一类混沌系统的同步与控制方法研究
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摘要
混沌系统的同步与控制在众多领域中有着广阔的应用前景。自从提出混沌同步与控制以来,它在信息科学的保密通讯和生命科学的生物医学工程中起潜在作用,使得它成为当前混沌研究中的一大热点。人们对混沌同步与控制的研究,虽然已经取得了一些进展,但是还没有完全认识和掌握,混沌同步与控制研究仍然是一个科学前沿。深入研究混沌同步与控制问题,有效利用混沌理论服务人类社会,是一个意义重大而又影响深远的课题,已引起人们的极大关注。本文针对混沌系统同步与控制做了深入研究,提出了一类新的复杂混沌系统及其吸引子,改进与发展了几种混沌系统同步与控制的方法,丰富了混沌控制理论的内容,具有一定的理论和应用价值。本论文的研究和创新工作主要包括以下内容：
1.提出了一个新双涡卷混沌过渡系统及其统一系统,给出了一个新的临界条件分类标准,发现了另一个简单的过渡系统-Lii-like系统。Lii-like系统在Loren系统和Chen系统之间,也可以实现了从一个系统到另一个系统的过渡。还引入一个简单的单参数新双涡卷统一混沌系统,统一了上面三个不同的混沌系统。最后,还详细研究了这两个新型混沌系统的动力学行为和特征。
2.提出并研究了一类新型三涡卷混沌系统及其统一系统。新型三涡卷混沌系统属于二次三维光滑的非线性自治系统族,但是他们都存在不同拓扑结构的三涡卷混沌吸引子。其中,对一些基本的系统混沌动力行为和特征,如Lyapunov指数、分数维、周期窗口、局部分岔、Poincare截面以及分形等都作了一些深入的分析和研究,构造由一个变量来控制三涡卷的统一混沌系统,实现了三涡卷混沌系统之间的顺利过渡。
3.提出了一个新型四维分数阶超级混沌系统,研究了它的基本动力学行为,结合Chen分数阶超级混沌系统和Lu分数阶超级混沌系统,深入研究了异结构分数阶超级混沌系统的同步问题,利用分数阶微积分理论和拉普拉斯变换理论,运用分数阶系统稳定性理论和计算机模拟技术,给出非线性状态控制器的设计方法,设计了非线性状态控制器,使得一类非线性分数阶异结构超级混沌系统达到同步。该同步方法理论严格、设计简单、便于实现、同步误差收敛速度快并且达到同步时间短。
4.研究了基于激活牵制技术的不确定新超级混沌系统和三涡卷混沌统一系统的同步与反同步控制。提出激活牵制控制策略,它仅仅控制新的不确定混沌系统的一个状态,即只需要一个控制器就可以实现驱动系统和响应系统之间的同步和反同步,并确保不确定新超级混沌系统和三涡卷混沌统一系统的同步和鲁棒稳定性。理论分析和数值模拟表明,该控制器在相当短的时间内实现不确定新超级混沌系统和三涡卷混沌统一系统的同步与反同步。
5.研究了一类反对称异结构超级混沌系统及其可调节尺度因子的广义投影同步控制,提出了一种新的控制设计方法,设计了稳定控制器来控制三对角结构混沌系统的同步。给出了系统稳定的充分条件,然后证明它是系统的误差状态渐近稳定的来源。使变尺度因子的广义投影同步在反对称结构和异结构超级混沌系统之间得以实现。随后,我们针对三对角结构混沌系统的同步稳定控制,也提出了一种新的控制设计方法。以递归设计三对角结构系统方法为基础,将原始非线性仿射系统转化为带特殊三对角结构的稳定系统,控制器作用于改造后的混沌系统,并使其产生同步。驱动系统和响应系统之间的耦合系数赋予了一个新而广泛的含义。该方法运用于许多实际非线性控制问题,可以降低控制成本。数值模拟验证了该方法的有效性和可靠性。
6.针对一种自适应控制方法和参数更新规则来稳定新型参数未知的超级混沌系统,提出参数未知的新型超级混沌系统的自适应同步控制器和参数更新规则,镇定该系统到一个不稳定的平衡点。基于Lyapunov稳定性理论和自适应控制理论,该同步控制在未知参数的两个相同或两个不同的超级混沌系统分别予以实现。随后,深入研究并实现了新型超级混沌系统的变尺度因子射影同步。数值仿真结果验证了所提出的新同步方法的灵活性和有效性。
The chaos synchronization and control have the broad application prospects in many research fields. Since the chaos synchronization and control have been proposed, they have the potentially important role in both communications of the information science and the bio-medical engineering of life sciences. These make the chaos theory become a hot spot of research. Currently people have made some progress on the chaos synchronization and control method of research, but the study of chaos synchronization and control is not yet understand and grasped fully, the chaos synchronization and control are still the science foreland. It is a significant and far-reaching issue that we will study the chaos synchroniza-tion and control in depth and use the chaos theory in human social services effectively. It has caused a great of concern from a large number of researchers. In this paper, we have studied the chaos synchronization and control in depth. Some works are including the proposing of a new class of complex and chaotic systems and their attractors and the development of im-proved various synchronization and control methods in several chaotic systems. The works have enriched the content of the chaos theory and have certain theoretical and application value. The main research and innovation in this paper are as follows.
