Lyapunov指数的非线性控制.ppt

第三章 Lyapunov指数的非线性控制,Instability Lyapunov exponents 不稳定性判据,1.1 初始条件敏感性 是导致系统不可预测的原因，其实是 系统固有的不稳定性在作用 1.2 指数分叉 （Exponents divergence) 李雅普诺夫指数 与等价的误差放大倍数 k 有： L.E.（李雅普诺夫指数 ）是1892年提出的， 直到近几年，才认识到它的重要性 它是判断有界非线性系统是否为混沌 系统的一个重要方法。,定义1,它是用来度量相空间中两条相邻轨迹随时 间变化按指数规律吸引或分离的程度。这 两条轨迹有不同的初始条件。一个正的李 雅普诺夫指数说明，在一个具有有界轨道 的动力学系统中，存在着混沌运动。 （王东升 1985）,定义2 (公式定义) wolf 1985,对一n维相空间中的连续动力学系统，考虑 一个以 为中心， 为半径的n维无 穷小超球面的长时间演变行为，其中 。随着时间的变化，由于流形的 局部变形的本质，球面会逐渐演化成为一超 椭球面。按椭球第i个主轴的长度 可定 义系统的第i个李雅普诺夫指数：,或可定义为 其中式（1-1）， 的单位为比特/时间，表示单 位时间内系统信息损失的多少，适用于从信息论 的角度来考虑李雅普诺夫指数的含义。,x0,p1(0),t - time flow,p2(t),p1(t),p2(0),x(t),Definition of Lyapunov Exponents,Given a continuous dynamical system in an n-dimensional phase space, we monitor the long-term evolution of an infinitesimal n-sphere of initial conditions. The sphere will become an n-ellipsoid due to the locally deforming nature of the flow. The i-th one-dimensional Lyapunov exponent is then defined as following:,一种实用的计算方法：即发现系统由 方向定义 的椭球主轴长度按 倍数增加，由 两个 方向定义的面积按 倍数增加，由 三个方向定义的体积按 倍数增加，因 此，这提示人们利用椭球主轴定义的i维体积的变 化来计算李雅普诺夫指数。表：运动类型与李 雅普诺夫指数关系,表：运动类型与李雅普诺夫指数关系 最大李雅普诺夫指数 返回,The sign of the Lyapunov Exponent,0 - the system attracts to a fixed point or stable periodic orbit. These systems are non conservative (dissipative) and exhibit asymptotic stability. =0 - the system is neutrally stable. Such systems are conservative and in a steady state mode. They exhibit Lyapunov stability. 0 - the system is chaotic and unstable. Nearby points will diverge irrespective of how close they are.,定义：最大李雅普诺夫指数,假设系统有n个Lyapunov exponents，将它们按 大小顺序排列起来，如： ，则称 这些数组成的集合为李雅普诺夫指数谱，记为 。其中 被称为最大李雅普诺夫指 数。 返回,Order: 1 2 n The linear extent of the ellipsoid grows as 21t The area defined by the first 2 principle axes grows as 2(1+2)t The volume defined by the first 3 principle axes grows as 2(1+2+3)t and so on The sum of the first j exponents is defined by the long-term exponential growth rate of a j-volume element.,由最大李雅普诺夫指数 可判断系统的运动性 质。 如当 时，由式（1-1）可知 即相轨迹上相邻的两点产生的两条轨道按指 数的规律发散。如果系统的变化是有界的， 系统为混沌运动。,当 时，同样由式（1-1）可知， 即相轨迹上相邻的两点产生的两条轨道的距离既不增加也不减少。 当 时，有 即相轨迹上相邻两点产生的两条轨道之间的距离在逐渐减小。随着时间的推移，两者间的距离可任意小。因此，系统最终收敛为一个极限环或一个点。 1.3 从数据中测量最大指数,1.4 混沌运动中的几个重要概念,1.平衡点（不动点，the Fixed Point ） 对差分方程 （状态变量 ）而言，满足 的点成为平衡点。 对微分方程 （状态变量 ）而言， 满足 的点称为不动点（平衡点）。 平衡点又称不动点、静止点、奇点、临界点。 2.拓扑可迁 （Topologically Transitiue） 满映射f:JJ，如果对于任意的一对开集 存在k0，使 其中f满足 则称满映射f为拓扑可迁。,3.初值敏感性（Sensitive Dependence on Initial Conditions ） 满映射f:JJ ，对任意 、任意的x的邻域N，存 在 ，整数n0 以及小量 0 ，使 ，则称满映射f具有初值敏 感性（f大于8，不能对系统进行长期有效预测）。 4.吸引子（Attractor） 指相空间中的一个点集或一个子空间，随时间的流 逝，在暂态消失之后所有的相轨线都趋向于它，吸 引子即是稳定平衡点。,5.奇怪吸引子（strange Attractor） 即：Chaotic Attractor，指相空间中吸引子的集 合，在该集合中混沌轨道在运行。此吸引子不是 平衡点，也不是极限环，也不是周期吸引子，而 是具有分数维的吸引子。 6.流形（Manifold） 指相空间中的一个子空间。凡是具有初始条件的 解位于此子空间中者，在微分或差分方程的作用 下，这个解仍在此空间中，这个子空间就叫流形。,7.混沌（Chaos） 如果满映射f:JJ满足下述条件，则称映射f为混沌的。 f具有初值敏感性不可长期预测 f为拓扑可迁的不可分为两个相关的子系统 f在定义域上周期轨道稠密表面随机，内有规律 混沌映射三要素： a不可预测性 Unpredictability b不可分解性 Indecomposability c含有规律性成分 Anelement of regularity,Lyapunov 指数的计算,一、一维离散非线性系统的李雅普诺夫指数的 计算 由Logistic映射虫口模型 推出：,二、高维离散非线性系统李雅普诺夫指数的计算,1.数学基础奇异值 设A为nm维的复空间（或实空间），则 均为非负定的矩阵，且 即矩阵A的秩为r，而r为不大于min(n,m)的非负整， 所以有 均有r个正特征值。 ， ， 的第i个特征 值。 的其余特征值为0。,奇异值（Singular Value）定义： 的特征值为 ，则称 为A的非零奇异值。如何求 解？见下分解定理。 对任何矩阵 ，设rank A=r, 为奇异值，则 存在n阶正交矩阵U，m阶正交矩阵V，使A=UDV。 其中 ， ， 2.Lyapunov指数的计算： 如离散非线性系统 给定,其中 且有界， 在研究范围内 连续可微，则上述系统在点Z处的Jacobian矩阵为 令 令 为矩阵 的第i个奇异值，其中i=1,2,，n. j=0,1,.，则系统由 点出发的轨道 对应的第i个 Lyapnov指数为,3.一类特殊系统的Lyapunov指数的计算 若在z点的Jacobian阵为常数对角阵 其中 为非0常数。则有,所以系统的奇异值为 所以 若 则Lypapunov 指数为,Lyapunov Exponents,Lyapunov A. M. (1857-1918),Alexander Lyapunov was born 6 June 1857 in Yaroslavl, Russia in the family of the famous astronomer M.V. Lypunov, who played a great role in the education of Alexander and Sergey. Aleksandr Lyapunov was a school friend of Markov and later a student of Chebyshev at Physics & Mathematics department of Petersburg University which he entered in 1976. He attended the chemistry lectures of D.Mendeleev. In 1985 he brilliantly defends his MSc diploma “On the equilibrium shape of rotating liquids”, which attracted the attention of physicists, mathematicians and