Order and Chaos in Systems of Coaxial Vortex Pairs

TL;DR

This paper investigates the complex dynamics of coaxial vortex pairs, deriving explicit formulas for leapfrogging periods, exploring chaotic behaviors, and analyzing the influence of initial conditions using neural networks.

Contribution

It provides explicit formulas for vortex leapfrogging periods, extends analysis to multiple vortex pairs, and employs neural networks for sensitivity analysis of initial conditions.

Findings

Explicit formula for leapfrogging period matches numerical results.

Conditions for chaotic, periodic, and escape behaviors identified.

Initial separation most influences leapfrogging period according to neural network analysis.

Abstract

Systems of coaxial vortex pairs in an inviscid flow give rise to complex dynamics, with motions ranging from ordered to chaotic. This complexity arises due to the problem's high nonlinearity and numerous degrees of freedom. We analyze the periodic interactions of two vortex pairs with the same absolute strength moving along the same axis and in the same direction. We derive an explicit formula for the leapfrogging period, considering different initial sizes and horizontal separations, and find excellent quantitative agreement with the numerically computed leapfrogging period. We then extend our study to three coaxial vortex pairs with differing strengths, exploring a broad range of initial geometric configurations, and identify conditions that lead to escape to infinity, periodic or quasi-periodic leapfrogging, and chaotic interactions. We also quantify the occurrence of periodic leapfrogging, revealing that the system transitions to two subsystems when vortex pairs have dissimilar strengths and sizes. By performing a sensitivity analysis using neural networks, we find that the initial horizontal separation between the vortex pairs has the most significant effect on the leapfrogging period.