285-317, 1985. If the Lyapunov exponent is zero, there is a bifurcation: a 1-cycle is doubling to a 2-cycle, a 2-cycle is. Terms offered: Fall 2009, Spring 2003, Spring 2000 Table-driven and retargetable code generators. Hence the Lyapunov exponent is a stability certiﬁcate for trajectories as opposed to an equilibrium. algorithm to compute the Lyapunov exponent and correlation dimension for quantitative observation of dynamic character-istics of the particles, and then analyze the relation between chaos and the swarm intelligent model. The underlying numerical method to. org) taught by Prof. Linearized dynamics _ = f(x0) f(x) = [ small )expand f(x0) around x] ˇ[f(x) + J(x)(x0 x)] f(x) = J(x) with stability matrix J(x) @[email protected] along x(t). Add to My List Edit this Entry Rate it: (1. We apply the method by M. Register management. So if we plot on an X Y coordinate system the depth of the bulge, X, on the X axis, and the Lyapunov exponent, at first flow is laminar and the Y value, ie the Lyapunov exponent stays 0. It is a well-known fact that close trajectories diverge exponentially fast in a chaotic system. Govorukhin, which is given in the folder lyapounov2. Lyapunov exponent calcullation for ODE-system. A term paper or project is. The set of Lyapunov exponents contains much physical information characteriz-ing the considered dynamical system, but calculation of the full spectrum demands much time and labor. Determining Maximum. 7 is a graph of the maximum Lyapunov exponent for a two-dimensional map and analytic curve. For the Lorenz system regions with chaotic, regular (oscillating) and vanishing dynamics can be identified. Even although others are free, they require the purchase of expensive commercial software on which they depend (e. In a code block in your script, type a relevant keyword, such as Lyapunov or Lyapunov exponent. Matlab codes to calculate lyapunov exponent of continuous data using Jacobean. 0 100 200 300 400 500 600 700 800 900 1000 −20 −15 −10 −5 0 5 10 Time Lyapunov Exponents Fig. The deformation matrix (deformation gradient tensor, Lyapunov matrix) M is de ned such that. Organizers. The Lyapnov exponents in a phase space of various plasma states (dilute, liquid and solid states) are measured with the use of SCOPE. Swinney, and J. Lyapunov exponent of the logistic map (Mathematica Code) Posted on May 15, 2015 by Neel In a previous post I’d shown a way to get the Lyapunov exponent from the time series data of any map. See the complete profile on LinkedIn and discover Yury’s connections and jobs at similar companies. Therefore to solve a higher order ODE, the ODE has to be ﬁrst converted to a set of ﬁrst order ODE’s. For r ∈ (2. In this exercise, we measure the sensitivity to initial conditions for the logistic map by introducing the Lyapunov exponent. This code corresponds to the formulas in Cochran and Ellner (1992). In those ﬂuids, explicit values have been computed for the maximum Lyapunov exponent as well as for the Kolmogorov-Sinai entropy per unit time which is equal to the sum. 01, N = 7, Kruskal–Wallis test). Some remarks are in order: Remark 1 The FTLE, , is a function of the state variable x at time t 0, but if we vary t 0, then it is also a function of time. ECE 120 Introduction to Computing credit: 4 Hours. , NY 10003, USA. In this paper the Benettin-Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo's derivative and the corresponding Matlab code are presented. Determining Maximum. (4) The Lyapunov dimension of the system is a fraction. (3) exists for all t>0 and satis es the exponential decay estimate jv(t)j Ce tjv(0)j for all t 0. - The median of the absolute value of the Lyapunov spectra. Compute the maximum Lyapunov exponent using a C-implemented estimator that uses the DOPRI853 integrator. Lyapunov exponent, chaos, Perron effects: time-varying linearization, stability and instability by the first approximation Author Kuznetsov N. If you have time series data, you can use this code. Lagrangian Coherent Structures and Finite-Time Lyapunov Exponents¶. Matlab codes to calculate lyapunov exponent of continuous data using Jacobean. Hence, only the largest Lyapunov exponent (LLE), which determines the predictability of the dynamical system, is frequently referred. the largest positive Lyapunov exponent by (x 0), or simply by. The relationship between the system state and the amplitude of the weak signal is defined by examining the Lyapunov exponents of the system. The values of z and r are used in computing the Lyapunov exponent according to the following definition. Roughly speaking the (maximal) Lyapunov exponent is the time constant, lambda, in the expression for the distance between two nearby orbits, exp (lambda * t). Ott "Chaos is Dynamical Systems", Cambridge. Danca, Nikolay Kuznetsov, International Journal of Bifurcation. Therefore, the graph tends toward the actual Lyapunov exponent. 