复杂混沌系统的存在性及动力学特性分析
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摘要
混沌,在经历了半个世纪的发展之后已经渗透到了不同的自然科学和工程领域,并且混沌应用已经逐步彰显出巨大的生命力。近年来,混沌研究从简单的认识、分析发展到了从理论层面深入地研究混沌的本质,进而在科学和工程技术领域中控制和利用混沌,例如混沌保密通信、流媒体加密等。对于混沌行为更加复杂的超混沌系统,由于其具有更强的随机性和不可预测性,从而拥有很大的应用潜力。本文在前人工作的基础上,在混沌系统的动力学分析和混沌性判定、超混沌系统的建模和理论分析方面进行了深入的研究。主要创新工作总结如下：
(1)基于非线性状态反馈的方法,提出了两个强的四维自治超混沌系统。从Lyapunov指数谱、分形图、相图、Poincare截面等方面进行了数值分析,新提出的系统具有非常丰富的动力学演化行为,可以经历周期、拟周期、混沌和超混沌状态,并且处于超混沌的参数范围较广、两个正Lyapunov指数值较大。此外,利用中心流形理论详细地分析了一个超混沌系统的叉式分岔和Hopf分岔行为。在另一个系统中观察到了一条由周期通向超混沌的间歇振荡道路,从而提供了一条新的通往超混沌的可能性道路。两个超混沌系统动态复杂、丰富,具有较强的理论意义和应用价值。
(2)研究了三个不同的三维连续混沌系统,即著名的Chen系统、Qi等提出的一个复杂混沌系统和一个流行病模型。利用拓扑马蹄理论,对这些系统中的典型混沌吸引子的存在性进行了严格的计算机辅助证明,不仅证明了它们具有拓扑马蹄,而且给出了系统拓扑熵的下界估计。Qi等提出的复杂混沌系统不仅具有较大的正lyapunov指数值,而且具有较大的拓扑熵下界估计值,说明了Qi等提出的混沌吸引子具有较为复杂的动力学行为。
(3)研究了两个典型的经济系统中的混沌动态。一个是由Kopel提出的微观经济模型,即古诺双寡头模型,利用拓扑马蹄引理证明了双寡头模型中混沌吸引子的存在性。同时也详细分析了该模型中出现的种种非线性现象,包括过渡混沌、非光滑商业周期、混沌吸引子共存、两种类型的间歇混沌等。还分析了混沌市场对竞争双方所获利润的影响,得到了混沌市场并不是完全有害的结论,即任何一个竞争者都可能从一个处于混沌状态的市场中获取更高的利润。最后又设计了一个状态反馈控制器来控制双寡头模型中的混沌,取得了很好的控制效果,即竞争者公平地共享市场。另一个模型是宏观经济模型,即商业周期模型,同样也利用计算机辅助证明的方法分析了该系统的拓扑马蹄动态,进而证明了混沌吸引子的存在性。
(4)对分数阶的Chen系统和两个改进的分数阶Chua电路进行了探讨和研究。首先证明了分数阶Chen系统中混沌吸引子的存在性,并对该系统进行了模拟电路实现研究。当经典的Chua系统中的非线性函数采用正弦波函数时,其分数阶系统可以产生单方向的卷数可调的多涡卷吸引子；当非线性函数采用某种形式的锯齿波函数时,改进的分数阶Chua系统可以产生网格状的横纵向卷数均可调的多涡卷吸引子。
Over the past half century, the world has witnessed the rapid development of Chaos theory. Chaos has penetrated into almost every science and engineering field, and its application has shown the tremendous vitality in many aspects of human life. Recently, researchers have turned their close attention from the simple analysis of chaos to deeply analyze the essence of chaos theoretically, so as to control and make use of chaos in engineering fields, such as the chaotic secure communication and the streaming media encryption. In particular, hyperchaos is more complex than chaos, and has stronger randomness and unpredictability. Therefore, hyperchaos has great potential applications in chaos-needed fields. In this dissertation, the existence of chaos and dynamical behavior analysis of some chaotic systems, and also the generation and theoretical analysis of two hyperchaotic systems are investigated systematically. The main work can be summarized as follows:
(1) Based on the state feedback control method, two strong four-dimensional autonomous continuous hyperchaotic systems are proposed. By analyzing the Lyapunov exponent spectra, bifurcation diagram, phase portraits, Poincare sections, it can be obtained that the new systems have very rich dynamical behaviors, and they can evolve into period, quasi-period, chaos and hyperchaos. The two new hyperchaotic systems both have wider hyperchaotic parameter region, two relatively larger positive Lyapunov exponents. Furthermore, the pitchfork bifurcation and Hopf bifurcation in one hyperchaotic system are analyzed in detail. Moreover, a new intermittent route from period to hyperchaos is observed in another hyperchaotic system. The two new hyperchaotic systems have complex and rich nonlinear dynamics, and thus have great theoretical significance and application value.
