复动力系统的空间分形控制与混沌同步
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摘要
复动力系统主要研究解析函数的迭代问题,它的主要研究对象Julia集一般具有分形结构,而产生Julia集的迭代动力系统具有混沌特性,因此它与混沌、分形等研究领域有着紧密的联系.本论文主要对复动力系统的定性理论、控制和同步进行了一系列的基础研究,主要包括空间Julia集的控制和同步以及混沌复系统的同步.
1.耦合映像格子模型的空间Julia集的控制与同步
基于平面分形集-Julia集的吸引不动点的理论,给出了耦合映像格子模型的空间Julia集不动平面的稳定条件.在不动平面已知的情况下,分别利用梯度控制和辅助参考控制的方法,使耦合映像格子模型的空间Julia集的不动平面达到稳定.但在很多情况下不动平面不易求出,对于这样的系统可以利用最优函数控制方法,控制其不稳定不动平面为稳定平面.继而,分别利用梯度控制和最优函数控制方法,控制两个不同的耦合映像格子模型的空间Julia集的轨迹实现同步,从而实现了空间Julia集的同步.最后,利用线性耦合的方法,实现两个不同的耦合映像格子模型的空间Julia集的轨迹的耦合同步,进而实现空间Julia集的耦合.
2.空间混沌Julia集的控制与同步
基于平面分形集-Julia集的基本性质,给出了空间混沌Julia集的性质和其不动平面稳定的复参数c的稳定区域.利用辅助参考控制的方法,根据不动平面稳定的条件,获得控制参数的范围,以实现空间混沌Julia集的控制.同时,给出了两个不同空间混沌Julia集的广义同步定义,利用线性反馈的方法,使其广义同步误差系统镇定到稳定的不动平面,实现了不同空间混沌Julia集的线性广义同步.通过构造多变量多项式的变换,并利用非线性反馈的方法,实现了不同空间混沌Julia集的非线性广义同步.
3.新的复类Lorenz系统的基本性质及其反同步
基于混沌实系统的判定和基本性质,构造了一个新的复类Lorenz系统,详细地分析了它的基本动态特性,并分别利用主动控制和反馈控制的方法实现了新的复类Lorenz系统的反同步.这两种方法具有控制器设计过程简单易行、计算量小等特点,但反馈控制优于主动控制方法,因为反馈控制更易于达到人们的目的,计算更简单.数值仿真验证了这两种方法实现新的复类Lorenz系统的反同步是可行的.
4.含有未知参数的混沌复系统的自适应反同步
用一般的数学表达式描述了带有未知参数的不确定混沌复系统.基于Lyapunov稳定理论,利用自适应控制方法,实现了这类系统的反同步,并给出了自适应控制器和未知参数估计的具体表达式,并将结论分别成功地应用于实现两个不同和相同的不确定混沌复系统的反同步.
5.基于状态观测器的混沌复系统的投影同步
首先假设了不确定混沌复系统的输出,并基于其输出构造了此混沌复系统的非线性状态观测器.根据极点配置方法和Lyapunov稳定理论,得到了反馈增益矩阵,使该混沌复系统和其状态观测器实现投影同步.将该结论成功地应用于两个带有不确定项的混沌复系统.
6.含有扰动的混沌复系统的鲁棒自适应全状态混合投影同步
基于混沌实系统的全状态混合投影同步的定义,给出了混沌复系统全状态混合投影同步的定义.用一般的数学表达式描述了带有扰动的不确定混沌复系统,然后基于Lyapunov稳定理论和构造的补偿器,利用非线性控制和自适应控制方法,设计了实现混合投影同步的控制器和参数自适应率,给出了具体的数学表达式,并实现了不确定混沌复系统的全状态混合投影同步,使其状态误差收敛到一个较小的界内.最后将这个结论分别成功地应用于实现两个不同和相同的带有外部扰动的不确定混沌复系统的全状态混合投影同步.
综上,本论文围绕空间分形集和混沌复系统展开了研究,将控制引入到空间分形,对进一步研究空间分形和解释复杂的现象具有重要的意义；实现了混沌复系统的多类同步,为进一步加强通讯安全奠定了理论基础.