1. A new double-scroll chaotic transition system and its unified system are proposed. A new classification of critical conditions is given. Another simple transition systems-Lu-like system has been found. The Lu-like system is also the transition system between the Loren system and Chen system. A simple new double scroll chaotic unified system with one parameter is also introduced, which unifies the above three different chaotic systems. And finally, a detailed study has been done on the dynamics and features of these two new chaotic systems.
2. A new type of three-scroll chaotic system and its unified system are presented and studied. The new three-scroll chaotic systems are three-dimensional smooth family of non-linear autonomous systems with squared items, but they all have the three scroll chaotic attractors with the different topologies. Among them, some basic behavior features of this type chaotic dynamical system have been made some in-depth analysis and research, such as Lyapunov index, fractal dimension, cycle window, the local bifurcation, Poincare cross section and fractals. Constructing a three-scroll chaotic unified system with a variable to control and achieving the smooth transition among the three-scroll chaotic systems.
3. Proposing a new type of four-dimensional fractional super-chaotic system and studing its basic dynamic behavior. Combined with Chen fractional super-chaotic systems and Lu fractional super-chaotic system, the different fractional super-chaotic system synchroniza-tion has been studied in depth. Based on the fractional calculus theory and Laplace transform theory, using fractional order systems stability theory and computer simulation techniques fully, presenting the design method of the nonlinear state controller and designing a nonlinear state controller. So a different class of nonlinear fractional order super-chaotic systems with different structure can be achieved synchronization. This synchronization method is strict. The design is simple, and easy to implement. The synchronization error can convergence fast and the time for achieving synchronization is short.
4. Study on the chaotic synchronization and anti-synchronization for the uncertain new super-chaotic system and three-scroll chaotic unified system based on the active pinning control technology. At the same time, proposing the active control technology strategy which only controls a state of uncertain chaotic systems. That is one controller needed to realize the synchronization and anti-synchronization between the drive system and the response system, and ensure the robust stability of the synchronization for the uncertain chaotic system and three-scroll chaotic unified system. The theoretical analysis and numerical simulations show that the chaotic synchronization and anti-synchronization can be achieved within a relatively short time by this controller in the uncertain new super-chaotic system and three-scroll chaotic unified system, respectively.
5. Study on the generalized projective synchronization with the adjusted scale factor about a class of antisymmetric chaotic systems with different structures. A new control design method is proposed, and a stable controller is designed to control the synchronization of chaotic systems with the tridiagonal structure. Giving the sufficient conditions for stability of the chaos system, and then proving that it is the stable source asymptotically for state error of the system. So that the general projective synchronization with the adjusted scale factor can be achieved between the antisymmetric chaotic systems with different structures. Subsequently, we focused on the stability control of chaotic synchronization for the systems with the tridiagonal structure, and also proposed a new control design method. Based on the design method recursively for the system with the tridiagonal structure, the original nonlinear affine system is transformed into the stable system with a special tridiagonal structure. The chaotic synchronization is achieved after the controller acting on the transformation chaos system. The coupling coefficient between the drive system and response system has a new broad interpretation. The method will be used in many practical nonlinear control problems, which can reduce the cost. Numerical simulations show the validity and reliability of the method.
6. An adaptive control method and parameters update law are designed to guarantee stabilize the new super-chaotic system with unknown parameters. Proposing the new adap-tive synchronization controller and parameters updating rules, and stabilizing the system to an unstable equilibrium point. Based Lyapunov stability theory and adaptive control the-ory, the chaos synchronization is achieved among the two same super-chaotic systems with unknown parameters or two different ones with unknown parameters. Subsequently, the projective synchronization has been achieved and studied in depth on a new type of super-chaotic system with variable scale factor. Furthermore, numerical simulations verified the flexibility and effectiveness of the new proposed synchronization method.