(2) The topological horseshoe dynamics are investigated in three different three-dimensional continuous systems, namely the famous Chen system, a complex chaotic system presented by Qi et al. and an epidemic disease model. By utilizing the recent famous topological horseshoe theory, the existence of topological horseshoe is verified in each system, and the topological entropy is also estimated during the process of proof. Thus, the existence of chaos is proved in a computer assisted manner. The chaotic system proposed by Qi et al. has not only bigger Lyapunov exponent but also bigger lower bound estimation of topological entropy. The result shows that the Qi chaotic attractor has very complex dynamical behaviors.
(3) Chaotic dynamics in two different economic systems are investigated in detail. One is a microeconomic model, that is, a Cournot duopoly model proposed by Kopel in 1996. The existence of horseshoe chaos is first verified in this economic model, and then many kinds of nonlinear dynamics, such as transient chaos, nonsmooth business cycle, co-existence of two chaotic attractors, and two different types of economic intermittency, are observed and analyzed. Besides, the influence of chaos on the profits of duopolists is analyzed. It is numerically demonstrated that chaotic market is not totally harmful, that is to say. either of the duopolists could be beneficial from a chaotic market. A state feedback controller is designed to control chaos to the Nash equilibrium, as a result of which the two duopolists share the market with good equity. A macroeconomic model, namely a business cycle model is also learned, the topological horseshoe dynamics are analyzed, and thus the existence of chaos in this model is proved in a computer assisted manner.
(4) The fractional-order Chen system and two modified fractional-order Chua circuits are analyzed. The computer-assisted proof for the existence of chaos in fractional-order Chen system is provided. The analog circuit implementation of fractional-order Chen system is also given. When the sine function is applied to replace the nonlinear function of Chua's circuit, and the integer order is replaced by a fractional order, the modified fractional order Chua system can generate n-scroll chaotic attractor. Another modified fractional order Chua system with the sawtooth and staircase function being the nonlinearity, can generate n×m-scroll chaotic attractor.
(1)基于非线性状态反馈的方法,提出了两个强的四维自治超混沌系统。从Lyapunov指数谱、分形图、相图、Poincare截面等方面进行了数值分析,新提出的系统具有非常丰富的动力学演化行为,可以经历周期、拟周期、混沌和超混沌状态,并且处于超混沌的参数范围较广、两个正Lyapunov指数值较大。此外,利用中心流形理论详细地分析了一个超混沌系统的叉式分岔和Hopf分岔行为。在另一个系统中观察到了一条由周期通向超混沌的间歇振荡道路,从而提供了一条新的通往超混沌的可能性道路。两个超混沌系统动态复杂、丰富,具有较强的理论意义和应用价值。
(2)研究了三个不同的三维连续混沌系统,即著名的Chen系统、Qi等提出的一个复杂混沌系统和一个流行病模型。利用拓扑马蹄理论,对这些系统中的典型混沌吸引子的存在性进行了严格的计算机辅助证明,不仅证明了它们具有拓扑马蹄,而且给出了系统拓扑熵的下界估计。Qi等提出的复杂混沌系统不仅具有较大的正lyapunov指数值,而且具有较大的拓扑熵下界估计值,说明了Qi等提出的混沌吸引子具有较为复杂的动力学行为。
(3)研究了两个典型的经济系统中的混沌动态。一个是由Kopel提出的微观经济模型,即古诺双寡头模型,利用拓扑马蹄引理证明了双寡头模型中混沌吸引子的存在性。同时也详细分析了该模型中出现的种种非线性现象,包括过渡混沌、非光滑商业周期、混沌吸引子共存、两种类型的间歇混沌等。还分析了混沌市场对竞争双方所获利润的影响,得到了混沌市场并不是完全有害的结论,即任何一个竞争者都可能从一个处于混沌状态的市场中获取更高的利润。最后又设计了一个状态反馈控制器来控制双寡头模型中的混沌,取得了很好的控制效果,即竞争者公平地共享市场。另一个模型是宏观经济模型,即商业周期模型,同样也利用计算机辅助证明的方法分析了该系统的拓扑马蹄动态,进而证明了混沌吸引子的存在性。
(4)对分数阶的Chen系统和两个改进的分数阶Chua电路进行了探讨和研究。首先证明了分数阶Chen系统中混沌吸引子的存在性,并对该系统进行了模拟电路实现研究。当经典的Chua系统中的非线性函数采用正弦波函数时,其分数阶系统可以产生单方向的卷数可调的多涡卷吸引子；当非线性函数采用某种形式的锯齿波函数时,改进的分数阶Chua系统可以产生网格状的横纵向卷数均可调的多涡卷吸引子。
