Bifurcation of Periodic Orbits and Chaos in Duffing Equation

Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated, The conditions of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, period-windows （period-2, 3 and 5） in chaotic regions.     

Bifurcations and chaos in Duffing equation with damping and external excitations

Duffng equation with damping and external excitations is investigated.By using Melnikov method and bifurcation theory,the criterions of existence of chaos under periodic perturbations are obtained.By using second-order averaging method,the criterions of existence of chaos in averaged system under quasi-periodic perturbations forff=nω+εσ,n=2,4,6(whereσis not rational toω)are investigated.However,the criterions of existence of chaos for n=1,3,5,7-20 can not be given.The numerical simulations verify the theoretical analysis,show the occurrence of chaos in the averaged system and original system under quasiperiodic perturbation for n=1,2,3,5,and expose some new complex dynamical behaviors which can not be given by theoretical analysis.In particular,the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations,and the period-doubling bifurcations to chaos has not been found under quasi-periodic perturbations.     

Duffing方程的静态与动态分岔特性研究

对于Duffing方程的静态和全局动态分岔,通过研究平均方程的全局行为得到了出现各种分岔的条件,揭示了Duffing方程周期解的变化过程及其具有的非线性动力学性质。     

非线性振动系统的异宿轨道分叉,次谐分叉和混沌

在参数激励与强迫激励联合作用下具有van der Pol阻尼的非线性振动系统,其动态行为是非常复杂的.本文利用Melnikov方法研究了这类系统的异宿轨道分叉、次谐分叉和混沌.对于各种不同的共振情况,系统将经过无限次奇阶次谐分叉产生Smale马蹄而进入混沌状态.最后我们利用数值计算方法研究了这类系统的混沌运动.所得结果揭示了一些新的现象.     

用加权残余法求解含大参数的Duffing方程

本文应用加权残余法分析了含大参数的 Duffing方程 ,并得到了整个区域内 (0 <ε<∞ )一致有效的近似解 ,得到的近似周期的最大相对误差小于 7.0 % ,当参数为小量时 (ε 1 ) ,得到的近似解和摄动解完全一致 .     

Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

A discrete predator-prey system with Holling type-IV functional response obtained by the Euler method is first investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-two bifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximate expressions for saddle-node, Hopfand homoclinic bifurcation sets near the Bogdanov-Takens bifurcation point. We also show the existence of degenerated fixed point with codimension three at least. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors such as the attracting invariant circle, period-doubling bifurcation from period-2,3,4 orbits.interior crisis, intermittency mechanic, and sudden disappearance of chaotic dynamic.     

多频激励软弹簧型Duffing系统中的混沌

研究了多频激励下的软弹簧型Duffing系统的混沌动力学,发现混沌产生的根本原因是系统相空间中横截异宿环面的存在．建立了双频激励情况下二维环面上的Poincaré映射、稳定流形和不稳定流形,应用Melnikov方法给出了稳定流形和不稳定流形横截相交的条件,并将此方法推广到激励包含有限多个频率的情形．推广了Melnikov方法在高维系统中的应用,给出了Smale马蹄意义下混沌存在的判据．同时证明,激励频率数目的增加扩大了参数空间上的混沌区域．     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.     

Duffing方程的奇异点方法

设g∈C2(R),p(t)为连续的2π周期函数.考虑Duffing方程x+g(x)=p(t),x(O)=x(2π),x(0)=x(2π),笔者应用奇点理论,证明了Duffing算子Fx(t)=x(t)+g(x(t)).当g(x)为严格凸且g’(x)渐近跨越第一共振点0时, F整体等价于Whitney意义下的fold映射,特别地,获得2π周期解的不存在性、唯一性与唯二性定理.     

Periodic Solutions of the Duffing DifferentialEquation Revisited via the Averaging Theory

We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations $ddot y+ a sin y= b sin t$ and $ddot y+a y-c y^3=bsin t$, where $a$, $b$ and $c$ are real parameters.     

一个Duffing方程的调和解和次调和解

本文证明了Duffing方程x+arctan x=p(t)的调和解和无穷多的次调和解的存在 性,其中周期的2π连续函数p(t)满足|p(t)|<π/2,(?)t∈R.     

The Lower Bounds of T-Periodic Solutions for the Forced Duffing Equation

This paper is devoted to the discussion of the number of T-periodic solutions for the forced Duffing equation, x″ + kx′ + g(t, x) = s(1 + h(t)), with g(t, x) being a continuous function by using the degree theory, upper and lower solutions method, and the twisting theorem.     

关于Dufing方程周期解的存在性与唯一性

采用Lyapunov-Schmidt约化结合Sturm特征值比较方法,将二阶Duffing方程转化为单个函数方程．通过研究函数的单调性,给出Duffing方程的周期解的存在性与唯一性．     

关于一类时滞Duffing型方程的周期解注记

利用重合度理论研究一类Duffing型方程的周期解问题,获得了有关周期解存在性的新的结果.     

Naimark-Sacker Bifurcations in the Euler Method for a Delay Differential Equation

We consider the application of the Euler method to a delay differential equation which has a Hopf bifurcation, and prove that Naimark-Sacker bifurcations, i.e., Hopf bifurcations of maps, occur in the discretization by using the center manifold theorem and the Naimark-Sacker theorem. We also present obstacles to the generalization of the result.     

一类具时滞耗散型Duffing方程的周期解

利用Mawhin重合度理论研究了一类耗散型时滞Duffing方程ax″+f[x′(t-τ1(t))]+cx+g(x(t-τ2(t)))=p(t)周期解的存在性,得到了该方程2π周期解存在的充分性定理.     

Boundedness of Solutions of a Quasi-periodic Sublinear Duffing Equation

The authors study the Lagrangian stability for the sublinear Duffing equations(x)+e(t)丨x丨α-1x = p(t)with 0＜α＜1,where e and p are real analytic quasi-periodic fu...     

Periodic Solutions of a Class of Duffing Differential Equations

In this work we study the existence of new periodic solutions for the well knwon class of Duffing differential equation of the form $x^{\prime\prime}+ c x^{\prime}+ a(t) x +b(t) x^3 = h(t)$, where $c$ is a real parameter, $a(t)$, $b(t)$ and $h(t)$ are continuous $T$--periodic functions. Our results are proved using three different results on the averaging theory of first order.