equation of motion for point vortices in multiply connected circular domains by takashi sakajo...

Equation of motion for point vortices in multiply connected circular domains

by Takashi Sakajo

Proceedings AVolume 465(2108):2589-2611

August 8, 2009

©2009 by The Royal Society

Contour plots of the Hamiltonian (3.4) in the circular domains with two obstacles that are symmetric with respect to the real axis.

Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611


Topological patterns of the contour lines in the upper semicircles corresponding to the contour plots given in figure 1.



Contour plots of the Hamiltonian (3.4) in the symmetric circular domains with the three obstacles (3.6) for a=0, b=0.8, r1=0.05 and c=0.



Contour plots of the Hamiltonian (3.4) in the symmetric circular domains with the three obstacles (3.6) for a=0, b=0.6, r1=0.1 and c=0.3.



Contour plots of the Hamiltonian (3.4) in the symmetric circular domains with the four obstacles of radius 0.1 (3.7), in which we change a with fixed b=0.5.



Contour plots of the Hamiltonian (3.4) and their corresponding topological patterns of the contour lines in the upper semicircles for (a) a=0.1935, b=0.5, (b) a=0.21269, b=0.5 and (c) a=0.15,

b=0.2838 in equation (3.7).



Schematic pictures for degenerate pinching bifurcations of the contour lines of the Hamiltonian around the centre points, which generate new saddle/centre points and homoclinic/heteroclinic

connections.



Contour plots of the Hamiltonian (3.4) in the symmetric circular domains with the four obstacles of radius 0.1 in equation (3.7), in which we change b with a=0.15.



Transition diagram of the topological patterns of the contour lines of the Hamiltonian in the upper semicircles found in figures 5, 6 and 8.



Contour plot of the Hamiltonian (3.8) for the single point vortex in the doubly connected circular domain with one obstacle of radius 0.1, whose centre is located at 0.5i.



Contour plots of the Hamiltonian (3.8) for the single point vortex in the triply connected circular domains with the two obstacles of equation (3.9) with b=0 and various a, whose topological

patterns for a=0.2, 0.3 and 0.5 are equivalent to (a) pattern VII,...



Motion of the two point vortices in the symmetric domain with the two obstacles (3.5) for a=0, b=0.4 and r=0.05.



(a) Evolutions of for the initial configurations z(0)=z1 (continuous line) and z2 (dashed line) with ε1=1.0×10−4.



equation of motion for point vortices in multiply connected circular domains by takashi sakajo...

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