## Nonlinear stability analysis of a regular vortex pentagon outside a circle.(English)Zbl 1252.76017

Summary: A nonlinear stability analysis of the stationary rotation of a system of five identical point vortices lying uniformly on a circle of radius $$R_0$$ outside a circular domain of radius $$R$$ is performed. The problem is reduced to the problem of stability of an equilibrium position of a Hamiltonian system with a cyclic variable. The stability of stationary motion is interpreted as Routh stability. Conditions for stability, formal stability and instability are obtained depending on the values of the parameter $$q = R^2/R^2_0$$.

### MSC:

 76B47 Vortex flows for incompressible inviscid fluids 34D20 Stability of solutions to ordinary differential equations 70K30 Nonlinear resonances for nonlinear problems in mechanics

### Keywords:

point vortices; stationary motion; stability; resonance
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### References:

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