09 m and l a = 0 , 35 m that respectively represents the position and sizing of piezoelectric actuator bonded on the flexible link. The Lyapunov exponent λ represents a measure of stable or unstable change in the nonlinear behavior of systems. We apply the method by M. Also, the source I found that equation from also states that the Lyapunov exponent ends up being $\ln\left(\dfrac{3+\sqrt{5}}2\right)$ which is just the natural log of one of the eigenvalues of the Jacobian. Theoretical studies of Oseledec  and numerical algorithm of Benettin et al. Lyapunov exponent, chaos, Perron effects: time-varying linearization, stability and instability by the first approximation Author Kuznetsov N. -Hai-Feng Liu, Zheng-Hua Dai, Wei-Feng Li, Xin Gong, Zun-Hong Yu(2005) Noise robust estimates of the largest Lyapunov exponent,Physics Letters A 341, 119Ò127 -Sprott,J. “There’s a value in chaos theory, a number called the Lyapunov exponent, which tells you how to predict a chaotic systems boundary – it’s knowledge horizon, if you like. For example, using lyapunovs and DynamicalSystem gives you the Lyapunov exponents of a dynamical system with known equations of motion. The underlying numerical method to solve. The largest Lyapunov exponents for the system. The underlying numerical method to. The Lyapnov exponents in a phase space of various plasma states (dilute, liquid and solid states) are measured with the use of SCOPE. These are videos from the Nonlinear Dynamics course offered on Complexity Explorer (complexity explorer. In this case, Lyapunov dimension is 1. (A) The largest Lyapunov exponent analysis of LFPs before, during and after ultrasound stimulation. May 13, 2020 Compiled on May 13,. the complete Lyapunov spectrum is written i lim N3 1 N ln i N i 0. We find that the maximum Lyapunov exponent of the critical behavior between fixed-point chaos and nonchaos always oscillates around zero. I learned a lot about calculating Lyapunov exponents and thinking practically about chaotic systems. It is common to refer to the largest one as the Maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. The 2 Lyapunov exponents for this ﬂlter are ¡ﬂﬁf, corresponding to a complex conjugate pair of eigenvalues. 1 State Space Reconstruction. numtraffic. The finite-time Lyapunov exponent, FTLE, which we will denote by , is a scalar value which characterizes the amount of stretching about the trajectory of point over the time interval [t, t + T]. A strictly positive maximal Lyapunov exponent is synonymous of exponential instability, but one should be warned that in some special cases, this may not be true (see, e. Contents: 6. Figure 1: Lyapunov exponents as we adjust p with step 0. In particular, the Lyapunov exponents (LEs) have been extensively. For r ∈ (2. nonlinearTseries Quickstart 2020-06-13. Approximate derivative. 3236 using the Matlab LET tool-box. This GUI can simulate and regulate a nonlinear dynamic system. λ = lim n → ∞ ⁡ ∑ i = 1 n ⁡ ln ⁡ r i ⁢ 1 − 2 ⁢ z i n. GaussLegendre, etc. 285-317, 1985. TABLE I: Summary of Lyapunov exponents Circuit QR Method Time Series Method Four Element 0. numerically obtained ﬂuid velocity ﬁelds to obtain the largest ﬁnite-time Lyapunov exponent (FTLE) ﬁeld. The table below summarizes our results. I am trying to find the maximal lyapunov exponent of a system described by 3 differential equations and one of the diff eq. The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. Jay Kominek's mpeg movie of what happens as you change tau in embeddings of data from the Lorenz system. Exactly the same code can be used, only input arguments should be changed to multiprecision type. Matlab source code nma_project_511. If you have a user account, you will need to reset your password the next time you login. org) taught by Prof. Superstable fixed points and superstable periodic points have a Lyapunov exponent of λ = −∞. Period-doubling route to chaos. E E 539 Advanced Topics in Solid-State Electronics (1-5, max. 2 Nonlinear Signal Processing. Flow analysis and global optimization methods. (A) The largest Lyapunov exponent analysis of LFPs before, during and after ultrasound stimulation. com and play around with the different constants to see what each does to the graph. Thus, there is a spectrum of Lyapunov exponents— equal in number to the dimensionality of the phase space. Figure 2: A comparison of resampled Monte{Carlo (J. ECE 120 Introduction to Computing credit: 4 Hours. The Lyapunov spectra of system (2)witht = 1000, starting from (1. 00 / 1 vote). The most used algorithm for that is one proposed in , but others have been proposed as well, such as Sato’s , which is the algorithm for Lyapunov exponent calculation in MATLAB compatible OpenTSTOOL software  based on non-linear time series algorithms and. , proposed the frequently used choice of calculating such ex-ponents using Gram-Schmidt orthonormalization process. of Lyapunov exponents from experimental time series. , NY 10003, USA. m SO_build_datasets. 