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 , 1998-2001.
• Visiting Assistant Professor, Rice Univ. and UCLA, Summer 2002
• Teaching Assistant, Michigan State University 1992-1998.

Contact Information

tel: 410 293 6708 (office) 

E-mail: popovici@usna.edu

Research Interests: Dynamical Systems and Image Analysis 
         

1. Agent- Based Dynamical Systems 

   I am currently interested in the rigurous study of the stability of swarms, which are high-dimensional 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 ODEs

   r_k’’= (1-|r_k’|²) r_k’-(r_k-R) where R=(r₁+r₂+… r_n)/n.

    In a more general context the motion takes place in dimension 2 or higher, agents can be heterogeneous (i.e. the self-propulsion of agent k is given by ∇ F_k(r’_k) rather than the identical propulsion (1-|r_k’|²) r_k’ ), the communication network is given by a matrix  ( a_jk  ) rather than the all-to-one coupling through the center of mass,  and the attraction-repulsion 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^Tr.) 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 Systems
   
   

   Annimations of the trajectories for r_k’’= (1-|r_k’|²) r_k’-(r_k-R) where R=(r₁+r₂+… 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 1-dimensional 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 multi-agent system satisfying a generalized Lienard equation
          On Spatial Cohesiveness of Second-Order Self-Propelled Swarming Systems
   

   
   Student Research 

   The 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 East-West 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 one-dimensional 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.

2. Piece-wise SmoothDynamical Systems
   
   

3. Image Analysis