Hamiltonian bifurcation perspective on two interacting vortex pairs: From symmetric to asymmetric leapfrogging, period doubling, and chaos

Abstract

In this work we study the dynamical behavior of two interacting vortex pairs, each one of them consisting of two point vortices with opposite circulation in the 2d plane. The vortices are considered as effective particles and their interaction can be desribed in classical mechanics terms. We first construct a Poincaré section, for a typical value of the energy, in order to acquire a picture of the structure of the phase space of the system. We divide the phase space in different regions which correspond to qualitatively distinct motions and we demonstrate its different temporal evolution in the "real" vortex-space. Our main emphasis is on the leapfrogging periodic orbit, around which we identify a region that we term the "leapfrogging envelope" which involves mostly regular motions, such as higher order periodic and quasi-periodic solutions. We also identify the chaotic region of the phase plane surrounding the leapfrogging envelope as well as the so-called walkabout and braiding motions. Varying the energy as our control parameter, we construct a bifurcation tree of the main leapfrogging solution and its instabilities, as well as the instabilities of its daughter branches. We identify the symmetry-breaking instability of the leapfrogging solution (in line with earlier works), and also obtain the corresponding asymmetric branches of periodic solutions. We then characterize their own instabilities (including period doubling ones) and bifurcations in an effort to provide a more systematic perspective towards the types of motions available to this dynamical system.

Collinear interaction of vortex pairs with different strengths—Criteria for leapfrogging

“…The criterion developed explicitly in (33) and (34) appears to be in general accord with the results obtained by a different analysis and given in Eckhardt and Aref appendix B [17].…”

“…We do not consider here any conditions of stability of the solutions presented, although we note that there is some computational consideration of this matter contained in Acheson [2]. The dynamical systems approach and stability issues are also pursued analytically in Berger [12], in Behring and Goodman [11], and in a numerical investigation in Whitchurch, Kevrekidis, and Koukouloyannis [33].…”

Stability of leapfrogging vortex pairs: A semi-analytic approach

“…In this section, we apply the method of harmonic balance (MHB) to the π-periodic differential equation (20). As noted in Sec.…”

“…To apply the method of harmonic balance, we write the periodic solution to system (20) as a Fourier series. The following two observations allow us to simplify the form of this series.…”

“…We found using a trigonometric basis here to be more natural than a complex exponential basis used in (26). Putting this ansatz into the Floquet system (20) and collecting coefficients of the harmonics formally result in an infinite-dimensional matrix problem…”