Professor Irina Popovici
Education:
 Ph.D., Michigan State University, 1998.
 M.S., Al. I. Cuza University, Romania, 1991.
Professional Experience:
 Professor, USNA, Mathematics Department
 Hedrick Assistant Professor, UCLA , 19982001.
 Visiting Assistant Professor, Rice Univ. and UCLA, Summer 2002
 Teaching Assistant, Michigan State University 19921998.
Contact Information
tel: 410 293 6708 (office)
Email: popovici@usna.edu
Research Interests: Dynamical Systems and Image Analysis

Agent Based Dynamical Systems
I am currently interested in the rigurous study of the stability of swarms, which are highdimensional autonomous nonlinear systems. The model of the particles' motion incorporates a self propeling term describing the way agents gain kinetic energy from the environment or from within, and a coupling term describing the mutual attraction and repulsion between agents. In the simplest presentation, identical agents move in the plane, with their postion vectors r_{k }satisfying the ODEsr_{k}’’= (1r_{k}’^{2}) r_{k}’(r_{k}R) where R=(r_{1}+r_{2}+… r_{n})/n.
In a more general context the motion takes place in dimension 2 or higher, agents can be heterogeneous (i.e. the selfpropulsion of agent k is given by ∇ F_{k}(r’_{k}) rather than the identical propulsion (1r_{k}’^{2}) r_{k}’ ), the communication network is given by a matrix ( a_{jk } ) rather than the alltoone coupling through the center of mass, and the attractionrepulsion is given by ∇U(r_{k} r_{j})(r’_{k}r’_{j}), where U is a general potential (rather than the parabolic potential U= r^{T}r.) The posted publications, ONR funding and presentations in this project were developed in collaboration with Prof. Medynets (with the exception of the earlier undertakings in flocking and oscillators listed under student research section).
Presentations
ONR Nonlinear Physics 2022 Annual Review
BIRS Complex & Nonlinear SystemsAnnimations of the trajectories for r_{k}’’= (1r_{k}’^{2}) r_{k}’(r_{k}R) where R=(r_{1}+r_{2}+… r_{n})/n.
Given that the propulsion is scaled to unit speed, one might expects that the system evolves toward states where particles move at unit speed. Necessarily, if all agents have unit speed, the system' center of mass has zero acceleration. Excluding 1dimensional motion, those configurations either have a stationary center of mass, leading to agents rotating about the center, or have the center of mass move reclilinearly with unit speed. The first annimation illustrates trajectories whose limit configuration has a stationary center of mass; the second with center of mass moving at unit speed; the last has R''≠0 (mixed state).
Rotating State:
Translating State:
Mixed State Oscillations
Publications
On the stability of a multiagent system satisfying a generalized Lienard equation
On Spatial Cohesiveness of SecondOrder SelfPropelled Swarming Systems
Student ResearchThe papers are available from the USNA intranet https://www.usna.edu/MathDept/academics/MidnResearch/honorspapers.php
"Dynamical Systems with Delayed Response" by Rachel Manhertz,
"Impulse Differential Equations with Applications to the Pulse Vaccination Strategy of the SIR Model" by Joe Spirnak,
"Synchronization of Coupled Nonlinear Oscillators: Exploring the Asymmetry of EastWest Jet Lag" by Hunter McGavran,
"A Proof of the Isoperimetric Inequality with Riemann Mapping" by Bennett Marston,
"A study of the periodic orbits of an onedimensional model of cardiac rhythm" by Michael Spoja,
"Qualitative Examinations of Systems of Ordinary Differential Equations with Applications in Human Physiology" by Michael B. Lemonick,
"The Mathematics of Flocking: Examining and Simulating Models for Emergent Behavior" by Thomas Cleary. 
Piecewise SmoothDynamical Systems

Image Analysis