两类动力系统的分支与混沌
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摘要
本文研究一个连续Josephson系统和一个离散Tinkerbell映射当参数变化时,动态的变化情况。
对于带参数激励的Josephson系统,我们把它分为两种未扰动系统进行研究,即和首先应用Melnikov方法,给出在周期扰动下系统产生混沌的条件。接着,应用二阶平均方法和次谐波Melnikov函数,分析系统在未扰动中心附近的谐波解,(2,3,n-阶)次谐波解和(2,3-阶)超谐波解的存在性和分支。并且,我们发现,对于第一种未扰动系统,应用平均化方法,没有二阶次谐波与二阶超谐波发生。通过数值模拟,包括二维参数平面和三维参数空间的分支图,最大Lyapunov指数图,相图,Poincare映射,验证理论分析的结果且进一步研究系统参数对动力学行为的影响。我们发现了复杂且有趣的动力学现象,比如：跳跃行为,周期轨的对称碰撞,瞬态混沌,混沌的产生,混沌突然转变为周期轨,内部危机,混沌吸引子,奇异的非混沌运动,非吸引的混沌集,混沌区间中连续的周期-n(n=6,8，9,等)轨的倍周期分支和连续的周期-n(n=4,8,12,等)轨的反倍周期分支,等等。特别地,当参数β增加时,我们观察到,经历一些奇异的非混沌运动之后,接着出现连续的周期轨倍周期分支到混沌这样一个过程。
对于Tinkerbell映射,应用中心流形定理和分支理论,第一次比较系统地导出了fold分支、flip分支和Hopf分支的存在条件并通过分析和数值模拟的方法证明了存在Marotto意义下的混沌。具体地讲,我们找到了Marotto意义下的混沌,发现了一条从不变环到瞬态混沌的路径,这些瞬态混沌中有周期窗口,包括周期2,7,8,9,10,13,17,19,2326等周期,还观察到了混沌的突然出现与消失,不变环变成周期1轨,对称碰撞,混沌区域中连续的倍周期分支,内部危机,混沌吸引子,2个、10个和13个分别共存的混沌集,两个共存的不变环,以及与一个非吸引的混沌集共存的两个吸引的混沌集等动力学现象。特别是,当参数变化时,没有明显的路线从倍周期分支到混沌,但有从周期1轨道到不变环,然后到瞬态混沌的线路。结合现有文献,以及本文的新结果,使我们对Tinkerbell映射有一个较完整的理解。
全文共分六章。
第一章,介绍本文的研究背景及现状、研究内容、方法和意义。
第二章,预备知识,简单介绍连续和离散动力系统的分支和混沌,包括中心流形定理,二阶平均方法,Mclnikov方法,混沌的定义、特征,分形维数以及通向混沌的道路。
第三章,应用Mclnikov方法,给出带参数激励的Josephson系统在周期扰动下产生混沌的条件。并且,通过数值模拟验证理论分析结果和系统参数对动力学性质的影响,发现更复杂的动态。
第四章,我们分析上述带参数激励的Joscphson系统的周期解分支。应用二阶平均方法和次谐波Melnikov函数,分析系统在未扰动中心附近的谐波解,(2,3,n-阶)次谐波解和(2.3-阶)超谐波解的存在性和分支。并用数值模拟验证理论结果和发现新的动态。
第五章研究Tinkcrbcll映射的动力学行为。主要包括应用中心流形定理和分支理论导出fold分支、flip分支和Hopf分支的存在条件以及Marotto意义下的混沌的存在条件。同时,通过数值模拟,验证我们所得到的理论结果以及观察新的有趣动力学性质。
第六章介绍本文中所观察到通往混沌的道路。
This thesis discusses the dynamics of the Josephson system and the Tin-dcrbell map as the parameters varying.
For the Josephson system with parametric excitation, it is analyzed that we propose two types of the unperturbed systems: and Firstly, by applying Melnikov method, the unperturbed heterocilinic and ho-moclinic bifurcations result in chaos under periodic perturbations. Secondly, by the second-order averaging method and the subharmonic Melnikov func-tion, we analyze bifurcations and the existence of harmonic,(2,3, n-order) subharmonic and (2,3-order) superharmonic solutions near the unperturbed centers. But, for the first unperturbed system, there are no2-order subhar-monic and supcrharmonic solutions. Thirdly, by using numerical simulations, including bifurcation diagrams in two-and three-dimensional spaces, the maxi-mum Lyapunov exponents, phase portraits and Poincare map, we demonstrate our theoretical analysis and further study the effect of the parameters on dy-namical behavior. We find the complex and interesting dynamics, such as the jumping behaviors, symmetry-breaking of periodic orbits, transient chaos, on-set of chaos, chaos suddenly converting to period orbits, interior crisis, chaotic attractors, some strange non-chaotic motions, non-attracting chaotic set, inter-locking period-doubling bifurcations from period-n(n=6,8,9, etc) orbits and interlocking reverse period-doubling bifurcations from period-n(n=4,8,12, etc) orbits in chaotic regions, and so on. In particular, we observe the process from interlocking period-doubling bifurcations of periodic orbits to chaos after some strange non-chaotic motions as the parameter β increases.
