正弦平方势与掺杂超晶格双稳态系统的全局分叉

Sine-squared Potential and Global Bifurcation for Dopping Superlattice as Bistable State Cell

掺杂超晶格是对同一材料交替掺入n-型和p-型杂质,形成n-i-p-i-n-i-p-i…一维阵列的周期结构。由于交替掺杂,衬底材料的导带受到周期调制形成一个个十分类似于正弦平方形式的量子阱。引入正弦平方势,在经典力学框架内,把粒子的运动方程化为具有阻尼项和受迫项的广义摆方程。用Jacobian椭圆函数和第一类全椭圆积分找到了无扰动系统的解和粒子振动周期,利用Melnikov方法分析了系统的全局分叉与Smale马蹄变换意义上的混沌行为,给出了系统通过级联分叉进入混沌的临界值。结果表明,对于异宿轨道,当参数满足条件πqε0-4μ02mV0qE0<πsechd m2/2V0ν时,系统出现了Smale马蹄变换意义上的混沌振荡。对于振荡型周期轨道,当参数满足条件πqε0-4μ02mV0qE0<πsechpK′(κ2)Kd(κm)/2V0ν[E(κ)-κ′2K(κ)]时,产生了奇阶振荡型次谐分叉。注意到系统进入混沌的临界条件与它的参数有关,只需适当调节这些参数就可以避免或控制混沌,为光学双稳态器件的设计提供了理论分析。

The called doping superlattice is that n-type and p-type impurity are doped alternately on a sub- strate material, thus forming 1-dimension periodic structure with n-i-p-i-n-i-p-i...arrange. The conductor band botton of the substrate material is modulated periodically by alternately dopping, leading to shape similar to sine-squared potential. The particle motion equation is reduced to the general pendulum equation with a dampping and a force terms in the classical mechanics frame based on the bistable state effect of the superlattice quantum well. The equations solution and the period of a particle motion are found strictily by Jacobian ellipse function and the 1 st kind ellipse integral for the non-perturbed system. The global bifurcation and a chaotic be- haours with the Smale horseshoe were analysed by Melnikov method. The critical condition through to a chaos based on a cascade bifurcation was found. It shown that the chaotic oscillation with the Smale horseshoe exits in the system for the separatrive orbit; odd-order subharmoric bifurcations found also for the oscillation periodic orbit. Noting to critical condition entered in a chaos being related to the parameters of the system, the chaos can be avoided or controlled. The theoretical analyse is provided to the design of the optical bistable state cell.