一維波方程系統中的等向與非等向混沌震動

Abstract

波方程在一維閉區間I的右端有梵德波形式的非線性邊界條件。而在閉區間I的左端，當參數 時其邊界條件為能量的輸入；當 時其邊界條件為齊次諾曼條件。這個波方程系統的解與一個區間函數的疊代有關，所以我們定義當這個區間函數有李—約克的渾沌現象時稱這個波方程系統有混沌震動。因為在我們討論的波方程系統中兩特徵線的運動速度為任意兩正數c₁與 c₂，所以系統中有著等向性與非等向性的混沌震動現象產生。在這篇論文中，我們將討論波方程系統中混沌震動現象的產生分別與參數 η, c₁, c₂ 變化之間的關聯

A wave equation on a one-dimensional interval I has a van der Pol type nonlinear boundary condition at the right end. At the left end, the boundary condition is energy-injecting if the parameter η>0 and is the homogeneous Neumann condition if η=0. The solution of the wave system is corresponding to the iteration of one interval map, so we say the wave system is chaotic if the interval map is chaotic in the sense of Li-Yorke. The system which we consider contains both isotropic and nonisotropic chaotic vibrations, since the two associated families of characteristics travel with two speeds c₁, c₂ for any given positive c₁, c₂. In this paper, we discuss that the chaotic vibrations of the 1D wave system occur when the paramerters η, c₁, c₂ vary separately.