1. Equation of motion for point vortices in multiply connected circular domains by Takashi Sakajo Proceedings A Volume 465(2108):2589-2611 August 8, 2009 ©2009 by The Royal Society

2. 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 ©2009 by The Royal Society

3. Topological patterns of the contour lines in the upper semicircles corresponding to the contour plots given in figure 1. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

4. 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. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

5. 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. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

6. 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. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

7. 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). Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

8. 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. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

9. 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. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

10. Transition diagram of the topological patterns of the contour lines of the Hamiltonian in the upper semicircles found in figures 5, 6 and 8. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

11. 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. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

12. 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,... Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

13. 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. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society

14. (a) Evolutions of for the initial configurations z(0)=z1 (continuous line) and z2 (dashed line) with ε1=1.0×10−4. Takashi Sakajo Proc. R. Soc. A 2009;465:2589-2611 ©2009 by The Royal Society