New Estimations of Bounds for the Chu Chaotic System

This paper is concerned with positive invariant set for the Chu chaotic system using a technique combing the generalized Lyapunov function theory and maximum principle. For this chaotic system, some new ellipsoid estimations and a cylindrical domain estimation of the globally exponentially attractive set for all the positive parameters values of the system are derived without existence assumptions via employing inequality techniques and the continuous extension principle.

Observation of quasi mono-energetic electron bunches in the new ellipsoid cavity model

In this work, we introduce a new ellipsoid model to describe bubble acceleration of electrons and discuss the required conditions of forming it. We have found that the electron trajectory is strongly related to background electron energy and cavity potential ratio. In the ellipsoid cavity regime, the quality of the electron beam is improved in contrast to other methods, such as that using periodic plasma wakefield, spherical cavity regime, and plasma channel guided acceleration. The trajectory of the electron motion can be described as hyperbola, parabola, or ellipsoid path. It is influenced by the position and energy of the electrons and the electrostatic potential of the cavity. In the experimental part of this work, a 20 TW power and 30 fs laser pulse was focused on a pulsed He gas jet. We have focused the laser pulse in the best matched point above the nozzle gas to obtain a stable ellipsoid bubble. The finding of the optimum points will be described in analytical details.

Extremum properties of the new ellipsoid

For a convex body $ K $ in $ {\mathbb R}^{n} $, Lutwak, Yang and Zhang defined a new ellipsoid $ \Gamma_{-2}K $, which is the dual analog of the Legendre ellipsoid. In this paper, we prove the following two results: (i) For any origin-symmetric convex body $ K $, there exist an ellipsoid $ E $ and a parallelotope $ P $ such that $ \Gamma_{-2}E \supseteq \Gamma_{-2}K \supseteq \Gamma_{-2}P $ and $ V(E)=V(K)=V(P) $; (ii) For any convex body $K$ whose John point is at the origin, then there exists a simplex $T$ such that $ \Gamma_{-2}K \supseteq \Gamma_{-2}T $ and $ V(K)=V(T) $.