7 shows the Lyapunov exponents using the lost digits technique (solid line) and the analytic solution (dashed line) for the largest Lyapunov exponent for this map. Intermediate Lyapunov exponents for systems with periodic orbit gluing property. four Lyapunov exponents is less than zero. This paper introduces a relative simple method to calculate the Lyapunov exponent for a system of ordinary differential equations. For finding out Lyapunov spectrum I already tried Tisean. First, it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. Exactly the same code can be used, only input arguments should be changed to multiprecision type. For p = −1, we calculate the Lyapunov exponents as shown in Fig. The Lyapunov exponent of LFPs in the LIPUS group. Many important techniques are not represented, for instance algorithms for estimating lyapunov exponents. Problem Set 9: Lyapunov exponents. The negative value of maximum Lyapunov exponent implies exponential convergence of nearby system trajectories. Lyapunov exponent of the logistic map (Mathematica Code) Posted on May 15, 2015 by Neel In a previous post I’d shown a way to get the Lyapunov exponent from the time series data of any map. For most flows of practical importance, the FTLE varies as a function of space and time. Zhang, “Estimation of Lyapunov Exponents From A Time Series For N-Dimensional State Space Using Nonlinear Mapping,” Nonlinear Dynamics, 69: 1493-1507, 2012 (2012 Impact factor: 3. I also know that Lyapunov exponents alone cannot be used to determine if a system is chaotic or periodic. The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. For this, one can plot ln d(k) vs k and apply a linear fit. For different perturbations of initial conditions we have different Lyapunov exponents often called as a Lyapunov spectrum. Nanda Kishore Reddy, Lyapunov exponents and eigenvalues of products of random matrices : 12:20-13:00: Arijit Chakrabarty, The Hadamard product and the free convolutions: 13:00-14:30: Lunch: 19:00-21:00: Conference dinner : Friday: 10:00-10:40: 10:40-11:20: Ayan Bhattacharya,Large deviation for extremes of branching random walk with regularly. Period-doubling route to chaos. This remarkable speeding up was possible because the balancing strategy implemented in MB05OD can reduce the matrix norm for triangular matrices and save matrix. Numerical methods and computer codes for fluid flow problems. 1 (1993): 117-134. The third line, corresponding to the chaotic case of E 3, indicates a positive value of the exponents. the Lyapunov exponents are the same then they can only take the value zero. 1] were ranked and listed in Tables 4 and 5. References:  Sandri, M. 1 and c = −2. Roughly speaking Lyapunov exponents can be thought of as the generalization of eigenvalues from linear systems to nonlinear systems. Figure 6(b) shows a diagram of maximum Lyapunov exponents for a constant input frequency of 62 rad/s. Come to Algebra-equation. Exponential Transformation is the detailed implementation handbook for becoming an Exponential Organization. Prerequisites. By "complete" I mean simply that the only additional things you need to use the software are 1) some data and 2) matlab. Finding p-cycles and showing their stability. 285-317, 1985. the maximum nonzero Lyapunov exponent. However, this procedure can be tedious for large-scale complex ﬂows of general interest. py Code example to make cobweb plots identifying fixed points and periodic orbits of iteratively applied one dimensional maps. Therefore to solve a higher order ODE, the ODE has to be ﬁrst converted to a set of ﬁrst order ODE’s. exponent greater than zero. 234 Microprocessor Systems 4 (3-3) Course Prerequisite: CPT S 121 with a C or better; E E 214 with a C or better. positive the Lyapunov Exponent be the more chaotic behaviour. I want to investigate the chaotic nature of Lorenz system when the parameter R varies. Maxime Fortier-Bourque. This "sensitivity to initial conditions" is generally taken as the definition of chaos. Calculating Lyapunov exponents with jitcode_lyap ¶ jitcode_lyap is a simple extension of jitcode that almost automatically handles calculating Lyapunov exponents by evolving tangent vectors. 2 Lyapunov Exponents. Superstable fixed points and superstable periodic points have a Lyapunov exponent of λ = −∞. max-s) and long-term (λ max-l) maximum finite-time Lyapunov exponents were calculated from measured trunk kinematics to assess stability. A MATLAB-based program for dynamical system investigation. However, it is hard to follow and there is no graphical output of the plot (and it is in C, more complex). Exactly the same code can be used, only input arguments should be changed to multiprecision type. We can solve for this exponent, asymptotically, by ˇln(jx n+1 y n+1j=jx n y nj) for two points x n;y nwhere are close to each other on the trajectory. If delta(0) is the distance between two Takens' vectors in the embedding. Lyapunov Exponents. I run the following code but I got an empty plot. Papachristodoulou A Converse Sum-of-Squares Lyapunov Result with a Degree Bound IEEE Transactions on Automatic Control, Vol 57, No. A positive MLE is usually taken as an indication that the system is chaotic. The phase-space diagram is that of a strange attractor. It is used as a quantitative measure for the sensitive dependence on initial conditions. ECE 120 Introduction to Computing credit: 4 Hours. DANCA Romanian Institute od Science and Technology, 400487 Cluj-Napoca, Romania, [email protected] Therefore to solve a higher order ODE, the ODE has to be ﬁrst converted to a set of ﬁrst order ODE’s. In this paper, the Benettin-Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. Govorukhin. tive Lyapunov exponents from finite amounts of experimental data. Select Estimate Lyapunov Exponent from the suggested command completions. In this sketch $\lambda$ is the maximum Lyapunov exponent (and in general a system has as many exponents as its dimensionality). This exponent can be calculated as the real components of the eigenvalue solutions to the differential equations describing a system. 