The complex dynamical system mainly studies the iteration of analytic func-tions. Its main object is the Julia set with a fractal structure generally, and the map used to produce Julia sets is chaotic. So it is closely linked with chaos and frac-tal. This paper focuses on the qualitative theory of the complex dynamical system and a series of basic researches on its control and synchronization, including con-trol and synchronization of spatial Julia sets and synchronization of chaotic complex systems.
1. Control and synchronization of Julia sets in coupled map lattice
Based on the stability theory of fixed points for the classical Julia set, the stable condition of the fixed plane for the Julia set in coupled map lattice was given. As the fixed plane was known, the gradient control and auxiliary reference control was respectively used to control the stability of the fixed plane. But in many practical circumstances, the fixed plane was not easily obtained. For such systems, the optimal function control is applied to control the stability. In addition, the synchronization of two different Julia sets in coupled map lattice was also achieved by synchronizing their movement trajectories using the gradient control and optimal function control respectively. At last, the coupling of two different Julia sets in coupled map lattice was also analyzed by coupling their movement trajectories using the linear coupling.
2. Control and synchronization of spatial chaotic Julia sets
Based on the basic properties of the classical Julia set, the properties of spa-tial chaotic Julia sets and the stable regions of the complex parameter c were given. According to the stable conditions of the fixed plane, the scope of the control pa-rameter was obtained by using the auxiliary reference control so as to control the spatial chaotic Julia set. Moreover, the definition of the generalized synchronization between two different spatial chaotic Julia sets was given. The linear generalized synchronization of spatial Julia sets was achieved by linear feedback control. In addition, the nonlinear generalized synchronization of spatial chaotic Julia sets was also analysed by constructing a multivariate polynomial transformation and using nonlinear feedback control.
3. Anti-synchronization of a new complex Lorenz-like system and its dynami-cal properties
Based on chaotic real systems and their basic properties, a new complex Lorenz-like system was constructed and its dynamical properties was also discussed. The anti-synchronization of the new complex Lorenz-like systems was separately inves-tigated by active control and nonlinear control methods. Although the both methods used to achieve the anti-synchronization of the new complex Lorenz-like system were simple, nonlinear control was preferable for personal purposes and simpler for computations. Numerical simulations verified that both methods are effective.
4. Adaptive anti-synchronization of chaotic complex systems with unknown parameters
The adaptive anti-synchronization of a class of chaotic complex systems with fully uncertain parameters, which were described by a united mathematical expres-sion was presented. Based on Lyapunov stability theory, we developed an adaptive control scheme and adaptive laws of parameters to anti-synchronize two unknown chaotic complex systems. The anti-synchronization of two identical complex Lorenz systems and two different complex Chen and Lu systems were taken as two exam-ples to verify the feasibility and effectiveness of the presented scheme.
5. Observer-based projective synchronization of chaotic complex systems
Based on the assumed output of the uncertain chaotic complex system, its ob-server was designed. According to the nonlinear state observer and the pole place-ment technique, we got the feedback gain matrix and achieved the projective syn-chronization between uncertain chaotic complex systems and its observer. The pro-posed synchronization scheme was confirmed by numerical simulations of two well known chaotic complex systems.
6. Robust adaptive full state hybrid projective synchronization of chaotic com- plex systems with unknown parameters and external disturbances
Based on the definition of full state hybrid projective synchronization (FSHPS) of chaotic real system, we gave the definition of FSHPS of chaotic complex system. By introducing a dynamic compensator and using nonlinear control and adaptive control, we proposed the robust adaptive FSHPS scheme, which can achieve adap-tive FSHPS of two different chaotic complex systems asymptotically with a small error bound. The adaptive laws of the unknown parameters were given, and the suf-ficient conditions of realizing FSHPS were derived as well. Finally, the proposed control scheme was successfully applied to two identical chaotic complex systems and two different chaotic complex systems.
