两类连续和离散动力系统的分支和混沌
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摘要
本文研究一个连续动力系统和一个离散动力系统当参数变化时,动态的变化状况。
对于带外力与阻尼激励,具有五次非线性项的Duffing方程,应用Melnikov方法给出在周期扰动下系统产生混沌运动的准则,应用二阶平均方法和Melnikov方法给出在拟周期扰动Ω=nw+(?)σ,n=2,4,6下,平均系统存在混沌的准则,但是证明当n=1,3,5,7-20时不能用平均方法给出混沌的存在准则(这里σ和ω不可约),利用数值模拟验证平均系统和与其取值相对应的原方程的混沌存在的一致性。并应用二阶平均方法、分支理论和混沌理论给出当三个频率满足Ω:w:w0=n:m:1(其中n,m=1,2,3,1／2)时系统的共振及分支情况,应用Melnikov方法和数值模拟,讨论当Ω:w:w0=n:m:1(m>3)时的共振情况。对于generalized Henon映射,应用中心流形定理和分支理论给出fold分支、flip分支和Hopf分支的存在条件,并从理论上证明在一定的参数条件下,映射存在Marotto意义下的混沌。通过数值模拟(包括不动点的分支图、同宿和异宿三维分支曲面、三维和二维分支图、最大Lyapunov指数图、相图、Poincare映射),验证理论分析结果的正确性和展示丰富而复杂的新的动力学行为。其中包括带周期窗口的暂态混沌、周期轨的跳跃行为、从周期-2(4)到周期-4(12)的bubble.混沌行为和周期行为(或拟周期行为)交替出现、混沌多次突然出现、混沌行为和不变环突然出现或突然消失、混沌突然消失到周期轨、倍周期分支和逆倍周期分支到混沌、周期轨的对称破缺、interior crisis现象、拟周期路径到混沌,和各种各样的的拟周期吸引子、混沌吸引子、非吸引的混沌集、奇怪的混沌吸引子和奇怪的非混沌吸引子。对吸引子的最大Lyapunov指数的计算进一步确定了混沌行为的发生。
全文共分四章。
第一章是关于动力系统的分支和混沌预备知识。简要介绍连续和离散动力系统的中心流形定理,二阶平均方法和Melnikov方法.对混沌的定义、特征以及通向混沌的道路也作了简要地介绍。
第二章应用定性理论、分支和混沌理论以及数值模拟讨论带外力与阻尼激励的Duffing方程的动力行为,应用Melnikov方法证明在周期扰动下混沌的存在性准则,应用二阶平均方法和Melnikov方法给出在拟周期扰动Ω=nω+(?)σ,n=2,4,6下,平均系统的混沌的存在准则,但是证明当n=1,3,5,7-20时不能用平均方法给出混沌的存在准则。并且,通过数值模拟验证理论分析结果的正确性和原方程的参数对动力性质的影响和发现新的动态。
第三章研究带周期外力与阻尼激励的Duffing方程的共振解及其分支情况。给出当三个频率ω0,ω和Ω满足某些共振关系时,系统的动态情况。应用二阶平均方法、分支和混沌理论研究该系统当Ω：ω：ω0=n：m：1(其中n,m=1,2,3,1／2)时的共振及分支情况,应用Melnikov方法和数值模拟,讨论当Ω:ω:ω0=n:m:1(m>3)时的共振情况。并用数值模拟验证理论结果和发现新的动态。
第四章研究generalized Henon映射的分支和混沌现象。应用中心流形定理和分支理论给出fold分支、flip分支和Hopf分支的存在条件,并从理论上证明在一定的参数条件下,映射存在Marotto意义下的混沌。并用数值模拟验证理论结果和发现新的动态。
This thesis discuss the dynamics of one continuous and one discrete dy-namical system as the parameters varying.
For the Duffing equation with damping excitation and one external forcing and fifth nonlinear-restoring force, we prove the criterion of existence of chaos under periodic perturbation by applying Melnikov method, and by second-order averaging method and Melnikov method give the criterion of existence of chaos of averaged system under quasi-periodic perturbation forΩ=nw+(?)σ,n= 2,4,6, and prove that we can't give the criterion of existence of chaos for n= 1,3,5,7 - 20 by applying averaging method, whereσis not rational to w, and show the existence of chaotic dynamics in averaged equation and original equation by numerical simulations. And under the three frequencies satisfyingΩ:w:W0= n:m:1 (n, m= 1,2,3,1/2), the resonances and bifurca-tions of the system are investigated by using qualitative theory, bifurcation and chaos theories, and by applying Melnikov method and numerical simulations discuss the resonances under the conditionsΩ:w:w0= n:m:1 (m> 3). For generalized Henon, the conditions of existence for fold bifurcation, flip bi-furcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, chaotic behavior in the sense of Marotto's definition is proved under parameters satisfying some conditions. Meanwhile, the numerical simulations, including bifurcation diagram of fixed points, homoclinic and hete-roclinic bifurcation surface, bifurcation diagrams in three-and two-dimensional spaces, the maximum Lyapunov exponent, phase portraits, Poincare maps, not only show the consistence with the theoretical analysis, but also exhibit the rich and new dynamical behaviors, including the transient chaos with com-plex windows, jumping behaviors of period-orbits, the bubble from period-2(4) to period-4(12), chaotic behavior and periodic motion (or quasi-periodic mo-tions) occurs alternately, onset of chaos which occurs more than once, chaotic behaviors and invariant cycles suddenly appear or disappear, chaos suddenly disappears to period orbits, period-doubling bifurcations and reverse period- doubling bifurcations to chaos, the symmetry-breaking of period-orbits, initial crisis, quasi-periodic route to chaos and many quasi-periodic attractors, chaotic attractors, non-attracting chaotic sets, the strange chaotic attractors and the strange non-chaotic attractors. The computation of maximum Lyapunov expo-nents confirm the chaotic behaviors.