38 Pages Posted: 15 Mar 2007. A positive MLE is usually taken as an indication that the system is chaotic. Optimization by program transformation. introduction of stretching quantiﬁers such as the diﬀerent types of Lyapunov exponents. 76×106 individual parameter points. positive the Lyapunov Exponent be the more chaotic behaviour. 5,200,[0 1 0],10); See files: lyapunov. If delta(0) is the distance between two Takens' vectors in the embedding. the maximum iteration. Lyapunov不等式与Riccati不等试是控制理论中广泛应用的两类线性矩阵不等式,其正定可行解问题的研究一直是控制理论中的核心问题,文中从矩阵不等式的基本描述出发对以上两种有直接联系且有重要应用意. The Lyapunov exponent of LFPs in the LIPUS group. Samuel Lelièvre. Theory and MATLAB code for depth estimation from stereo image pairs using the simple procedure of block-matching. In a code block in your script, type a relevant keyword, such as Lyapunov or Lyapunov exponent. Also, in the case of the logistic map, any system with a Lyapunov exponent greater than zero is a chaotic system. Lyapunov exponents characterize the dynamics of a system near its attractor. 6493ቍ �ݢ ቌ߫,߭,߬ቍ ൞ ቌ0. Charles Fougeron. 1: Lyapunov exponents of a smooth map f on R with x 0 as initial point be defined as λ (x 0) = [ ln ( f (x ) ln ( f (x ) ln ( f (x ) ln ( f (x ) ] n 1 Lim 0 2 n 1 provided the limit exists. The method used is based on a fitting of the time series with an analytical function through a least square minimization, after which the Lyapunov exponents are determined from the stability of the trajectories calculated using this analytical function. is a delay differential equation including a parameter K. Journal of Computational Physics 319 , 9-27. In particular, u 0 is an asymptotically stable solution of (1). (image Depends on: Gram-Schmidt function. A Lyapunov exponent is associated with each trajectory Γ 0 and each initial perturbation δΓ 0. You can choose and change arbitrary the number of iteration. Charles Fougeron code for Lyapunov exponents of strata coverings (see trac ticket #16102) Chris Swierczewski Riemann surfaces package. 7 shows the Lyapunov exponents using the lost digits technique (solid line) and the analytic solution (dashed line) for the largest Lyapunov exponent for this map. (34), using Matlab software. The next step is to extend the Lyapunov exponents to three-dimensional flows. the complete Lyapunov spectrum is written i lim N3 1 N ln i N i 0. Govorukhin. Govorukhin, which is given in the folder lyapounov2. 5 As noted above, negative values of λ here indicate stability in the logistic map. Before computing The Largest Lyapunov Exponent, you must find the minimum embedding dimension(m), time delay(tao) and mean period parameters. Charles Fougeron code for Lyapunov exponents of strata coverings (see trac ticket #16102) Chris Swierczewski Riemann surfaces package. This approach allows for calculation of the full spectrum of Lyapunov exponents through local. The theory is then applied to real-world problems from a broad range of disciplines including population dynamics, biology, economics, nonlinear optics, and neural networks. [Note the new due date!] Homework 3, due Thursday, March 14, in class! Homework 4, due Thursday, April 18. DANCA Romanian Institute od Science and Technology, 400487 Cluj-Napoca, Romania, [email protected] For higher dimensions, you can define this to be the Euclidean distance and modify the code accordingly. Following posts gives a very basic example to hel user use the lyapunov function in Matlab. We'll measure the spectrum of the three Lyapunov exponents of the Lorenz chaotic attractor in three dimensions. The underlying numerical method to solve. The plane goes through an point P[2 5 3] with an normal vector of [0. If you change delta_r to 0. New Directions in Statistical Signal Processing: From Systems to. Largest Lyapunov exponent: LLE. • Maximum Lyapunov exponent. numerically obtained ﬂuid velocity ﬁelds to obtain the largest ﬁnite-time Lyapunov exponent (FTLE) ﬁeld. 09 m and l a = 0 , 35 m that respectively represents the position and sizing of piezoelectric actuator bonded on the flexible link. Computing exactly, or approximating, Lyapunov exponents of heaps of pieces, and more gener- ally, of products of max-plus matrices, is a long standing problem [15, 44, 41, 1, 28, 23, 12, 3, 4, 25,. In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. That’s the maximum limit of our foreknowledge!. I am researching about Lyapunov exponents, and that the Rosenstein algorithm can be used to calculate the maximum exponent. The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in MATLAB's {\tt expm} function. It plots out a graph of AVERAGES of the log of f'(x) over a trajectory of length N. We plot the above points in order to get the graph of First Lyapunov Exponent as elicited below: 2. The syntax for calculating Lyapunov exponents is: [T,Res]=tc_explyapunov(f,to,dt,tf,ystartutp) [T,Res]=tc_explyapunov2(f,g,to,dt,tf,ystart,ioutp) [T,Res]=tc_explyapunov3(f,g,h,to,dt,tf,ystart,ioutp). Vastano, "Determining Lyapunov Exponents from a Time Series," Physica D, Vol. Flow analysis and global optimization methods. The maximum Lyapunov exponent is 0 throughout the angle range of 14}603, indicating quasi-periodic motion. One of the features of chaos is exponential divergence (sensitivity to initial conditions). By "complete" I mean simply that the only additional things you need to use the software are 1) some data and 2) matlab. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Table (1) shows a calculation of the CLEs for various subsystems. Sterk Second supervisor: H. Danny Kaplan's Matlab code for Nonlinear Time Series Analysis; Danny Kaplan's Electronic Supplement to Understanding Nonlinear Dynamics; Artificial Noise Generators in Matlab (white, pink and brown noise). Lyapunov exponents found with our method agreed well with those obtained using the Matlab code by V. � However, if lambda is positive, then the distance between nearby orbits grows exponentially in time, and the system exhibits sensitive dependence on initial conditions. The code, as you will see, becomes substantially more complex. Orientation to electrical and computer engineering via hands-on exercises and projects; introduction to circuits, components, instrumentation, and electronic prototyping; computer productivity tools; hardware and software trouble shooting. When a = 4, b = −0. Samuel Lelièvre. Lyapunov Exponents for ODE; Referenced in 4 articles Calculation Lyapunov Exponents for ODE, open source MATLAB code: Lyapunov exponent calcullation for ODE-system this m-file for determining Lyapunov exponents was proposed in A. The nonlinearTseries package provides functionality for nonlinear time series analysis. The Lyapunov characteristic exponent [LCE] gives the rate of exponential divergence from perturbed initial conditions. 25ቍ(Sprott, 2007). For the logistic map f(x)=ax(1-x), write a program to compute the Lyapunov exponent and reproduce Fig. Translated to the universe example, we would look at two almost identical universes and see whether their states diverge from each other. By "complete" I mean simply that the only additional things you need to use the software are 1) some data and 2) matlab. py Code example to make cobweb plots identifying fixed points and periodic orbits of iteratively applied one dimensional maps. chaos lyapunov henon-map lorenz-attractor. Charles Fougeron. the maximum nonzero Lyapunov exponent. Also, in the case of the logistic map, any system with a Lyapunov exponent greater than zero is a chaotic system. Before computing The Largest Lyapunov Exponent, you must find the minimum embedding dimension(m), time delay(tao) and mean period parameters. 99, the maximum Lyapunov exponent by numerical computation is [[lambda]. The Lyapunov exponents are the natural logs of the eigenvalues of the Jacobian matrix for the map and are given by the following equation: L±=ln{1+r/2±[(1+r/2) 2 −1] 1/2}. The objective of this thesis is to nd the parameter values for a system that determines chaos via the Lyapunov exponents. A version of MATLAB m-code is provided in the appendix. Lok S2393263 ais the max-imum rate of population growth and K represents a sort of "ideal" population or "carrying capacity", which is basically the maximum size of the population. java, Lyapunov. few weeks were devoted to measuring stock volatility with Lyapunov exponents. Jordan, and A. Bifurcation diagrams, chaos, intermittency, Lyapunov exponents, periodicity, quasiperiodicity, and universality are some of the topics introduced. Since then, he has helped organizations disrupt their own industries—by applying Exponential Organization (ExO) principles. We develop Lyapunov theory for these mixed discrete-continuous dynamics and use Sum-of-Squares to construct proofs of stability and exponential convergence. Also, the source I found that equation from also states that the Lyapunov exponent ends up being $\ln\left(\dfrac{3+\sqrt{5}}2\right)$ which is just the natural log of one of the eigenvalues of the Jacobian. The slope will be an estimate for the Lyapunov exponent. For p = −0. Lyapunov exponents found with our method agreed well with those obtained using the Matlab code by V. "A practical method for calculating largest Lyapunov exponents from small data sets. It plots out a graph of AVERAGES of the log of f'(x) over a trajectory of length N. The Estimate Lyapunov Exponent task lets you interactively estimate the Lyapunov exponent of a uniformly sampled signal. