Synchronization in Networks of Coupled Harmonic Oscillat.pptVIP

Synchronization in Networks of Coupled Harmonic Oscillators with Stochastic Perturbation and Time Delays 尚轶伦 上海交通大学 数学系 Outline Introduction ● Backgrounds ● Problem formulation Main result ● Synchronization of coupled harmonic oscillators Methods of proof Numerical examples Synchronized oscillators Cellular clocks in the brain Pacemaker cells in the heart Pedestrians on a bridge Electric circuits Laser arrays Oscillating chemical reactions Bubbly fluids Neutrino oscillations Synchronous firings of male fireflies Kuramoto model We want to study synchronization conditions for coupled harmonic oscillators over general directed topologies with noise perturbation and communication time delays. Basic definitions For a matrix A, let ||A||=sup{ ||Ax||: ||x||=1}. ||.|| is the Euclidean norm. Let G=(V, E, A) be a weighted digraph with vertex set V={1, 2,..., n} and edge set E. An edge (j, i) ∈ E if and only if the agent j can send information to the agent i directly. The in-degree neighborhood of the agent i : Ni ={ j∈ V : (j, i) ∈ E}. A=(aij) ∈Rn×n is the weighted adjacency matrix of G. aij 0 if and only if j ∈ Ni. D=diag(d1,..., dn) with di=|Ni|. The Laplacian matrix L=(lij) =D-A. Our model Consider n coupled harmonic oscillators connected by dampers and each attached to fixed supports by identical springs with spring constant k. The dynamical system is described as xi’’+kxi+∑j∈Niaij(xi’-xj’)=0 for i=1,…, n where xi denotes the position of the ith oscillator, k is a positive gain, and aij characterizes interaction between oscillators i and j. Here, we study a leader-follower version of the above system. Communication time delay and stochastic noises during the propagation of information from oscillator to oscillator are introduced. xi’’(t)+kxi(t)+∑j∈Niaij(xi’(t-s)-xj’(t-s))+bi(xi’(t