Chaos is a general phenomenon in nonlinear dynamical systems. Accelerators { storage rings in particular { in which particles are stored for 10 revolutions constitute a particularly intricate nonlinear dynamical system. (In comparison, the earth has revolved around the sun for only 10 turns.) Storage rings therefore provide an ideal testing ground for chaos physics. In fact, it is the chaos phe...

Chaos is a general phenomenon in nonlinear dynamical systems. Accelerators { storage rings in particular { in which particles are stored for 10 revolutions constitute a particularly intricate nonlinear dynamical system. (In comparison, the earth has revolved around the sun for only 10 turns.) Storage rings therefore provide an ideal testing ground for chaos physics. In fact, it is the chaos phe...

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We describe periodic coherent structures of the CGLE, called modulated amplitude waves (MAWs). MAWs of various periods P occur in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period, pairs of MAWs cease to exist via a ...

<p style='text-indent:20px;'>The notion of propagation chaos for large systems interacting particles originates in statistical physics and has recently become a central many areas applied mathematics. The present review describes old new methods as well several important results the field. models considered include McKean-Vlasov diffusion, mean-field jump Boltzmann models. first part this...

<p style='text-indent:20px;'>The notion of propagation chaos for large systems interacting particles originates in statistical physics and has recently become a central many areas applied mathematics. The present review describes old new methods as well several important results the field. models considered include McKean-Vlasov diffusion, mean-field jump Boltzmann models. first part this...

In this paper we present an adaptive multi-element generalized polynomial chaos (ME-gPC) method, which can achieve hp-convergence in random space. ME-gPC is based on the decomposition of random space and generalized polynomial chaos (gPC). Using proper numerical schemes to maintain the local orthogonality on-the-fly, we perform gPC locally and adaptively. The key idea is to combine the polynomi...