In conclusion, this dissertation focuses on the controls of spatial fractal sets and synchronization of chaotic complex systems. The control was introduced successful-ly into the spatial Fractal, which had important practical significance to further study the spatial fractal and explain the more complex phenomena. All kinds of synchro-nization of chaotic complex systems was achieved firstly, which provided theoretical basis for further enhancing the security of communications.
1.耦合映像格子模型的空间Julia集的控制与同步
基于平面分形集-Julia集的吸引不动点的理论,给出了耦合映像格子模型的空间Julia集不动平面的稳定条件.在不动平面已知的情况下,分别利用梯度控制和辅助参考控制的方法,使耦合映像格子模型的空间Julia集的不动平面达到稳定.但在很多情况下不动平面不易求出,对于这样的系统可以利用最优函数控制方法,控制其不稳定不动平面为稳定平面.继而,分别利用梯度控制和最优函数控制方法,控制两个不同的耦合映像格子模型的空间Julia集的轨迹实现同步,从而实现了空间Julia集的同步.最后,利用线性耦合的方法,实现两个不同的耦合映像格子模型的空间Julia集的轨迹的耦合同步,进而实现空间Julia集的耦合.
2.空间混沌Julia集的控制与同步
基于平面分形集-Julia集的基本性质,给出了空间混沌Julia集的性质和其不动平面稳定的复参数c的稳定区域.利用辅助参考控制的方法,根据不动平面稳定的条件,获得控制参数的范围,以实现空间混沌Julia集的控制.同时,给出了两个不同空间混沌Julia集的广义同步定义,利用线性反馈的方法,使其广义同步误差系统镇定到稳定的不动平面,实现了不同空间混沌Julia集的线性广义同步.通过构造多变量多项式的变换,并利用非线性反馈的方法,实现了不同空间混沌Julia集的非线性广义同步.
3.新的复类Lorenz系统的基本性质及其反同步
基于混沌实系统的判定和基本性质,构造了一个新的复类Lorenz系统,详细地分析了它的基本动态特性,并分别利用主动控制和反馈控制的方法实现了新的复类Lorenz系统的反同步.这两种方法具有控制器设计过程简单易行、计算量小等特点,但反馈控制优于主动控制方法,因为反馈控制更易于达到人们的目的,计算更简单.数值仿真验证了这两种方法实现新的复类Lorenz系统的反同步是可行的.
4.含有未知参数的混沌复系统的自适应反同步
用一般的数学表达式描述了带有未知参数的不确定混沌复系统.基于Lyapunov稳定理论,利用自适应控制方法,实现了这类系统的反同步,并给出了自适应控制器和未知参数估计的具体表达式,并将结论分别成功地应用于实现两个不同和相同的不确定混沌复系统的反同步.
5.基于状态观测器的混沌复系统的投影同步
首先假设了不确定混沌复系统的输出,并基于其输出构造了此混沌复系统的非线性状态观测器.根据极点配置方法和Lyapunov稳定理论,得到了反馈增益矩阵,使该混沌复系统和其状态观测器实现投影同步.将该结论成功地应用于两个带有不确定项的混沌复系统.
6.含有扰动的混沌复系统的鲁棒自适应全状态混合投影同步
基于混沌实系统的全状态混合投影同步的定义,给出了混沌复系统全状态混合投影同步的定义.用一般的数学表达式描述了带有扰动的不确定混沌复系统,然后基于Lyapunov稳定理论和构造的补偿器,利用非线性控制和自适应控制方法,设计了实现混合投影同步的控制器和参数自适应率,给出了具体的数学表达式,并实现了不确定混沌复系统的全状态混合投影同步,使其状态误差收敛到一个较小的界内.最后将这个结论分别成功地应用于实现两个不同和相同的带有外部扰动的不确定混沌复系统的全状态混合投影同步.
综上,本论文围绕空间分形集和混沌复系统展开了研究,将控制引入到空间分形,对进一步研究空间分形和解释复杂的现象具有重要的意义；实现了混沌复系统的多类同步,为进一步加强通讯安全奠定了理论基础.
