Instability calculations of leapfrogging similar to those reported by Acheson. 6 Two different perturbations are shown. Vertical lines indicate the initial positions. (a) A “disintegration” instability for α = 0.25 perturbed by ξ− = η+ = −10−6. (b) A “walkabout” instability for α = 0.30 perturbed by ξ− = 0, η+ = 10−5 (see text for precise definitions).
Positive vortices are at in the complex plane, negative vortices at . Also shown are ζ±, Eqs. (2a) .
Level curves of the Hamiltonian (9a) .
Floquet exponents as a function of α shown for α0 < α ⩽ α2. See the text for these values of α. The dashed curve shows 2|μ+|, the thick grey curve shows T/3, cf. (11) , and the solid curve shows |μ+|T. Note that μ+ is purely real for α < α2 and purely imaginary for α2 < α < 1, indicating that the monodromy matrix has complex conjugate eigenvalues of unit modulus in the latter range.
Detail of the Floquet exponent μ+ around α = α2 = ϕ2. For α < α2, where μ+ is imaginary, we show μ+/i. The dashed line at α = 0.382 shows Acheson's numerically determined value for the crossover from unstable to stable leapfrogging. 6 The text provides further discussion.
Scattering of two identical vortex pairs showing intermediate states consisting of leapfrogging and “walkabout” motions. The vertical bar marks the starting leapfrogging configuration, α = 0.25, (ξ−, η+) = (−1, 1) × 10−5, from which time is integrated forwards and backwards. The computation has been checked by reverse integration, and the variation of the integrals of motion is of negligible order (10−12).
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