astronomers of the world. The same year he starts to work in Kharkov University at the Department of Mechanics. He gives lectures on Theoretical Mechanics, ODE, Probability. In 1892 defends PhD. In 1902 was elected to Science Academy. After wifes death 31.10.1928 committed suicide and died 3.11.1918.,What is “chaos”?,Chaos is a aperiodic long-time behavior arising in a deterministic dynamical system that exhibits a sensitive dependence on initial conditions.,Trajectories which do not settle down to fixed points, periodic orbits or quasiperiodic orbits as t,The system has no random or noisy inputs or parameters the irregular behavior arises from systems nonliniarity,The nearby trajectories separate exponentially fast,Lyapunov Exponent 0,Non-wandering set,- a set of points in the phase space having the following property: All orbits starting from any point of this set come arbitrarily close and arbitrarily often to any point of the set. Fixed points: stationary solutions; Limit cycles: periodic solutions; Quasiperiodic orbits: periodic solutions with at least two incommensurable frequencies; Chaotic orbits: bounded non-periodic solutions.,Appears only in nonlinear systems,Attractor,A non-wandering set may be stable or unstable Lyapunov stability: Every orbit starting in a neighborhood of the non-wandering set remains in a neighborhood. Asymptotic stability: In addition to the Lyapunov stability, every orbit in a neighborhood approaches the non-wandering set asymptotically. Attractor: Asymptotically stable minimal non-wandering sets. Basin of attraction: is the set of all initial states approaching the attractor in the long time limit. Strange attractor: attractor which exhibits a sensitive dependence on the initial conditions.,Sensitive dependence on the initial conditions,Definition: A set S exhibits sensitive dependence if r0 s.t. 0 and xS y s.t |x-y|r for some n.,pendulum A: = -140, d/dt = 0 pendulum B: = -1401, d/dt = 0,Demonstration,The sensitive dependence of the trajectory on the initial conditions is a key element of deterministic chaos!,However,Sensitivity on the initial conditions also happens in linear systems,SIC leads to chaos only if the trajectories are bounded (the system cannot blow up to infinity). With linear dynamics either SIC or bounded trajectories. With nonlinearities could be both.,xn+1= 2xn,But this is “explosion process”, not the deterministic chaos!,Why? There is no boundness.,There is no folding without nonlinearities!,The Lyapunov Exponent,A quantitative measure of the sensitive dependence on the initial conditions is the Lyapunov exponent . It is the averaged rate of divergence (or convergence) of two neighboring trajectories in the phase space. Actually there is a whole spectrum of Lyapunov exponents. Their number is equal to the dimension of the phase space. If one speaks about the Lyapunov exponent, the largest one is meant.,x0,p1(0),t - time flow,p2(t),p1(t),p2(0),x(t),Definition of Lyapunov Exponents,Given a continuous dynamical system in an n-dimensional phase space, we monitor the long-term evolution of an infinitesimal n-sphere of initial conditions. The sphere will become an n-ellipsoid due to the locally deforming nature of the flow. The i-th one-dimensional Lyapunov exponent is then defined as following:,On more formal level,The Multiplicative Ergodic Theorem of Oseledec states that this limit exists for almost all points x0 and almost all directions of infinitesimal displacement in the same basin of attraction.,Order: 1 2 n The linear extent of the ellipsoid grows as 21t The area defined by the first 2 principle axes grows as 2(1+2)t The volume defined by the first 3 principle axes grows as 2(1+2+3)t and so on The sum of the first j exponents is defined by the long-term exponential growth rate of a j-volume element.,Signs of the Lyapunov exponents,Any continuous time-dependent DS without a fixed point will have 1 zero exponents. The sum of the Lyapunov exponents must be negative in dissipative DS at least one negative Lyapunov exponent. A positive Lyapunov exponent reflects a “direction” of stretching and folding and therefore determines chaos in the system.,The signs of the Lyapunov exponents provide a qualitative picture of a systems dynamics,1D maps: ! 