For the Tinkerbell map, it is the first time systematically to discuss the existence of fold bifurcation, flip bifurcation and Hopf bifurcation and chaos in the sense of Marotto by both analytical and numerical methods. More pre-cisely, for the Tinderbell map, this thesis reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2,7,8,9,10,13,17,19,23,26, and so on, and suddenly appearing or disappearing chaos, an invariant circle turning to a period-one orbit., symmetry-breaking of periodic orbits, inter-locking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2,10,13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on. In particular, it is found that there is no obvious route from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invari-ant circle and then to transient chaos as the parameters vary. Combining the existing results in the literature with the new results reported in this thesis, a more complete understanding of the Tinkerbell map is obtained.
This thesis consists of six chapters as following:
In Chapter1, we introduce the background and the present situation of the related research field, and the content, methods and significance of my thesis.
Chapter2presents the preparatory knowledge of bifurcations and chaos on continuous and discrete systems, including the center manifold theorem, the second-order averaging method, the Melnikov method, definitions and charac-teristics of chaos, fractal dimension, and the route to chaos.
In Chapter3, by applying the Melnikov method, we give the conditions resulting in chaos under periodic perturbation in Josephson system with para-metric excitation Moreover using numerical simulations, we demonstrate the theoretical results, discuss the influence of the parameters on dynamics, and find more complex dynamics.
In chapter4, we analyze bifurcations of periodic solutions in the above Josephson system with parametric excitation. By using second-order averaging method and subharinonic Mclnikov function, we analyze bifurcations and the existence of harmonic,(2.3.n-order) subharmonic and (2,3-order) superhar-monic solutions near the unperturbed centers. The numerical simulations show the consistence with the theoretical analysis and exhibit some new properties.
In chapter5, dynamical behavior of the Tinkerbell map are investigated in detail. Some sufficient conditions are derived on the existence of fold bifurcation, flip bifurcation and Hopf bifurcation, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations exhibit new and interesting dynamical behavior.
In chapter6, we introduce the observed route to chaos in the thesis.
对于带参数激励的Josephson系统,我们把它分为两种未扰动系统进行研究,即和首先应用Melnikov方法,给出在周期扰动下系统产生混沌的条件。接着,应用二阶平均方法和次谐波Melnikov函数,分析系统在未扰动中心附近的谐波解,(2,3,n-阶)次谐波解和(2,3-阶)超谐波解的存在性和分支。并且,我们发现,对于第一种未扰动系统,应用平均化方法,没有二阶次谐波与二阶超谐波发生。通过数值模拟,包括二维参数平面和三维参数空间的分支图,最大Lyapunov指数图,相图,Poincare映射,验证理论分析的结果且进一步研究系统参数对动力学行为的影响。我们发现了复杂且有趣的动力学现象,比如：跳跃行为,周期轨的对称碰撞,瞬态混沌,混沌的产生,混沌突然转变为周期轨,内部危机,混沌吸引子,奇异的非混沌运动,非吸引的混沌集,混沌区间中连续的周期-n(n=6,8，9,等)轨的倍周期分支和连续的周期-n(n=4,8,12,等)轨的反倍周期分支,等等。特别地,当参数β增加时,我们观察到,经历一些奇异的非混沌运动之后,接着出现连续的周期轨倍周期分支到混沌这样一个过程。
对于Tinkerbell映射,应用中心流形定理和分支理论,第一次比较系统地导出了fold分支、flip分支和Hopf分支的存在条件并通过分析和数值模拟的方法证明了存在Marotto意义下的混沌。具体地讲,我们找到了Marotto意义下的混沌,发现了一条从不变环到瞬态混沌的路径,这些瞬态混沌中有周期窗口,包括周期2,7,8,9,10,13,17,19,2326等周期,还观察到了混沌的突然出现与消失,不变环变成周期1轨,对称碰撞,混沌区域中连续的倍周期分支,内部危机,混沌吸引子,2个、10个和13个分别共存的混沌集,两个共存的不变环,以及与一个非吸引的混沌集共存的两个吸引的混沌集等动力学现象。特别是,当参数变化时,没有明显的路线从倍周期分支到混沌,但有从周期1轨道到不变环,然后到瞬态混沌的线路。结合现有文献,以及本文的新结果,使我们对Tinkerbell映射有一个较完整的理解。
全文共分六章。
第一章,介绍本文的研究背景及现状、研究内容、方法和意义。
第二章,预备知识,简单介绍连续和离散动力系统的分支和混沌,包括中心流形定理,二阶平均方法,Mclnikov方法,混沌的定义、特征,分形维数以及通向混沌的道路。
第三章,应用Mclnikov方法,给出带参数激励的Josephson系统在周期扰动下产生混沌的条件。并且,通过数值模拟验证理论分析结果和系统参数对动力学性质的影响,发现更复杂的动态。
第四章,我们分析上述带参数激励的Joscphson系统的周期解分支。应用二阶平均方法和次谐波Melnikov函数,分析系统在未扰动中心附近的谐波解,(2,3,n-阶)次谐波解和(2.3-阶)超谐波解的存在性和分支。并用数值模拟验证理论结果和发现新的动态。
第五章研究Tinkcrbcll映射的动力学行为。主要包括应用中心流形定理和分支理论导出fold分支、flip分支和Hopf分支的存在条件以及Marotto意义下的混沌的存在条件。同时,通过数值模拟,验证我们所得到的理论结果以及观察新的有趣动力学性质。
第六章介绍本文中所观察到通往混沌的道路。
This thesis discusses the dynamics of the Josephson system and the Tin-dcrbell map as the parameters varying.