The complex dynamical system mainly studies the iteration of analytic func-tions. Its main object is the Julia set with a fractal structure generally, and the map used to produce Julia sets is chaotic. So it is closely linked with chaos and frac-tal. This paper focuses on the qualitative theory of the complex dynamical system and a series of basic researches on its control and synchronization, including con-trol and synchronization of spatial Julia sets and synchronization of chaotic complex systems.
1. Control and synchronization of Julia sets in coupled map lattice
Based on the stability theory of fixed points for the classical Julia set, the stable condition of the fixed plane for the Julia set in coupled map lattice was given. As the fixed plane was known, the gradient control and auxiliary reference control was respectively used to control the stability of the fixed plane. But in many practical circumstances, the fixed plane was not easily obtained. For such systems, the optimal function control is applied to control the stability. In addition, the synchronization of two different Julia sets in coupled map lattice was also achieved by synchronizing their movement trajectories using the gradient control and optimal function control respectively. At last, the coupling of two different Julia sets in coupled map lattice was also analyzed by coupling their movement trajectories using the linear coupling.
2. Control and synchronization of spatial chaotic Julia sets
Based on the basic properties of the classical Julia set, the properties of spa-tial chaotic Julia sets and the stable regions of the complex parameter c were given. According to the stable conditions of the fixed plane, the scope of the control pa-rameter was obtained by using the auxiliary reference control so as to control the spatial chaotic Julia set. Moreover, the definition of the generalized synchronization between two different spatial chaotic Julia sets was given. The linear generalized synchronization of spatial Julia sets was achieved by linear feedback control. In addition, the nonlinear generalized synchronization of spatial chaotic Julia sets was also analysed by constructing a multivariate polynomial transformation and using nonlinear feedback control.
3. Anti-synchronization of a new complex Lorenz-like system and its dynami-cal properties
Based on chaotic real systems and their basic properties, a new complex Lorenz-like system was constructed and its dynamical properties was also discussed. The anti-synchronization of the new complex Lorenz-like systems was separately inves-tigated by active control and nonlinear control methods. Although the both methods used to achieve the anti-synchronization of the new complex Lorenz-like system were simple, nonlinear control was preferable for personal purposes and simpler for computations. Numerical simulations verified that both methods are effective.
4. Adaptive anti-synchronization of chaotic complex systems with unknown parameters
The adaptive anti-synchronization of a class of chaotic complex systems with fully uncertain parameters, which were described by a united mathematical expres-sion was presented. Based on Lyapunov stability theory, we developed an adaptive control scheme and adaptive laws of parameters to anti-synchronize two unknown chaotic complex systems. The anti-synchronization of two identical complex Lorenz systems and two different complex Chen and Lu systems were taken as two exam-ples to verify the feasibility and effectiveness of the presented scheme.
5. Observer-based projective synchronization of chaotic complex systems
Based on the assumed output of the uncertain chaotic complex system, its ob-server was designed. According to the nonlinear state observer and the pole place-ment technique, we got the feedback gain matrix and achieved the projective syn-chronization between uncertain chaotic complex systems and its observer. The pro-posed synchronization scheme was confirmed by numerical simulations of two well known chaotic complex systems.
6. Robust adaptive full state hybrid projective synchronization of chaotic com- plex systems with unknown parameters and external disturbances
Based on the definition of full state hybrid projective synchronization (FSHPS) of chaotic real system, we gave the definition of FSHPS of chaotic complex system. By introducing a dynamic compensator and using nonlinear control and adaptive control, we proposed the robust adaptive FSHPS scheme, which can achieve adap-tive FSHPS of two different chaotic complex systems asymptotically with a small error bound. The adaptive laws of the unknown parameters were given, and the suf-ficient conditions of realizing FSHPS were derived as well. Finally, the proposed control scheme was successfully applied to two identical chaotic complex systems and two different chaotic complex systems.
In conclusion, this dissertation focuses on the controls of spatial fractal sets and synchronization of chaotic complex systems. The control was introduced successful-ly into the spatial Fractal, which had important practical significance to further study the spatial fractal and explain the more complex phenomena. All kinds of synchro-nization of chaotic complex systems was achieved firstly, which provided theoretical basis for further enhancing the security of communications.
引文
属性不符
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