Over the past half century, the world has witnessed the rapid development of Chaos theory. Chaos has penetrated into almost every science and engineering field, and its application has shown the tremendous vitality in many aspects of human life. Recently, researchers have turned their close attention from the simple analysis of chaos to deeply analyze the essence of chaos theoretically, so as to control and make use of chaos in engineering fields, such as the chaotic secure communication and the streaming media encryption. In particular, hyperchaos is more complex than chaos, and has stronger randomness and unpredictability. Therefore, hyperchaos has great potential applications in chaos-needed fields. In this dissertation, the existence of chaos and dynamical behavior analysis of some chaotic systems, and also the generation and theoretical analysis of two hyperchaotic systems are investigated systematically. The main work can be summarized as follows:
(1) Based on the state feedback control method, two strong four-dimensional autonomous continuous hyperchaotic systems are proposed. By analyzing the Lyapunov exponent spectra, bifurcation diagram, phase portraits, Poincare sections, it can be obtained that the new systems have very rich dynamical behaviors, and they can evolve into period, quasi-period, chaos and hyperchaos. The two new hyperchaotic systems both have wider hyperchaotic parameter region, two relatively larger positive Lyapunov exponents. Furthermore, the pitchfork bifurcation and Hopf bifurcation in one hyperchaotic system are analyzed in detail. Moreover, a new intermittent route from period to hyperchaos is observed in another hyperchaotic system. The two new hyperchaotic systems have complex and rich nonlinear dynamics, and thus have great theoretical significance and application value.
(2) The topological horseshoe dynamics are investigated in three different three-dimensional continuous systems, namely the famous Chen system, a complex chaotic system presented by Qi et al. and an epidemic disease model. By utilizing the recent famous topological horseshoe theory, the existence of topological horseshoe is verified in each system, and the topological entropy is also estimated during the process of proof. Thus, the existence of chaos is proved in a computer assisted manner. The chaotic system proposed by Qi et al. has not only bigger Lyapunov exponent but also bigger lower bound estimation of topological entropy. The result shows that the Qi chaotic attractor has very complex dynamical behaviors.
(3) Chaotic dynamics in two different economic systems are investigated in detail. One is a microeconomic model, that is, a Cournot duopoly model proposed by Kopel in 1996. The existence of horseshoe chaos is first verified in this economic model, and then many kinds of nonlinear dynamics, such as transient chaos, nonsmooth business cycle, co-existence of two chaotic attractors, and two different types of economic intermittency, are observed and analyzed. Besides, the influence of chaos on the profits of duopolists is analyzed. It is numerically demonstrated that chaotic market is not totally harmful, that is to say. either of the duopolists could be beneficial from a chaotic market. A state feedback controller is designed to control chaos to the Nash equilibrium, as a result of which the two duopolists share the market with good equity. A macroeconomic model, namely a business cycle model is also learned, the topological horseshoe dynamics are analyzed, and thus the existence of chaos in this model is proved in a computer assisted manner.
(4) The fractional-order Chen system and two modified fractional-order Chua circuits are analyzed. The computer-assisted proof for the existence of chaos in fractional-order Chen system is provided. The analog circuit implementation of fractional-order Chen system is also given. When the sine function is applied to replace the nonlinear function of Chua's circuit, and the integer order is replaced by a fractional order, the modified fractional order Chua system can generate n-scroll chaotic attractor. Another modified fractional order Chua system with the sawtooth and staircase function being the nonlinearity, can generate n×m-scroll chaotic attractor.
引文
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