For the Josephson system with parametric excitation, it is analyzed that we propose two types of the unperturbed systems: and Firstly, by applying Melnikov method, the unperturbed heterocilinic and ho-moclinic bifurcations result in chaos under periodic perturbations. Secondly, by the second-order averaging method and the subharmonic Melnikov func-tion, we analyze bifurcations and the existence of harmonic,(2,3, n-order) subharmonic and (2,3-order) superharmonic solutions near the unperturbed centers. But, for the first unperturbed system, there are no2-order subhar-monic and supcrharmonic solutions. Thirdly, by using numerical simulations, including bifurcation diagrams in two-and three-dimensional spaces, the maxi-mum Lyapunov exponents, phase portraits and Poincare map, we demonstrate our theoretical analysis and further study the effect of the parameters on dy-namical behavior. We find the complex and interesting dynamics, such as the jumping behaviors, symmetry-breaking of periodic orbits, transient chaos, on-set of chaos, chaos suddenly converting to period orbits, interior crisis, chaotic attractors, some strange non-chaotic motions, non-attracting chaotic set, inter-locking period-doubling bifurcations from period-n(n=6,8,9, etc) orbits and interlocking reverse period-doubling bifurcations from period-n(n=4,8,12, etc) orbits in chaotic regions, and so on. In particular, we observe the process from interlocking period-doubling bifurcations of periodic orbits to chaos after some strange non-chaotic motions as the parameter β increases.
For the Tinkerbell map, it is the first time systematically to discuss the existence of fold bifurcation, flip bifurcation and Hopf bifurcation and chaos in the sense of Marotto by both analytical and numerical methods. More pre-cisely, for the Tinderbell map, this thesis reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2,7,8,9,10,13,17,19,23,26, and so on, and suddenly appearing or disappearing chaos, an invariant circle turning to a period-one orbit., symmetry-breaking of periodic orbits, inter-locking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2,10,13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on. In particular, it is found that there is no obvious route from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invari-ant circle and then to transient chaos as the parameters vary. Combining the existing results in the literature with the new results reported in this thesis, a more complete understanding of the Tinkerbell map is obtained.
This thesis consists of six chapters as following:
In Chapter1, we introduce the background and the present situation of the related research field, and the content, methods and significance of my thesis.
Chapter2presents the preparatory knowledge of bifurcations and chaos on continuous and discrete systems, including the center manifold theorem, the second-order averaging method, the Melnikov method, definitions and charac-teristics of chaos, fractal dimension, and the route to chaos.
In Chapter3, by applying the Melnikov method, we give the conditions resulting in chaos under periodic perturbation in Josephson system with para-metric excitation Moreover using numerical simulations, we demonstrate the theoretical results, discuss the influence of the parameters on dynamics, and find more complex dynamics.
In chapter4, we analyze bifurcations of periodic solutions in the above Josephson system with parametric excitation. By using second-order averaging method and subharinonic Mclnikov function, we analyze bifurcations and the existence of harmonic,(2.3.n-order) subharmonic and (2,3-order) superhar-monic solutions near the unperturbed centers. The numerical simulations show the consistence with the theoretical analysis and exhibit some new properties.
In chapter5, dynamical behavior of the Tinkerbell map are investigated in detail. Some sufficient conditions are derived on the existence of fold bifurcation, flip bifurcation and Hopf bifurcation, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations exhibit new and interesting dynamical behavior.
In chapter6, we introduce the observed route to chaos in the thesis.
引文
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