1、2023-5-91第一章第一章 有限差分方程有限差分方程 一、线性有限差分方程:0021201NRNNRRNNRNNttttRNN1几个概念：方程（线性）系统参数：系统参数：R初始条件：初始条件：N0 2023-5-92 N0=100,R0 衰减（decay）R=0.9 递增（growth）R=1.08 稳态（steady-state）R=12023-5-93N0=100,R0衰减（decay）R=-0.9递增（growth）R=-1.08稳态（steady-state）R=-12023-5-94吸引子（attractor）:随着时间的演化，系统的一种状态趋势 0R1:Nt 分叉点分叉点（bi
2、furcation point）:以某个参数值为分界，系统进入不同的状态 R=12023-5-95二、非线性的有限差分方程1、Logistic Equation:)x1(Rxxtt1t R11x0 x)x(fxtt 系统参数：R 初始条件：x0 固固 定定 点：点：（fixed point）2023-5-96系统参数：系统参数：R，初始条件：初始条件：x0，取0 x0 1，x0=0.1（有生态学意义）0 R 1 xt 0 (attractor)2023-5-97 1R3 R=1.5 单调逼近固定点 x*=0.333 R=2.9 交替逼近固定点 x*=0.655 xt 1 1/R2023-5-9
3、82023-5-99问题1：1、x0取不同值时，上述几种情况如何？2、x0=0.5,R分别为1.25,2,2.75，画出轨线 t-xt2023-5-910 3R3.449 R=3.3 周期2(period-2)t2txx 2023-5-911 3.449 R3.5699 R=3.52 周期4 周期8 周期16 周期倍增周期倍增（period-doubling）2023-5-9123.5699 4 轨线最终逃逸（escape）到无穷。问题2：1.How many iterations dose it take for the trajectories to get with 0.001of th
4、e final value x=0.3333 for R=1.5?2.What happens for R4?2023-5-916小结：小结：系统表现出的不同行为 稳定状态、周期、混沌稳定状态、周期、混沌 系统参数(R)的不同给系统带来的影响 初始状态(x0)的不同对系统的影响2023-5-917 分叉图(bifurcation diagram)2023-5-918三、稳定状态(steady state)和稳定性（stability）研究三个问题：1、系统是否存在固定点(fixed point)？2、系统是否在固定点处存在局部稳定性？局部稳定性局部稳定性(locally stable)3、系统
5、是否在固定点处存在全局稳定性？全局稳定性全局稳定性(globally stable)2023-5-919 局部稳定性局部稳定性 locally stable:If the initial condition happens to be near a fixed point,sequent iterates approach the fixed point,we say the fixed point is locally stable.(locally asymptotic stability)全局稳定性全局稳定性 globally stable:If the fixed point is a
6、pproached by all initial conditions,we say the fixed point is globally stable.2023-5-9201、固定点(fixed point)：R11x0 xxx)x1(Rxxttt1ttt1t)x(fxtt 2023-5-9212、固定点的局部稳定性 线性系统：固定点 R 1:不稳定0 R 1:稳定R=0：稳定 R=1：稳定0 xt 2023-5-922 -1 R 0 R -1R=-1 不稳定 2023-5-923非线性系统：固定点)x(fxtt xdxdfmt交替远离固定点交替远离固定点单调远离固定点单调远离固定点交替逼
7、近固定点交替逼近固定点单调逼近固定点单调逼近固定点:1m1munstablex1m0m11m0stablex1mtt 2023-5-924以逻辑方程为例分析：几种情况：)x1(Rxxtt1t )1(5.11tttxxx)1(9.21tttxxx)1(3.31tttxxx)1(52.31tttxxx)1(41tttxxx周期2固定点周期4混沌2023-5-925两个概念两个概念渐近（asymptotic dynamics）：The term asymptotic dynamics refers to the dynamics as time goes to infinity.暂态（transie
8、nt）:Behavior before the asymptotic dynamics is called transient2023-5-9263、固定点的全局稳定性线性系统 A locally stable fixed point is also globally stable.非线性系统 When multiple fixed point are present,none of the fixed points can be globally.2023-5-927吸引域（basin of attraction）The set of initial conditions that even
9、tually leads to a fixed point is called basin of attraction多稳定性（multi-stability）If multiple fixed points are locally stable we say there is multi-stability.2023-5-928四、周期的稳定性tntxx tt1tx)x1(3.3x )()1(3.3112ttttxffxxx2个固定点：0，0.6974个固定点：0,0.479,0.697,0.823以逻辑方程，R=3.3为例2023-5-929Conclusion:（考虑周期n）If the
10、re is stable cycle of period n,there must be at least n fixed points associated with the stable cycle,where the slope at each of the fixed points is equal and the absolute value of the slope a each of the fixed points is less than 1.2023-5-930 For 3.0000R3.4495,there is stable cycle of period 2 For
11、3.4495R3.5441,there is stable cycle of period 4 For 3.5441R3.5644,there is stable cycle of period 8 For 3.5644R 3.570,there are narrow ranges of periodic solutions as well as aperiodic behavior The period-doubling route to chaos2023-5-931混沌状况：在周期2、在周期3、在周期4的图中，固定点斜率的绝对值均大于1考虑一个极端的例子：因此，进入混沌状态混沌状态。tt
12、1tx)x1(4x 2023-5-932五、混沌（chaos）混沌的定义：Be aperiodic bounded dynamics in a deterministic system with sensitive dependence on initial conditions.混沌系统的性质 Aperiodic Bounded Deterministic Sensitive dependence on initial condition2023-5-933 Feigenhaums number:4.6692 定义：n the range of R values that give a pe
13、riod-n cycle.6692.4limn2nn 2023-5-934 分叉图(bifurcation diagram)2023-5-935六、准周期性（Quasi-periodicity）x t+1=f(xt)=xt+b (mod 1)其中，b为无理数 非周期性:x t+n xt 有界：在 xt 周围的固定范围内 The route to chaos:Quasi-periodicity2023-5-936一个例子：非周期性：有界性：11x,0 x1x,0 xt1t1tt 1xxt1t (mod 1)nxxtnt (mod 1)1/1-1/2023-5-937作业：用计算机实现分叉图(bifurcation diagram)（p31）计算Feigenhaums number 进一步找到周期3的R值 研究自相似性
