TY - JOUR

T1 - Transition to instability of the leapfrogging vortex quartet

AU - Goodman, Roy H.

AU - Behring, Brandon M.

N1 - Funding Information:
We dedicate this article to the memory of Denis Blackmore, an esteemed colleague who served with wit and generosity on the dissertation committee of BMB. This research arises from the dissertation research of BMB, supported by NJIT. RG gratefully acknowledges support from the NSF under DMS–2206016. We thank Panos Kevrekidis, who insisted that the question of linearized stability was not fully answered in [10] and encouraged us to do the perturbation theory properly. Finally, we must acknowledge how the solution (23) was found. After failing to find an exact periodic orbit to system (12) with [Formula presented] using both Mathematica and pencil and paper, we posted a question to the Math Overflow discussion board to ask for help. Steve Israel responded with a solution he found using Maple [16]. We are very grateful to him and would be interested to learn how Maple solved it. All other difficult integrals performed in this note were accomplished using Mathematica, without which this work would have been impossible.
Funding Information:
We dedicate this article to the memory of Denis Blackmore, an esteemed colleague who served with wit and generosity on the dissertation committee of BMB. This research arises from the dissertation research of BMB, supported by NJIT. RG gratefully acknowledges support from the NSF under DMS–2206016 . We thank Panos Kevrekidis, who insisted that the question of linearized stability was not fully answered in [10] and encouraged us to do the perturbation theory properly. Finally, we must acknowledge how the solution (23) was found. After failing to find an exact periodic orbit to system (12) with using both Mathematica and pencil and paper, we posted a question to the Math Overflow discussion board to ask for help. Steve Israel responded with a solution he found using Maple [16] . We are very grateful to him and would be interested to learn how Maple solved it. All other difficult integrals performed in this note were accomplished using Mathematica, without which this work would have been impossible.
Publisher Copyright:
© 2023 Elsevier Ltd

PY - 2023/2

Y1 - 2023/2

N2 - The point-vortex system is a system of longstanding interest in nonlinear dynamics, describing the motion of a two-dimensional inviscid fluid that is irrotational except at a discrete set of moving point vortices, at which the vorticity diverges. The leapfrogging orbit consists of two rotating pairs of like-signed vortices which, taken as a quartet, propagate at constant velocity. It is known that if the two pairs are initially widely separated, the motion is stable, while if they are closer together it becomes unstable, with this relation represented by a dimensionless parameter α defined in the text. We here demonstrate analytically that the transition from stability to instability happens at a critical value α=ϕ−2, where ϕ is the golden ratio. This value had been hypothesized based on careful numerics by Tophøj and Aref, and by the present authors using a semi-analytic argument but not previously demonstrated through exact analysis.

AB - The point-vortex system is a system of longstanding interest in nonlinear dynamics, describing the motion of a two-dimensional inviscid fluid that is irrotational except at a discrete set of moving point vortices, at which the vorticity diverges. The leapfrogging orbit consists of two rotating pairs of like-signed vortices which, taken as a quartet, propagate at constant velocity. It is known that if the two pairs are initially widely separated, the motion is stable, while if they are closer together it becomes unstable, with this relation represented by a dimensionless parameter α defined in the text. We here demonstrate analytically that the transition from stability to instability happens at a critical value α=ϕ−2, where ϕ is the golden ratio. This value had been hypothesized based on careful numerics by Tophøj and Aref, and by the present authors using a semi-analytic argument but not previously demonstrated through exact analysis.

KW - Bifurcation theory

KW - Hamiltonian mechanics

KW - Point vortices

UR - http://www.scopus.com/inward/record.url?scp=85147607892&partnerID=8YFLogxK

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U2 - 10.1016/j.mechrescom.2023.104068

DO - 10.1016/j.mechrescom.2023.104068

M3 - Article

AN - SCOPUS:85147607892

SN - 0093-6413

VL - 128

JO - Mechanics Research Communications

JF - Mechanics Research Communications

M1 - 104068

ER -