强流加速器中束晕—混沌的小波函数控制和超混沌电路的广义同步的研究
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摘要
本文在综述混沌控制和混沌同步方法研究概况的基础上，主要做了两方面的工作。
一方面，研究了强流加速器中离子运动表现出的一种非线性极强的时空混沌—“束晕-混沌”的控制问题，在给出强流离子束在周期性聚焦磁场通道中的有关动力学方程和相关物理概念基础上，通过选择下列两种小波函数构造非线性反馈控制器。
· 。广 。 飞 -((—^f
人‘(<)嚣三11—三(<—凸)’卜 。 (1)
at a j
，? —生止 ，
人‘(()=e ‘ (2)提出了用统计量离子束均方根半径r_rms作为反馈变量，并取如下形式的函数控制器G：
G=扰人bk一)—人‘(／bo)1 (3)利用式(1)，式(2)分别构造两类小波函数反馈控制器，控制器函数加在离子径向所受束自生电场力方程的右边，使成
／r=—gV毋‘卜，5)+G (4)并借助用于模拟周期性聚焦磁场通道离子束运动的多离子数值模拟程序(PIC程序)进行控制模拟。本文又引入统计平均量来衡量控制效果。我们首先研究了用式(1)构造的小波函数控制器对束晕-混沌进行连续控制，接着又研究了用式(2)构造的小波函数反馈控制器实现多周期间隔反馈控制束晕-混沌的方法。经过大量的数值模拟，模拟结果表明，在连续控制情况下，只要选择适当的控制参数，能对五种初始分布不同的离子束的束晕-混沌进行很好地控制。在多周期间隔控制条件下，只要选择适当的控制参数和间隔周期数，以控制初始条件遵从Kapchinskij-Vladimirskij分布(简称K-V分布)的离子束为例进行模拟，也能获得令人满意的控制效果。无论连续控制还是多周期间隔控制都能减少离子横向运动的能量，大大抑制离子向外逃逸的倾向和克服用传统挡板准直方法控制束晕存在的束晕再生现象的缺陷，即能控制束晕一直为零。这一理想数值大大优于文[73]晕度控制在0.1078左右的结果，也明显满足强流加速器10~(-4)数量级的实际最低限制。本文的研究结果为实用强流加速器的研制提供了一定价值的参考。
另一方面，本文又研究了超混沌电路的广义同步。理论分析，只要从驱动系统的状态方程出发，构造出一个特征值实部全为负数的矩阵和一个与之相对易的矩阵用以构造响应系统，通过信号驱动，就能实现驱动系统与响应系统的超混沌广义同步。本文以一个四维超混
饨振荡电路为例，给出了广义同步的数值模拟结果。研究结果可为实际的超混淹电路实现广
，义同步，提高混饨保密通信的抗破译能力提供有益的参考。
In this paper, we study two works on the basis of giving summary of the development in controlling chaos and chaos synchronization.
In the first part, we study the problem of controlling "beam halo-chaos" for high intensity ion beam in period-focusing channel .Some dynamic equations and correlative physical concerts of describing high intensity ion beam in period-focusing channel are introduced first , then two kinds of wavelet functions are selected ,whose terms written as follows respectively, to build nonlinear feedback control functions.
(~ b)2
a a J
fah(4=e a (2)
The beam root-mean-square radius (rr,.,,s) is taken as feedback variable ,the general form of our control function G is presented as follows:
(3)
two types of wavelet feedback control functions can be achieved by mean of the equation (I) and the equation (2). G is added into the right of ion radial self-electric force equation .so
Fr =-qV~(r.s)--G (4)
Using multi-particle code (particle-in-cell(PIC)code)for simulating the motion of the ion beam in the period-focusing channel controlled or not .Wè employ some statistical physical quantities to describe the control effects .Firstlv. we study continpous control with the control function corresponded to the equation (1); secondly. we explore the multi-periodical intervals control method with the control function corresponded to the equation (2) .The simulation results demonstrate that five different kinds of primary ion beams can be controlled well by selecting control parameters of continuous control .In the condition of multi-periodical intervals control ,the beam with the Kapchiskij-Vladimirskij(K-V)distribution is taken as a typical example ,good control result can also be achieved by selecting the proper control parameters and the proper number of control multi-periodical intervals.
Both the continuous control and the multi-periodical intervals feedback control can reduce the energy of radial motion restrict its tendency to run away from the channel , and overcome the
shortcoming that the halo regeneration phenomena as scraping halos by some conventional methods, e.g., by the means of collimator or shelves system, namely, zero halo intensity can be obtain always .This perfect result is much better than the approximate amount of 0.1087 that provided in reference [73],our result is agree with the minimum limitation( I 04)of halo intensity of factual high intensity accelerator .The research result offers a valuable reference for the application in practicable intensity accelerator.
In the second part ,we study the generalized synchronization of hyperchaotic circuit .Theoretical analysis demonstrates that driven system can be achieved if a matrix whose eigenvalues' real part is negative and its commutative matrix can be built .When transmitting the signal to the driven system ,generalized synchronization of driving system and driven system can be realized ,A 4-D hyperchaotic oscillator is taken as a typical example .The results of numerical simulation are presented ,the research results offer valuable reference for realizing the practical generalized synchronization of hyperchaotic circuit ,improving the degree of security in the field of chaotic secure communication.
