Periodic solutions of the N-vortex Hamiltonian system in planar domains

We investigate the existence of collision-free nonconstant periodic solutions of the N  -vortex problem in domains Ω⊂CΩ⊂C. These are solutions z(t)=(z1(t),…,zN(t))∈ΩNz(t)=(z1(t),…,zN(t))∈ΩN of the first order Hamiltonian systemz˙k(t)=−i∇zkHΩ(z(t)),k=1,…,N, where the Hamiltonian HΩHΩ has the formHΩ(z1,…,zN)=12π∑j,k=1j≠kNlog⁡1|zj−zk|−F(z). The function F:ΩN→RF:ΩN→R depends on the regular part of the hydrodynamic Green's function and is unbounded from above: F(z)→∞F(z)→∞ if zk→∂Ωzk→∂Ω for some k  . The Hamiltonian is unbounded from above and below, it is singular, not integrable, energy surfaces are not compact and not known to be of contact type. Using singular perturbation techniques and Conley index theory we prove the existence of a family of periodic solutions zr(t)zr(t), 0