2010/02/03 by J. J. P. Veerman, Veerman, J. J. P., F. M. Tangerman +1 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #Control and Stability of Dynamical Systems #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS) #Quantum chaos and dynamical systems #nlin.AO #nlin.PS
paper · pdf · doi:10.48550/arxiv.1002.0782
13 pages, 1 figure
arxiv created 2010/02/03 · openalex publication_date 2010/02/03 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a string of N+1 damped oscillators moving on the line of which the motion of the first (called the "leader") is independent of the others. Each of the followers `observes' the relative velocity and position of only its nearest neighbors. Inasmuch as these are different from 0, this information is then used to determine its own acceleration. Fix all parameters except the number N in such a way that the system is asymptotically stable. Now as N tends tends we consider the following problem. At t=0 the leader gets kicked and starts moving with unit velocity away from the flock. Due to asymptotic stability the followers will eventually fall in behind the leader and travel each at its own predetermined distance from the leader. In this note we conjecture that before equilibrium ensues, the perturbations to the orbit of the last oscillator grow exponentially in N except when there is a symmetry in the interactions and the growth is then linear in N. There are two cases. We prove the conjecture in one case, and give a strong heuristic argument in the other.