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 3 Write a Matlab function starting with the line function h = lyapexp(r, x0, Npre, n); which computes the Lyapunov exponent of the logistic map with the parameter r. E E 539 Advanced Topics in Solid-State Electronics (1-5, max. If the linear range of curve is determined,code can calculate the largest lyapunov exponent. The values of z and r are used in computing the Lyapunov exponent according to the following definition. Alternatively, by using numericallyapunov and Dataset you can approximate the maximum Lyapunov exponent of a measured trajectory. Learn more about nlinfit, lyapunov exponents, nonlinear regression. Expanding up to t 1 {\displaystyle t^{1}} gives P 0 (x) = 1 , P 1 (x) = x. ATM S 591:Source Code. Automatic ©Matlab application for fitting log-normal and 'power-law' distribution to empirical data, following the goodnes-of-fit based appr Code for Fig. A PMU-based online waveform stability monitoring technique is proposed based on the Maximum Lyapunov Exponent (MLE). For this purpose, the Lyapunov exponent values and bifurcation behaviours of two hyperchaotic systems are analysed for weak signal detection applications. is a delay differential equation including a parameter K. 01 in the code and plot dots instead of circles -- change plot(X,Y,'ko') to plot(X,Y,'k. "Numerical Calculation of Lyapunov Exponents. Homework 2, due Thursday, February 28. Lyapunov exponents of the Mandelbrot set (The mini-Mandelbrot) - Matlab. This NLD feature quantifies the exponential divergence between neighbor trajectories in the embedded space (the phase space or the attractor). Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Some remarks are in order: Remark 1 The FTLE, , is a function of the state variable x at time t 0, but if we vary t 0, then it is also a function of time. For this, one can plot ln d(k) vs k and apply a linear fit. Introduction to digital logic, computer systems, and computer languages. m convolve2. This plot is coarse-grained (for lack of a better term) so you can more easily see what's being plotted. Sterk Second supervisor: H. E E 539 Advanced Topics in Solid-State Electronics (1-5, max. 1 Naive numerical evaluation of 1 A naive approach is to solve the dynamical system x_ = f(x) numerically for two trajectories starting at x(0) and x(0) + (0). , the so-called Perron effect) (Leonov and Kuznetsov 2006) A strictly positive maximal Lyapunov exponent is often considered as a definition of deterministic chaos. The Lyapunov exponents are the natural logs of the eigenvalues of the Jacobian matrix for the map and are given by the following equation: L±=ln{1+r/2±[(1+r/2) 2 −1] 1/2}. Consistently ranked among the top programs in the US, our department is active in many areas of Computer Science, Applied Mathematics, Communication & Networks, and Control & Systems, and is engaged in a broad range of interdisciplinary initiatives: our research projects frequently involve connections with other disciplines such as Applied Physics, Bioengineering, Biology, Economics. The Matlab code for the simulations is available to the maximum possible population size Ground-truth largest Lyapunov exponents were calculated using code provided in Dynamical. • Numerical (maximum) Lyapunov exponent of a timeseries. It was originally called expgui; today it is known as expshow. In this paper we do not deal with other Lyapunov exponents, and, for the sake of simplicity, we omit the word largest. Liz Bradley. Before computing The Largest Lyapunov Exponent, you must find the minimum embedding dimension(m), time delay(tao) and mean period parameters. Lyapunov number is the exponent of Lyapunov exponent and is given by ( ) 0. On 1983, Gerry bases that as long as the maximum Lyapunov exponent is greater than zero, it certainly is in chaos. Use MatLab or Octave to compute the eigenvalues of your matrix. One the major stability technique for non linear dynamic system is called lyapunov. java, Lyapunov. It can be seen, say, that when a=2 , the LE is positive and chaos is in action [Check the bifurcation diagram to get the same intuition for that value of a ]. Alan Wolf's code to study the Lyapunov exponents of some time series (link below). The running estimates of Lyapunov exponents are computed and stored in the output file. 两类基于MATLAB的Lyapunov与Riccati线性矩阵不等式可行解的算法分析与验证. Divergence of nearest trajectories can be seen on the graph. The averaged exponent that determines the divergence rate is called the Lyapunov exponent (usually denoted with lambda). Abstract: This M-file calculates Lyapunov exponents with minimum RMSE neural network. In particular, u 0 is an asymptotically stable solution of (1). The syntax for calculating Lyapunov exponents is: [T,Res]=tc_explyapunov(f,to,dt,tf,ystartutp) [T,Res]=tc_explyapunov2(f,g,to,dt,tf,ystart,ioutp) [T,Res]=tc_explyapunov3(f,g,h,to,dt,tf,ystart,ioutp). jl are listed in the Contents page. A MATLAB-based program for dynamical system investigation. " Physica D: Nonlinear Phenomena 65. Here is a link to the code. (1986) to estimate the whole spectrum of Lyapunov exponents. Figures 11 and 12 show the objective function that depends on l a and x a variables with α = 300. Lyapunov exponent calcullation for ODE-system. Now, we will calculate the conditional Lyapunov exponents (CLEs) numerically. Its (positive) maximum Lyapunov exponent was optimized and the feasible solutions (combinations) of the coefficient values for a, b, c, [d. Result can be compared to the Matlab