【作者】 黄翔；

【导师】 井竹君；

【作者基本信息】 湖南师范大学 ， 应用数学， 2009， 硕士

【摘要】 本文应用动力系统的分支理论,二阶平均方法,Melnikov方法和混沌理论,研究带非线性恢复力和外力激励的Duffing-Van der Pol方程随系统参数变化的复杂动态行为,我们给出了谐波介和其分支存在的条件,以及在周期扰动下系统产生混沌的准则,也给出了在ω₂=nω₁+εv,n=1,2,3,4,5,7的拟周期扰动下平均系统产生混沌的准则,数值模拟验证了理论结果正确性,而用平均方法不能给出在ω=nw₁+εv,n=6,8,9-15（这里ω₁与v不为有理数）的拟周期扰动下产生混沌的准则,但数值模拟显示了原系统出现混沌。同时,用数值模拟（包括同宿和异宿分支曲面,动态分支图,最大Lyapunov指数图,相图,Poincared映射图）发现了许多新的复杂动态。我们发现周期倍分支和来自周期一和周期三轨的逆周期倍分支到一个混沌吸引子或到两个分离的混沌吸引子,混沌行为和周期窗口的交替出现,混沌的突然出现和突然收敛到周期二轨,带有复杂周期窗口和不带周期窗口的大范围混沌区域,不带周期窗口的不变环,具有复杂不变环的混沌区域以及内部危机。本文研究具有非线性恢复力和外力激励的Duffing-Van der Pol方程是前人未研究过的,这研究将丰富动力系统的内容,并具有一定的应用前景。全文共分三章。第一章是关于连续动力系统的分支理论、二阶平均方法、Melnikov方法与混沌理论的预备知识。第二章简单介绍了Duffing-Van der Pol方程的一些历史背景知识。第三章应用二阶平均方法和Melnikov理论研究具有非线性恢复力和外力激励的Duffing-Van der Pol方程,给出了谐波介和其分支存在条件以及周期扰动下系统产生混沌的准则,也给出了在ω=nω₁+εv,n=1,2,3,4,5,7的拟周期扰动下平均系统产生混沌的准则,而不能给出在ω=nω₁+εv,n=6,8,9-15的拟周期扰动下产生混沌的准则,但数值模拟显示了原系统出现混沌,用数值模拟我们给出十个系统参数变化时系统动态的变化。更多还原

【Abstract】 The complex dynamical behaviours as the system’s parametrics varing in Duffing-Van der Pol equation with nonlinear restoring and external excitations are investigated by using bifurcation theory,second-order averaging methods,Melnikov methods and chaotic theory.Wo prove that conditions of existence of harmoniz and its bifurcations,and The threshold values of existence of chaotic motion under the periodic perturbation.By applying the second-order averaging method and Melnikov method,we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω₂ = nω₁ +∈ν,n=1,2,3,4,5,7,and cannot prove the criterion of existence of chaos in second-order averaged system under quasiperiodic perturbation forω₂ = nω₁ +∈ν,n = 6,8,9 - 15,whereω₁ is rational toν, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams,Lyapunov exponent,phase portraits and Poincare（?） map,not only show the consistence with the theoretical analysis but also exhibit some new complex dynamics,we found that periodic doubling bifurcation and the reversed periodic from period-one-three doubling bifurcation leading to chaos,and the interleaving occurrences of chaotic behaviors and periodic windows,the onset of chaos and chaos suddenly conventing to periodic-two ubits,the large chaotic regions with complex period-windows and without period-windows,the chaotic regious with complex invariant torus,the region of invariant torus without period-windows,and the interior crisis.The paper consists of three chapters,as the following,Chapter 1 is the preparation knowledge.A brief review of second-order averaging methods and Melnikov methods,chaos and some routes to chaos for continual dynamical system is presented.In chapter 2,we briefly introduce the backgrounds and histories of Duffing Duffing-Van der Pol equations.In chapter 3,we study Duffing-Van der Pol equation with nonlinear restoring and external excitations.we prove that the conditions of harmoniz and its bifurcation, and the threshold values of existence of chaotic motion under the periodic perturbation,and the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω= nω₁ +∈ν,n = 1,2,3,4,5,7,and cannot prove the criterion of existence of chaos under quasi-periodic perturbation forω₂= nω₁ +∈ν,n = 6,8,9 - 15,but can show the occurrence of chaos in original system by numerical simulation.we find many new complex dynamics as the ten-system’s parametries varing by numerical simulation.更多还原