1.提出了一个新双涡卷混沌过渡系统及其统一系统,给出了一个新的临界条件分类标准,发现了另一个简单的过渡系统-Lii-like系统。Lii-like系统在Loren系统和Chen系统之间,也可以实现了从一个系统到另一个系统的过渡。还引入一个简单的单参数新双涡卷统一混沌系统,统一了上面三个不同的混沌系统。最后,还详细研究了这两个新型混沌系统的动力学行为和特征。
2.提出并研究了一类新型三涡卷混沌系统及其统一系统。新型三涡卷混沌系统属于二次三维光滑的非线性自治系统族,但是他们都存在不同拓扑结构的三涡卷混沌吸引子。其中,对一些基本的系统混沌动力行为和特征,如Lyapunov指数、分数维、周期窗口、局部分岔、Poincare截面以及分形等都作了一些深入的分析和研究,构造由一个变量来控制三涡卷的统一混沌系统,实现了三涡卷混沌系统之间的顺利过渡。
3.提出了一个新型四维分数阶超级混沌系统,研究了它的基本动力学行为,结合Chen分数阶超级混沌系统和Lu分数阶超级混沌系统,深入研究了异结构分数阶超级混沌系统的同步问题,利用分数阶微积分理论和拉普拉斯变换理论,运用分数阶系统稳定性理论和计算机模拟技术,给出非线性状态控制器的设计方法,设计了非线性状态控制器,使得一类非线性分数阶异结构超级混沌系统达到同步。该同步方法理论严格、设计简单、便于实现、同步误差收敛速度快并且达到同步时间短。
4.研究了基于激活牵制技术的不确定新超级混沌系统和三涡卷混沌统一系统的同步与反同步控制。提出激活牵制控制策略,它仅仅控制新的不确定混沌系统的一个状态,即只需要一个控制器就可以实现驱动系统和响应系统之间的同步和反同步,并确保不确定新超级混沌系统和三涡卷混沌统一系统的同步和鲁棒稳定性。理论分析和数值模拟表明,该控制器在相当短的时间内实现不确定新超级混沌系统和三涡卷混沌统一系统的同步与反同步。
5.研究了一类反对称异结构超级混沌系统及其可调节尺度因子的广义投影同步控制,提出了一种新的控制设计方法,设计了稳定控制器来控制三对角结构混沌系统的同步。给出了系统稳定的充分条件,然后证明它是系统的误差状态渐近稳定的来源。使变尺度因子的广义投影同步在反对称结构和异结构超级混沌系统之间得以实现。随后,我们针对三对角结构混沌系统的同步稳定控制,也提出了一种新的控制设计方法。以递归设计三对角结构系统方法为基础,将原始非线性仿射系统转化为带特殊三对角结构的稳定系统,控制器作用于改造后的混沌系统,并使其产生同步。驱动系统和响应系统之间的耦合系数赋予了一个新而广泛的含义。该方法运用于许多实际非线性控制问题,可以降低控制成本。数值模拟验证了该方法的有效性和可靠性。
6.针对一种自适应控制方法和参数更新规则来稳定新型参数未知的超级混沌系统,提出参数未知的新型超级混沌系统的自适应同步控制器和参数更新规则,镇定该系统到一个不稳定的平衡点。基于Lyapunov稳定性理论和自适应控制理论,该同步控制在未知参数的两个相同或两个不同的超级混沌系统分别予以实现。随后,深入研究并实现了新型超级混沌系统的变尺度因子射影同步。数值仿真结果验证了所提出的新同步方法的灵活性和有效性。
The chaos synchronization and control have the broad application prospects in many research fields. Since the chaos synchronization and control have been proposed, they have the potentially important role in both communications of the information science and the bio-medical engineering of life sciences. These make the chaos theory become a hot spot of research. Currently people have made some progress on the chaos synchronization and control method of research, but the study of chaos synchronization and control is not yet understand and grasped fully, the chaos synchronization and control are still the science foreland. It is a significant and far-reaching issue that we will study the chaos synchroniza-tion and control in depth and use the chaos theory in human social services effectively. It has caused a great of concern from a large number of researchers. In this paper, we have studied the chaos synchronization and control in depth. Some works are including the proposing of a new class of complex and chaotic systems and their attractors and the development of im-proved various synchronization and control methods in several chaotic systems. The works have enriched the content of the chaos theory and have certain theoretical and application value. The main research and innovation in this paper are as follows.