code written by V. 7 is a graph of the maximum Lyapunov exponent for a two-dimensional map and analytic curve. It is common to refer to the largest one as the Maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. I am looking for a code or a program package to compute the complete spectrum of Lyapunov exponents from time series of experimental data. Analysis of the maximum finite time Lyapunov exponent in time domain using data from torso stability tests @inproceedings{Dong2015AnalysisOT, title={Analysis of the maximum finite time Lyapunov exponent in time domain using data from torso stability tests}, author={Chaoke Dong}, year={2015} }. Participants. The syntax for calculating Lyapunov exponents is: [T,Res]=tc_explyapunov(f,to,dt,tf,ystartutp) [T,Res]=tc_explyapunov2(f,g,to,dt,tf,ystart,ioutp) [T,Res]=tc_explyapunov3(f,g,h,to,dt,tf,ystart,ioutp). If ExpansionRange is specified as a scalar M, then the range is set to be [1, M]. Application of the Lyapunov exponents technique fails, in general, to reveal ﬁne {Pmax(q)}=1/(1 in Matlab by using various built-in functions ranging. Compute the magnitude of the eigenvalues. I have find out embedding dimension by FNN method and delay by mutual information method. ro NIKOLAY KUZNETSOV Dept. Official source code of Arrhythmia Detection. Catalog Program Pages Referencing EGM 7819. the maximum iteration. Many important techniques are not represented, for instance algorithms for estimating lyapunov exponents. Maxime Fortier-Bourque. m convolve2. Matlab code for Lyapunov exponents of fractional order systems, Marius-F. Swinney, and J. Reset your password. A practical method for calculating largest Lyapunov exponents from small data sets. Stochastic systems biology: a paper by Golightly and Wilkinson. A couple of papers looking at seasonal effects in Measles. 2020-07-12. Lyapunov exponent calcullation for ODE-system. A Lyapunov function candidate is derived that links the energy function and the transient stability of the integrated system with the wide-area controller. Since then, he has helped organizations disrupt their own industries—by applying Exponential Organization (ExO) principles. These are designed to work with Matlab v7. Terms offered: Fall 2009, Spring 2003, Spring 2000 Table-driven and retargetable code generators. I run the following code but I got an empty plot. Note that the Hopf algorithms have since been significantly extended see the Hopf tutorial What is bifurcation Bifurcation examples in. of Applied Cybernetics, Saint-Petersburg State University, Russia and Dept. Timothy Behrens. Rosenstein, Michael T. Govorukhin, which is given in the folder lyapounov2. It gives a quantitative measure of the average separation of two initially close orbits as the system evolves in time. Amplitude modulation: DSB, SSB, AM, QAM and VSB; phase and frequency modulation and the basic design of a FM transmitter. When for at least one fixed point in a cycle both the Lyapunov exponents are zero we prove that the entire projective bundle is a single chain recurrent component. The motivation is that hyperchaotic systems are preferable for use in chaotic data protection systems due to their more complex behaviour, but very few such models are known. I have adapted this code to estimate Lyapunov spectra for continuous-time systems like the Lorenz attractor and Rossler attractor. _____ Some of my scientific programming code is publicly available through GitHub here. We used the attributes in Figure 5c for computing the FTLE. Numerical computation of Jacobians: Code Limit cycles: finding them using root find, Poincare maps, and stability: Code Parametric excitation: Mathieu's equations, tongues of instability: Code Chaos in the logistic map. Govorukhin. 1, 0, 0, -1. A practical method for calculating largest Lyapunov exponents from small data sets. [Luis Barreira; Ya B Pesin] -- Designed to work as a reference and as a supplement to an advanced course on dynamical systems, this book presents a self-contained and comprehensive account of modern smooth ergodic theory. Papachristodoulou A Converse Sum-of-Squares Lyapunov Result with a Degree Bound IEEE Transactions on Automatic Control, Vol 57, No. I'm using Prof. If you have time series data, you can use this code. A less general procedure [6, 11-14] for estimat- ing only the dominant Lyapunov exponent in ex- perimental systems has been used for some time. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5105-5118. I recently find the LCE program that is running in mathematica, but I cant run it for my problem; Dose anyone have matlab code of LCE? Thank you very much Eric Maik. The non-linear dynamic examples studied here are the Lorenz and the Rosler systems as well as the non-isothermal chemical system of Tomlin and Scott. MATLAB中文论坛MATLAB/Simulink 资源共享 (File Exchange)板块发表的帖子：计算混沌系统的Lyapunov指数谱GUI工具箱。此工具箱用Wolf法计算以下混沌系统的Lyapunov指数谱：1. EE 322 can be taken concurrently with this course. Lyapunov exponent calcullation for ODE-system. Application of the Lyapunov exponents technique fails, in general, to reveal ﬁne {Pmax(q)}=1/(1 in Matlab by using various built-in functions ranging. 