This thesis consists of four chapters as following:
Chapter 1 is the preparation knowledge. A brief review of center maniold theorem, second-order averaging method and Melnikov method for continuous and discrete dynamical system are presented. Some definitions and character-istics of chaos as well as some routes to chaos are mentioned.
In Chapter 2, by using qualitative theory, bifurcation and chaos theory and numerical simulations, the complex dynamics of the Duffing equation with damping excitation and one external forcing is investigated. Applying Melnikov method, the criterions of existence of chaos under periodic perturbation and applying second-order averaging method and Melnikov method, the criterions of existence of chaos of averaged system under quasi-periodic perturbationΩ= nw+(?)σ,n= 2,4,6 are given, but the criterion of existence of chaos for n= 1,3,5,7 - 20 can't been proven. And using the numerical simulations to prove the theoretical analysis and to study the influence of the parameters in original equation on dynamics, and find more complex dynamics.
In chapter 3, Duffing equation with damping excitation and one external periodic excitation is also investigated. The dynamics of the system under the three frequencies satisfying some resonances conditions are given. It is easy to obtain the resonances and bifurcations under the conditionsΩ:w:w0= n:m: 1 (m,n= 1,2,3,1/2) by second-order averaging method and bifurcation and chaos theories. The Melnikov's method and numerical simulations are applied to take into account the resonances when some specific resonant conditions among these three different frequenciesΩ:w:w0= n:m:1 (m> 3). The numerical simulations show the consistence with the theoretical analyses and exhibit some specific properties.
In chapter 4, the bifurcations and chaos phenomenons of generalized Henon map are investigated. The conditions of existence for fold bifurcation, flip bi-furcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, chaotic behavior in the sense of Marotto's definition is proved. And numerical simulation results show the consistence with the theo-retical analyses and display the new and interesting dynamical behaviors.
对于带外力与阻尼激励,具有五次非线性项的Duffing方程,应用Melnikov方法给出在周期扰动下系统产生混沌运动的准则,应用二阶平均方法和Melnikov方法给出在拟周期扰动Ω=nw+(?)σ,n=2,4,6下,平均系统存在混沌的准则,但是证明当n=1,3,5,7-20时不能用平均方法给出混沌的存在准则(这里σ和ω不可约),利用数值模拟验证平均系统和与其取值相对应的原方程的混沌存在的一致性。并应用二阶平均方法、分支理论和混沌理论给出当三个频率满足Ω:w:w0=n:m:1(其中n,m=1,2,3,1／2)时系统的共振及分支情况,应用Melnikov方法和数值模拟,讨论当Ω:w:w0=n:m:1(m>3)时的共振情况。对于generalized Henon映射,应用中心流形定理和分支理论给出fold分支、flip分支和Hopf分支的存在条件,并从理论上证明在一定的参数条件下,映射存在Marotto意义下的混沌。通过数值模拟(包括不动点的分支图、同宿和异宿三维分支曲面、三维和二维分支图、最大Lyapunov指数图、相图、Poincare映射),验证理论分析结果的正确性和展示丰富而复杂的新的动力学行为。其中包括带周期窗口的暂态混沌、周期轨的跳跃行为、从周期-2(4)到周期-4(12)的bubble.混沌行为和周期行为(或拟周期行为)交替出现、混沌多次突然出现、混沌行为和不变环突然出现或突然消失、混沌突然消失到周期轨、倍周期分支和逆倍周期分支到混沌、周期轨的对称破缺、interior crisis现象、拟周期路径到混沌,和各种各样的的拟周期吸引子、混沌吸引子、非吸引的混沌集、奇怪的混沌吸引子和奇怪的非混沌吸引子。对吸引子的最大Lyapunov指数的计算进一步确定了混沌行为的发生。
全文共分四章。
第一章是关于动力系统的分支和混沌预备知识。简要介绍连续和离散动力系统的中心流形定理,二阶平均方法和Melnikov方法.对混沌的定义、特征以及通向混沌的道路也作了简要地介绍。
第二章应用定性理论、分支和混沌理论以及数值模拟讨论带外力与阻尼激励的Duffing方程的动力行为,应用Melnikov方法证明在周期扰动下混沌的存在性准则,应用二阶平均方法和Melnikov方法给出在拟周期扰动Ω=nω+(?)σ,n=2,4,6下,平均系统的混沌的存在准则,但是证明当n=1,3,5,7-20时不能用平均方法给出混沌的存在准则。并且,通过数值模拟验证理论分析结果的正确性和原方程的参数对动力性质的影响和发现新的动态。
第三章研究带周期外力与阻尼激励的Duffing方程的共振解及其分支情况。给出当三个频率ω0,ω和Ω满足某些共振关系时,系统的动态情况。应用二阶平均方法、分支和混沌理论研究该系统当Ω：ω：ω0=n：m：1(其中n,m=1,2,3,1／2)时的共振及分支情况,应用Melnikov方法和数值模拟,讨论当Ω:ω:ω0=n:m:1(m>3)时的共振情况。并用数值模拟验证理论结果和发现新的动态。
第四章研究generalized Henon映射的分支和混沌现象。应用中心流形定理和分支理论给出fold分支、flip分支和Hopf分支的存在条件,并从理论上证明在一定的参数条件下,映射存在Marotto意义下的混沌。并用数值模拟验证理论结果和发现新的动态。
This thesis discuss the dynamics of one continuous and one discrete dy-namical system as the parameters varying.
