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Posts

Blog Post number 4

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This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 3

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This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 2

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This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 1

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

publications

The global phase space of the three-vortex interaction system and its application to vortex-dipole scattering

Published in Ph.D. dissertation, New Jersey Institute of Technology, 2026

A global reduction and phase-space analysis of the three-vortex problem, including bifurcations, vortex-dipole scattering, and extensions to an integrable four-vortex regime.

Recommended citation: Atul Anurag. The global phase space of the three-vortex interaction system and its application to vortex-dipole scattering. Ph.D. dissertation, New Jersey Institute of Technology.
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Phase portraits and the bifurcation set for the three-vortex interaction system

Published in Communications in Nonlinear Science and Numerical Simulation, 163, 110428, 2026

A global, singularity-free reduction of the three-vortex problem that organizes phase portraits and the bifurcation set across circulation regimes.

Recommended citation: Atul Anurag and Roy H. Goodman. Phase portraits and the bifurcation set for the three-vortex interaction system. Communications in Nonlinear Science and Numerical Simulation, 163, 110428.
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talks

Generalization of Leapfrogging Orbits of Point Vortices

Published:

Point vortex motion arises in the study of concentrated vorticity in an ideal, incompressible fluid described by Euler’s equations. The two-dimensional Euler equations of fluid mechanics, a partial differential equation (PDE) system, support a solution where the vorticity is concentrated at a single point. Helmholtz derived a system of ordinary differential equations (ODEs) that describe the motion of a set of interacting vortices that behave as discrete particles, which approximates the fluid motion in the case that the vorticity is concentrated in very small regions. This system of equations has continued to provide interesting questions for over 150 years. We will discuss the special class of relative periodic orbits known as the leapfrogging orbits. The relative periodic of a four-point vortex problem, two positive and two negative point vortices, all of the same absolute circulation arranged as co-axial vortex pairs, is known as the leapfrogging orbit. This dissertation will present the generalizations to the leapfrogging motion of point vortices and vortex rings, including their stability and dynamics. More specifically, we will study the leapfrogging motion of 2N vortices, with circulations half positive and half negative.

Point Vortex Dipole Scattering

Published:

We investigate three problems in point-vortices dynamics within a two-dimensional, inviscid, incompressible fluid. We derive a new reduction of a system of three vortices. The integrable reduced system has an easily visualized phase plane that illuminates the dynamics. We apply it to explain the scattering of the point-vortex dipole with a third vortex in two cases. We then add a fourth vortex and use the reduced dynamics of the three-vortex system as the basis for the perturbative study of dipole-dipole scattering.

The Phase Space of the Three-Vortex Problem

Published:

The motion of three point-vortices in a 2D inviscid, incompressible fluid has been widely studied. Gröbli (1877) derived a closed system for the evolution of the side lengths of the triangle formed by the vortices. This system has been the basis of most studies of this problem. These coordinates have a few disadvantages. First, the coordinates must satisfy the triangle inequality, so not all points in $\mathbb{R}^{3}$ are physical. Second, the system introduced non-physical singularities because collinear arrangements lie on the boundary of the triangle inequality. Third, these coordinates break the useful Hamiltonian structure and make phase-plane reasoning difficult. We introduce a coordinate system for this problem that overcomes these disadvantages using a sequence of systematic and standard reductions. We first use Jacobi coordinates, a common technique in n-body systems, and further apply a Nambu Bracket reduction. The method avoids creating non-physical singularities and makes the system’s phase-space geometry and topology plain. A previous Nambu-bracket formulation based on Gröbli’s reduction inherited its triangle-inequality-related disadvantages. Depending on the circulations, the phase space may be a sphere or one sheet of a two-sheeted hyperboloid. Prior attempts to classify the dynamics focused solely on the stability of the relative fixed points; this approach allows us to explain the results more clearly using the entire phase space.

Global Phase Plane Analysis of the three-vortex problem

Published:

We investigate two problems in point-vortex dynamics within a two-dimensional, inviscid, incompressible fluid. We derive a new reduction of a system involving three vortices, initially employing Jacobi coordinates followed by Nambu brackets. First, we conduct a global phase analysis of a three-vortex problem with arbitrary circulations. Second, we generalize the reduction method to study the dynamics of four vortices with vanishing total circulation. The novel reduction method eliminates coordinate singularities that made understanding the dynamics difficult.

The Global Phase Plane Analysis of Three Vortex Interactions

Published:

We investigate global phase planes in point-vortex dynamics in a two-dimensional, inviscid, incompressible fluid. We derive a symplectic reduction of a system involving three vortices, initially employing Jacobi coordinates followed by Lie-Poisson reduction. We conduct a global phase analysis of a three-vortex problem with arbitrary circulations with novel bifurcation analysis. This reduction method eliminates coordinate singularities that made understanding the dynamics challenging.

teaching

Ramapo College of New Jersey

Teaching, Ramapo College of New Jersey, 2026

Courses and selected teaching materials from Ramapo College of New Jersey.