arXiv subjects

Irina Popovici

Publications and source records attributed to Irina Popovici.

3 recordsLinked to original sources

arXiv2024

Stability of Translating States for Self-propelled Swarms with Quadratic Potential

The main result of this paper is proving the stability of translating states (flocking states) for the system of $n$-coupled self-propelled agents governed by $\ddot r_k = (1-|\dot r_k|^2)\dot r_k - \frac{1}{n}\sum_{j=1}^n(r_k-r_j)$, $r_k\in \mathbb R^2$. A flocking state is a solution where all agents move with identical velocity, of magnitude one. Numerical explorations have shown that for a large set of initial conditions, after some drift, the particles' velocities align, and the distance between agents tends to zero. We prove that every solution starting near a translating state asymptotically approaches a translating state nearby, an asymptotic behavior exclusive to swarms in the plane. We quantify the rate of convergence for the directional drift, the mean field speed, and the oscillations in the direction normal to the motion. The latter decay at a rate of $1/ \sqrt t$, mimicking the oscillations of some systems with almost periodic coefficients and cubic nonlinearities. We give sufficient conditions for that class of systems to have an asymptotically stable origin.

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arXiv2022

On Spatial Cohesiveness of Second-Order Self-Propelled Swarming Systems

The study of emergent behavior of swarms is of great interest for applied sciences. One of the most fundamental questions for self-organizing swarms is whether the swarms disperse or remain in a spatially cohesive configuration. In the paper we study dissipativity properties and spatial cohesiveness of the swarm of self-propelled particles governed by the model $\ddot r_k = -p_k(|\dot r_k|)\dot r_k - \sum_m a_{k,m}r_m$, where $r_k\in \mathbb R^d$, $k=1,\ldots,n$, and $A = \{a_{k,m}\}$ is a symmetric positive-semidefinie matrix. The self-propulsion term is assumed to be continuously differentiable and to grow faster than $1/z$, that is, $p_k(z)z\to\infty $ as $z\to\infty$. We establish that the velocity and acceleration of the particles are ultimately bounded. We show that when $\ker (A)$ is trivial, the positions of the particles are also ultimately bounded. For systems with $\ker (A)\neq \{0\}$, we show that, while the system might infinitely drift away from its initial location, the particles remain within a bounded distance from the generalized center of mass of the system, which geometrically coincides with the weighted average of agent positions. The weights are determined by the coefficients of the projection matrix onto $\ker (A)$. We also include the proof of the ultimate boundedness of velocities and accelerations for systems with bounded coupling, including systems coupled via the Morse potential. In our proof we switch to the velocity-acceleration coordinates and focus on the study of dissipativity properties for a more general class of Liénard systems $\ddot x_k = -\mathbb F_k(x_k)\cdot \dot x_k -\sum_{m} a_{k,m}x_m$, $k=1,\ldots,n$, $\mathbb F_k(x) = \nabla F_k(x)$ with $F_k: \mathbb R^d\rightarrow \mathbb R^d$ given by $F_k(x) = p_k(|x|)x$.

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arXiv2021

On the stability of Rotating States in Second-Order Self-Propelled Multi-Particle Systems

In this paper, we study the dynamics of a system of $n$ coupled, self-propelled particles: $\ddot r_k = (\alpha-\beta |\dot r_k|^2)\dot r_k - \frac{\gamma}{n}\sum_{m=1}^n(r_k-r_m)$, $r_k\in \mathbb R^2.$ Numerical experiments indicate that, for a large set of initial conditions, after an initial drift, the center of mass converges to a stationary point, with each particle eventually rotating around it with constant angular velocity. The distribution of particles on the circle need not be uniform. These limit configurations, where all particles rotate in the same direction, are termed {\it rotating states} . We prove that rotating states are stable and that every solution that starts sufficiently close, asymptotically approaches a rotating state, exponentially fast if $n$ is odd, or at a rate that may be exponential or $\frac{1}{\sqrt t} $ if $n$ is even. The proof uses a new approximation technique for the flow on the center manifold in the presence of non-isolated fixed points.

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