混沌与超混沌系统模型分析及模拟电路研究
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摘要
混沌是非线性动力学系统的一种运动形式,混沌现象在自然界中无处不在,自从1963年美国气象学家Lorenz在确定性耗散系统中发现了第一个混沌吸引子以来,关于混沌的研究吸引了越来越多的关注。近年来,混沌理论与应用均得到快速发展,混沌系统是一种复杂的非线性系统,要进一步深入研究这种复杂系统,一方面需要深入的理论分析,如动力学特性分析和混沌存在性证明；另一方面需要对这种系统研究其物理实现,即通过模拟电路验证其特性。
本文针对混沌系统的理论分析和电路实现方面开展了如下创新工作。
首先,对由陈增强等提出的一个四翼混沌吸引子进行了数值仿真分析。在Poincare映射的基础上,又利用Yang等提出的拓扑马蹄引理,借助于计算机辅助证明的方法,对该四翼混沌系统进行了拓扑马蹄分析,进而从理论上证明了其混沌吸引子的存在；并为该四翼混沌系统设计了模拟电路,通过示波器可以观测到该系统的各吸引子相轨迹与数值仿真的结果是一致的,进一步从物理层面上验证了该四翼混沌系统的特性。另外,还利用拓扑马蹄引理分析了一个Qi四翼混沌系统中拓扑马蹄的存在。
近几年来,超混沌的生成和应用已经成为混沌研究的一个热点问题。本文对一类已有的超混沌系统进行了模拟电路实现研究,并在此基础上,提出了一个具有更大参数范围和两个更大的正Lyapunov指数的超混沌系统。同时也对该系统进行了模拟电路实现研究。
Lu系统是统一的Lorenz系统族中介于Lorenz系统和Chen系统之间的一个过渡系统,本文也提出了一个由它产生的单平衡点的超混沌Lu系统,并应用中心流形定理对其进行了局部分岔分析。并为该系统设计了一个模拟电路进行研究。
Chaos is one of the modes of motion in nonlinear system, and the phenomenon of chaos is ubiquitous in nature. Since the American meteorologist Lorenz found the fist chaotic attractor in the deterministic dissipative system in 1963, the study on chaotic attractor has attracted more and more interests. Recently, chaos theory and its application have been developed rapidly. Chaotic systems are complex nonlinear systems. To deeply study the complex chaotic systems, one should not only analyze them theoretically, such as dynamical behavior analysis and proof for the existence of chaos, but also investigate the characteristics of them by the physical implementation, namely the analog circuits.
With respect to theoretical analysis and circuit implementation of chaotic systems, the innovation work in the thesis is summarized as follows.
This thesis firstly does numerical simulation study on a four-wing chaotic systems reported by Chen Zengqiang et al. Based on Poincare map, by using the topological horseshoe theory presented by Yang et al. and computer-assisted proof, it is verified that topological horseshoes exist in the system, and thus the existence of chaos is proved theoretically. An analog hardware circuit is also made for the four-wing chaotic system, and the results of circuit experiment observed by oscillation are well consistent with those of simulation. Furthermore, the characteristics of the four-wing chaotic system are verified physically. In addition, it is also proved that topological horseshoe exists in a Qi four-wing chaotic system by utilizing the topological horseshoe theorem.
In recent years, the generation and the application of hyper-chaos have become a hot topic of chaos. The thesis also focuses on the hardware implementation of the presented hyperchaotic systems. Based on deeply analyzing those hyperchaotic systems, a new hyperchaotic system which possesses a larger parameter range and two bigger positive Lyapunov exponents is also presented. It is implemented it by an analog circuit.
Lii system is a transition system between the Lorenz system and the Chen system in generalized Lorenz system. The thesis also presents a novel hyper-chaotic Lii system with only one equilibrium. The local bifurcation is analyzed by virtue of center manifold theory. An analog circuit is designed to study the hyperchaotic attractor.
本文针对混沌系统的理论分析和电路实现方面开展了如下创新工作。
首先,对由陈增强等提出的一个四翼混沌吸引子进行了数值仿真分析。在Poincare映射的基础上,又利用Yang等提出的拓扑马蹄引理,借助于计算机辅助证明的方法,对该四翼混沌系统进行了拓扑马蹄分析,进而从理论上证明了其混沌吸引子的存在；并为该四翼混沌系统设计了模拟电路,通过示波器可以观测到该系统的各吸引子相轨迹与数值仿真的结果是一致的,进一步从物理层面上验证了该四翼混沌系统的特性。另外,还利用拓扑马蹄引理分析了一个Qi四翼混沌系统中拓扑马蹄的存在。
近几年来,超混沌的生成和应用已经成为混沌研究的一个热点问题。本文对一类已有的超混沌系统进行了模拟电路实现研究,并在此基础上,提出了一个具有更大参数范围和两个更大的正Lyapunov指数的超混沌系统。同时也对该系统进行了模拟电路实现研究。
Lu系统是统一的Lorenz系统族中介于Lorenz系统和Chen系统之间的一个过渡系统,本文也提出了一个由它产生的单平衡点的超混沌Lu系统,并应用中心流形定理对其进行了局部分岔分析。并为该系统设计了一个模拟电路进行研究。
Chaos is one of the modes of motion in nonlinear system, and the phenomenon of chaos is ubiquitous in nature. Since the American meteorologist Lorenz found the fist chaotic attractor in the deterministic dissipative system in 1963, the study on chaotic attractor has attracted more and more interests. Recently, chaos theory and its application have been developed rapidly. Chaotic systems are complex nonlinear systems. To deeply study the complex chaotic systems, one should not only analyze them theoretically, such as dynamical behavior analysis and proof for the existence of chaos, but also investigate the characteristics of them by the physical implementation, namely the analog circuits.
With respect to theoretical analysis and circuit implementation of chaotic systems, the innovation work in the thesis is summarized as follows.
This thesis firstly does numerical simulation study on a four-wing chaotic systems reported by Chen Zengqiang et al. Based on Poincare map, by using the topological horseshoe theory presented by Yang et al. and computer-assisted proof, it is verified that topological horseshoes exist in the system, and thus the existence of chaos is proved theoretically. An analog hardware circuit is also made for the four-wing chaotic system, and the results of circuit experiment observed by oscillation are well consistent with those of simulation. Furthermore, the characteristics of the four-wing chaotic system are verified physically. In addition, it is also proved that topological horseshoe exists in a Qi four-wing chaotic system by utilizing the topological horseshoe theorem.
In recent years, the generation and the application of hyper-chaos have become a hot topic of chaos. The thesis also focuses on the hardware implementation of the presented hyperchaotic systems. Based on deeply analyzing those hyperchaotic systems, a new hyperchaotic system which possesses a larger parameter range and two bigger positive Lyapunov exponents is also presented. It is implemented it by an analog circuit.
Lii system is a transition system between the Lorenz system and the Chen system in generalized Lorenz system. The thesis also presents a novel hyper-chaotic Lii system with only one equilibrium. The local bifurcation is analyzed by virtue of center manifold theory. An analog circuit is designed to study the hyperchaotic attractor.
引文
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