一方面，研究了强流加速器中离子运动表现出的一种非线性极强的时空混沌—“束晕-混沌”的控制问题，在给出强流离子束在周期性聚焦磁场通道中的有关动力学方程和相关物理概念基础上，通过选择下列两种小波函数构造非线性反馈控制器。
· 。广 。 飞 -((—^f
人‘(<)嚣三11—三(<—凸)’卜 。 (1)
at a j
，? —生止 ，
人‘(()=e ‘ (2)提出了用统计量离子束均方根半径r_rms作为反馈变量，并取如下形式的函数控制器G：
G=扰人bk一)—人‘(／bo)1 (3)利用式(1)，式(2)分别构造两类小波函数反馈控制器，控制器函数加在离子径向所受束自生电场力方程的右边，使成
／r=—gV毋‘卜，5)+G (4)并借助用于模拟周期性聚焦磁场通道离子束运动的多离子数值模拟程序(PIC程序)进行控制模拟。本文又引入统计平均量来衡量控制效果。我们首先研究了用式(1)构造的小波函数控制器对束晕-混沌进行连续控制，接着又研究了用式(2)构造的小波函数反馈控制器实现多周期间隔反馈控制束晕-混沌的方法。经过大量的数值模拟，模拟结果表明，在连续控制情况下，只要选择适当的控制参数，能对五种初始分布不同的离子束的束晕-混沌进行很好地控制。在多周期间隔控制条件下，只要选择适当的控制参数和间隔周期数，以控制初始条件遵从Kapchinskij-Vladimirskij分布(简称K-V分布)的离子束为例进行模拟，也能获得令人满意的控制效果。无论连续控制还是多周期间隔控制都能减少离子横向运动的能量，大大抑制离子向外逃逸的倾向和克服用传统挡板准直方法控制束晕存在的束晕再生现象的缺陷，即能控制束晕一直为零。这一理想数值大大优于文[73]晕度控制在0.1078左右的结果，也明显满足强流加速器10~(-4)数量级的实际最低限制。本文的研究结果为实用强流加速器的研制提供了一定价值的参考。
另一方面，本文又研究了超混沌电路的广义同步。理论分析，只要从驱动系统的状态方程出发，构造出一个特征值实部全为负数的矩阵和一个与之相对易的矩阵用以构造响应系统，通过信号驱动，就能实现驱动系统与响应系统的超混沌广义同步。本文以一个四维超混
饨振荡电路为例，给出了广义同步的数值模拟结果。研究结果可为实际的超混淹电路实现广
，义同步，提高混饨保密通信的抗破译能力提供有益的参考。
In this paper, we study two works on the basis of giving summary of the development in controlling chaos and chaos synchronization.
In the first part, we study the problem of controlling "beam halo-chaos" for high intensity ion beam in period-focusing channel .Some dynamic equations and correlative physical concerts of describing high intensity ion beam in period-focusing channel are introduced first , then two kinds of wavelet functions are selected ,whose terms written as follows respectively, to build nonlinear feedback control functions.
(~ b)2
a a J
fah(4=e a (2)
The beam root-mean-square radius (rr,.,,s) is taken as feedback variable ,the general form of our control function G is presented as follows:
(3)
two types of wavelet feedback control functions can be achieved by mean of the equation (I) and the equation (2). G is added into the right of ion radial self-electric force equation .so
Fr =-qV~(r.s)--G (4)
Using multi-particle code (particle-in-cell(PIC)code)for simulating the motion of the ion beam in the period-focusing channel controlled or not .Wè employ some statistical physical quantities to describe the control effects .Firstlv. we study continpous control with the control function corresponded to the equation (1); secondly. we explore the multi-periodical intervals control method with the control function corresponded to the equation (2) .The simulation results demonstrate that five different kinds of primary ion beams can be controlled well by selecting control parameters of continuous control .In the condition of multi-periodical intervals control ,the beam with the Kapchiskij-Vladimirskij(K-V)distribution is taken as a typical example ,good control result can also be achieved by selecting the proper control parameters and the proper number of control multi-periodical intervals.
Both the continuous control and the multi-periodical intervals feedback control can reduce the energy of radial motion restrict its tendency to run away from the channel , and overcome the
shortcoming that the halo regeneration phenomena as scraping halos by some conventional methods, e.g., by the means of collimator or shelves system, namely, zero halo intensity can be obtain always .This perfect result is much better than the approximate amount of 0.1087 that provided in reference [73],our result is agree with the minimum limitation( I 04)of halo intensity of factual high intensity accelerator .The research result offers a valuable reference for the application in practicable intensity accelerator.
In the second part ,we study the generalized synchronization of hyperchaotic circuit .Theoretical analysis demonstrates that driven system can be achieved if a matrix whose eigenvalues' real part is negative and its commutative matrix can be built .When transmitting the signal to the driven system ,generalized synchronization of driving system and driven system can be realized ,A 4-D hyperchaotic oscillator is taken as a typical example .The results of numerical simulation are presented ,the research results offer valuable reference for realizing the practical generalized synchronization of hyperchaotic circuit ,improving the degree of security in the field of chaotic secure communication.
引文
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