1=: =0 a marginally stable orbit; 0 chaos. 3D continuous dissipative DS: (1,2,3) (+,0,-) a strange attractor; (0,0,-) a two-torus; (0,-,-) a limit cycle; (-,-,-) a fixed point.,The sign of the Lyapunov Exponent,0 - the system attracts to a fixed point or stable periodic orbit. These systems are non conservative (dissipative) and exhibit asymptotic stability. =0 - the system is neutrally stable. Such systems are conservative and in a steady state mode. They exhibit Lyapunov stability. 0 - the system is chaotic and unstable. Nearby points will diverge irrespective of how close they are.,Computation of Lyapunov Exponents,Obtaining the Lyapunov exponents from a system with known differential equations is no real problem and was dealt with by Wolf. In most real world situations we do not know the differential equations and so we must calculate the exponents from a time series of experimental data. Extracting exponents from a time series is a complex problem and requires care in its application and the interpretation of its results.,Calculation of Lyapunov spectra from ODE (Wolf et al.),A “fiducial” trajectory (the center of the sphere) is defined by the action of nonlinear equations of motions on some initial condition. Trajectories of points on the surface of the sphere are defined by the action of linearized equations on points infinitesimally separated from the fiducial trajectory. Thus the principle axis are defined by the evolution via linearized equations of an initially orthonormal vector frame e1,e2,en attached to the fiducial trajectory,Problems in implementing:,Principal axis diverge in magnitude. In a chaotic system each vector tends to fall along the local direction of most rapid growth. (Due to the finite precision of computer calculations, the collapse toward a common direction causes the tangent space orientation of all axis vectors to become indistinguishable.),Solution:,Gram-Schmidt reorthonormalization (GSR) procedure!,GSR:,The linearized equations act on e1,e2,en to give a set v1,v2,vn. Obtain v1,v2,vn. Reorthonormalization is required when the magnitude or the orientation divergences exceeds computer limitation.,GSR never affects the direction of the first vector, so v1 tends to seek out the direction in tangent space which is most rapidly growing, |v1| 21t; v2 has its component along v1 removed and then is normalized, so v2 is not free to seek for direction, however v1,v2 span the same 2D subspace as v1,v2, thus this space continually seeks out the 2D subspace that is most rapidly growing |S(v1,v2)|2(1+2)t |S(v1,v2,vk)|2(1+2+k)t k-volume So monitoring k-volume growth we can find first k Lyapunov exponents.,Example for Henon map,An orthonormal frame of principal axis vectors (0,1),(1,0) is evolved by applying the product Jacobian to each vector:,Result of the procedure:,Consider: J multiplies each current axis vector, which is the initial vector multiplied by all previous Js. The magnitude of each current axis vector diverges and the angle between the two vectors goes to zero. GSR corresponds to the replacement of each current axis vector. Lyapunov exponents are computed from the growth of length of the 1-st vector and the area.,1=0.603; 2=-2.34,Lyapunov spectrum for experimental data(Wolf et al.),Experimental data usually consist of discrete measurements of a single observable. Need to reconstruct phase space with delay coordinates and to obtain from such a time series an attractor whose Lyapunov spectrum is identical to that of the original one.,Procedure for 1,Procedure for 1+2,Implementation Details,Selection of embedding dimension and delay time: emb.dim2*dim. of the underlying attractor, delay must be checked in each case. Evolution time between replacements: maximizing the propagation time, minimizing the length of the replacement vector, minimizing orientation error. Henon attractor: the positive exponent is obtained within 5% with 128 points d