TY - GEN

T1 - Nonintegrable perturbations of two vortex dynamics

AU - Blackmore, Denis

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2008

Y1 - 2008

N2 - The governing equations of motion of two point vortices in an ideal fluid in the plane has a Hamiltonian formulation that is completely integrable, so the dynamics are regular in the sense that one has quasiperiodic solutions confined to invariant two-dimensional tori accompanied by periodic orbits. Moreover, it is well known that the same is true of the dynamics of two point vortices in an ideal fluid in a standard half-plane (with a straight line boundary). It is natural to ask if this is also the case for half-planes whose boundaries are perturbations of a straight line. We prove here that there are such Hamiltonian perturbations of two vortex dynamics in the half-plane that generate chaotic - and a fortiori nonintegrable - dynamics, thereby answering an open question of rather long standing. Our proof, like most demonstrations of this kind, is based on Melnikov's method.

AB - The governing equations of motion of two point vortices in an ideal fluid in the plane has a Hamiltonian formulation that is completely integrable, so the dynamics are regular in the sense that one has quasiperiodic solutions confined to invariant two-dimensional tori accompanied by periodic orbits. Moreover, it is well known that the same is true of the dynamics of two point vortices in an ideal fluid in a standard half-plane (with a straight line boundary). It is natural to ask if this is also the case for half-planes whose boundaries are perturbations of a straight line. We prove here that there are such Hamiltonian perturbations of two vortex dynamics in the half-plane that generate chaotic - and a fortiori nonintegrable - dynamics, thereby answering an open question of rather long standing. Our proof, like most demonstrations of this kind, is based on Melnikov's method.

KW - Chaos

KW - Hamiltonian dynamics

KW - Integrability

KW - Melnikov's method

KW - Transverse heteroclinic orbits

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U2 - 10.1007/978-1-4020-6744-0_29

DO - 10.1007/978-1-4020-6744-0_29

M3 - Conference contribution

AN - SCOPUS:84861139497

SN - 9781402067433

T3 - Solid Mechanics and its Applications

SP - 331

EP - 340

BT - IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence - Proceedings of the IUTAM Symposium

PB - Springer Verlag

T2 - IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence

Y2 - 25 August 2006 through 30 August 2006

ER -