics, K YUSHU U NIVERSITY Three, four and five point vortices which exhibit the relaxation oscillation – p.14/19