1. A new double-scroll chaotic transition system and its unified system are proposed. A new classification of critical conditions is given. Another simple transition systems-Lu-like system has been found. The Lu-like system is also the transition system between the Loren system and Chen system. A simple new double scroll chaotic unified system with one parameter is also introduced, which unifies the above three different chaotic systems. And finally, a detailed study has been done on the dynamics and features of these two new chaotic systems.
2. A new type of three-scroll chaotic system and its unified system are presented and studied. The new three-scroll chaotic systems are three-dimensional smooth family of non-linear autonomous systems with squared items, but they all have the three scroll chaotic attractors with the different topologies. Among them, some basic behavior features of this type chaotic dynamical system have been made some in-depth analysis and research, such as Lyapunov index, fractal dimension, cycle window, the local bifurcation, Poincare cross section and fractals. Constructing a three-scroll chaotic unified system with a variable to control and achieving the smooth transition among the three-scroll chaotic systems.
3. Proposing a new type of four-dimensional fractional super-chaotic system and studing its basic dynamic behavior. Combined with Chen fractional super-chaotic systems and Lu fractional super-chaotic system, the different fractional super-chaotic system synchroniza-tion has been studied in depth. Based on the fractional calculus theory and Laplace transform theory, using fractional order systems stability theory and computer simulation techniques fully, presenting the design method of the nonlinear state controller and designing a nonlinear state controller. So a different class of nonlinear fractional order super-chaotic systems with different structure can be achieved synchronization. This synchronization method is strict. The design is simple, and easy to implement. The synchronization error can convergence fast and the time for achieving synchronization is short.
4. Study on the chaotic synchronization and anti-synchronization for the uncertain new super-chaotic system and three-scroll chaotic unified system based on the active pinning control technology. At the same time, proposing the active control technology strategy which only controls a state of uncertain chaotic systems. That is one controller needed to realize the synchronization and anti-synchronization between the drive system and the response system, and ensure the robust stability of the synchronization for the uncertain chaotic system and three-scroll chaotic unified system. The theoretical analysis and numerical simulations show that the chaotic synchronization and anti-synchronization can be achieved within a relatively short time by this controller in the uncertain new super-chaotic system and three-scroll chaotic unified system, respectively.
5. Study on the generalized projective synchronization with the adjusted scale factor about a class of antisymmetric chaotic systems with different structures. A new control design method is proposed, and a stable controller is designed to control the synchronization of chaotic systems with the tridiagonal structure. Giving the sufficient conditions for stability of the chaos system, and then proving that it is the stable source asymptotically for state error of the system. So that the general projective synchronization with the adjusted scale factor can be achieved between the antisymmetric chaotic systems with different structures. Subsequently, we focused on the stability control of chaotic synchronization for the systems with the tridiagonal structure, and also proposed a new control design method. Based on the design method recursively for the system with the tridiagonal structure, the original nonlinear affine system is transformed into the stable system with a special tridiagonal structure. The chaotic synchronization is achieved after the controller acting on the transformation chaos system. The coupling coefficient between the drive system and response system has a new broad interpretation. The method will be used in many practical nonlinear control problems, which can reduce the cost. Numerical simulations show the validity and reliability of the method.
6. An adaptive control method and parameters update law are designed to guarantee stabilize the new super-chaotic system with unknown parameters. Proposing the new adap-tive synchronization controller and parameters updating rules, and stabilizing the system to an unstable equilibrium point. Based Lyapunov stability theory and adaptive control the-ory, the chaos synchronization is achieved among the two same super-chaotic systems with unknown parameters or two different ones with unknown parameters. Subsequently, the projective synchronization has been achieved and studied in depth on a new type of super-chaotic system with variable scale factor. Furthermore, numerical simulations verified the flexibility and effectiveness of the new proposed synchronization method.
引文
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[68]Zhou W., Pan L., Li Z., and Halang W. A., "Non-linear feedback control of a novel chaotic system," International Journal of Control, Automation, and Systems,2009, vol.7, no.6, pp.939-944.
[69]Pan L., Zhou W., Fang J., and Li D., "Analysis of linear and adaptive feedback syn-chronization in a new unified chaotic system," International journal of adaptive con-trol and signal processing, Jan 2010, no. DOI:10.1002/acs.1164. Accepted In Press.
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