For the Duffing equation with damping excitation and one external forcing and fifth nonlinear-restoring force, we prove the criterion of existence of chaos under periodic perturbation by applying Melnikov method, and by second-order averaging method and Melnikov method give the criterion of existence of chaos of averaged system under quasi-periodic perturbation forΩ=nw+(?)σ,n= 2,4,6, and prove that we can't give the criterion of existence of chaos for n= 1,3,5,7 - 20 by applying averaging method, whereσis not rational to w, and show the existence of chaotic dynamics in averaged equation and original equation by numerical simulations. And under the three frequencies satisfyingΩ:w:W0= n:m:1 (n, m= 1,2,3,1/2), the resonances and bifurca-tions of the system are investigated by using qualitative theory, bifurcation and chaos theories, and by applying Melnikov method and numerical simulations discuss the resonances under the conditionsΩ:w:w0= n:m:1 (m> 3). For generalized Henon, the conditions of existence for fold bifurcation, flip bi-furcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, chaotic behavior in the sense of Marotto's definition is proved under parameters satisfying some conditions. Meanwhile, the numerical simulations, including bifurcation diagram of fixed points, homoclinic and hete-roclinic bifurcation surface, bifurcation diagrams in three-and two-dimensional spaces, the maximum Lyapunov exponent, phase portraits, Poincare maps, not only show the consistence with the theoretical analysis, but also exhibit the rich and new dynamical behaviors, including the transient chaos with com-plex windows, jumping behaviors of period-orbits, the bubble from period-2(4) to period-4(12), chaotic behavior and periodic motion (or quasi-periodic mo-tions) occurs alternately, onset of chaos which occurs more than once, chaotic behaviors and invariant cycles suddenly appear or disappear, chaos suddenly disappears to period orbits, period-doubling bifurcations and reverse period- doubling bifurcations to chaos, the symmetry-breaking of period-orbits, initial crisis, quasi-periodic route to chaos and many quasi-periodic attractors, chaotic attractors, non-attracting chaotic sets, the strange chaotic attractors and the strange non-chaotic attractors. The computation of maximum Lyapunov expo-nents confirm the chaotic behaviors.
This thesis consists of four chapters as following:
Chapter 1 is the preparation knowledge. A brief review of center maniold theorem, second-order averaging method and Melnikov method for continuous and discrete dynamical system are presented. Some definitions and character-istics of chaos as well as some routes to chaos are mentioned.
In Chapter 2, by using qualitative theory, bifurcation and chaos theory and numerical simulations, the complex dynamics of the Duffing equation with damping excitation and one external forcing is investigated. Applying Melnikov method, the criterions of existence of chaos under periodic perturbation and applying second-order averaging method and Melnikov method, the criterions of existence of chaos of averaged system under quasi-periodic perturbationΩ= nw+(?)σ,n= 2,4,6 are given, but the criterion of existence of chaos for n= 1,3,5,7 - 20 can't been proven. And using the numerical simulations to prove the theoretical analysis and to study the influence of the parameters in original equation on dynamics, and find more complex dynamics.
In chapter 3, Duffing equation with damping excitation and one external periodic excitation is also investigated. The dynamics of the system under the three frequencies satisfying some resonances conditions are given. It is easy to obtain the resonances and bifurcations under the conditionsΩ:w:w0= n:m: 1 (m,n= 1,2,3,1/2) by second-order averaging method and bifurcation and chaos theories. The Melnikov's method and numerical simulations are applied to take into account the resonances when some specific resonant conditions among these three different frequenciesΩ:w:w0= n:m:1 (m> 3). The numerical simulations show the consistence with the theoretical analyses and exhibit some specific properties.
In chapter 4, the bifurcations and chaos phenomenons of generalized Henon map are investigated. The conditions of existence for fold bifurcation, flip bi-furcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, chaotic behavior in the sense of Marotto's definition is proved. And numerical simulation results show the consistence with the theo-retical analyses and display the new and interesting dynamical behaviors.
引文
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