2: Estimating Area Under a Curve eTool (Desmos) Lesson 2. 1 Description of computer codes Unpacking the zip archive ( le name: SOinC_source_files_4_Oct_2012. The Lyapunov exponents are the natural logs of the eigenvalues of the Jacobian matrix for the map and are given by the following equation: L±=ln{1+r/2±[(1+r/2) 2 −1] 1/2}. We get the saddle node bifurcation has positive Lyapunov exponent if µ≤−1,for all the domain. dim-dimensional space, we expect that the distance after a time t between the two trajectories. Now, for every R, I want to take the Lyapunov exponent at the end time. "Stabilizability of Linear Time-Varying Systems," Qualifying Examination Report, GT, USA, 1987. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. Roughly speaking Lyapunov exponents can be thought of as the generalization of eigenvalues from linear systems to nonlinear systems. We used a MATLAB code based on the Eckmann and Ruelle QR decomposition technique to calculate the exponents (Eckmann & Ruelle, 1985). This package permits the computation of the most-used nonlinear statistics/algorithms including generalized correlation dimension, information dimension, largest Lyapunov exponent, sample entropy and Recurrence Quantification Analysis (RQA), among others. m - algorithm lorenz_ext - Lorenz system extended rhs run_lyap - example of calling and result visualization Govorukhin V. The exponential divergence of phase-space trajectories (as described with the Lyapunov exponent) is something seen in the comparison of two (nearby) trajectories in a dynamical system. The Deep Time-Delay Reservoir Computing concept utilizes unidirectionally connected systems with time-delays for supervised learning. My problem is that I don't know first Lyapunov exponent is average of all the lyapunovs when k = 1 or average of all the lyapunovs for the first time that data[i] - data[j] < eps? Is this right implementation for Lyapunov exponent? And this is the Numerical Calculation of Lyapunov Exponent. like to thank Dr. The following Matlab project contains the source code and Matlab examples used for largest lyapunov exponent with rosenstein's algorithm. m, the MATLAB code rhs. The techniques included emphasize nonlinear prediction and surrogate data, as well as my own delta-epsilon technique. Lyapunov exponent calcullation for ODE-system. The non-linear dynamic examples studied here are the Lorenz and the Rosler systems as well as the non-isothermal chemical system of Tomlin and Scott. values and eigenvectors  and ﬁnite-time Lyapunov exponents and vectors . Pat Hooper. University of Auckland. VOICEBOX: Speech Processing Toolbox for MATLAB Introduction. I'm using Prof. Liz Bradley. 2: Estimating Area Under a Curve eTool (Desmos) Lesson 2. In this case quasi-periodic motion occurs at a angle of 28)53 and continues for all values up to 413 as indicated by the zero maximum. 8 is a graph of the maximum Lyapunov exponent for the Lorenz equations averaged over nine exponents. More information's about Lyapunov exponents and nonlinear dynamical systems can be found in many textbooks, see for example: E. m Cobwebs for logistic map, without transients logistic_orbit. Roughly speaking the (maximal) Lyapunov exponent is the time constant, lambda, in the expression for the distance between two nearby orbits, exp (lambda * t). The code has been tested with results of Rosenstein's article. Figures 11 and 12 show the objective function that depends on l a and x a variables with α = 300. We used the attributes in Figure 5c for computing the FTLE. Lyapunov Exponents. max-s) and long-term (λ max-l) maximum finite-time Lyapunov exponents were calculated from measured trunk kinematics to assess stability. See the official syllabus. I am using the code which was previously posted on Mathworks. pdf), Text File (. May 13, 2020 Compiled on May 13,. Provides the algorithm of Rosenstein et al. 01 in the code and plot dots instead of circles -- change plot(X,Y,'ko') to plot(X,Y,'k. 38424-largest-lyapunov-exponent-with-rosenstein-s-algorithm), MATLAB Central File Exchange. Exactly the same code can be used, only input arguments should be changed to multiprecision type. Lyapunov exponent calcullation for ODE-system. The method of using thrust and CUDA for parameter studies is very efficient since many systems are solved in parallel. The minimum and maximum value of ExpansionRate is used to estimate the local expansion rate to calculate the Lyapunov exponent. This allows you to estimate the Lyapunov exponent of a scalar map by only knowing the. For the logistic map f(x)=ax(1-x), write a program to compute the Lyapunov exponent and reproduce Fig. like to thank Dr. java, Lyapunov. Jonathan Vahe - oFusers day - May 18th 2016 Development of mixing indicators: Finite Time Lyapunov Exponents 1 Jonathan Vahe May 18th 2016 Sc = 1 Sc = 150. Lyapunov number is the exponent of Lyapunov exponent and is given by ( ) 0. A Matlab version of the Lyapunov exponent estimation algorithm of Wolf et al. The table below summarizes our results. Note: In order to get a good estimate of the Lyapunov exponent, your initial guess should already lie on the strange attractor. Also plot (just below) the bifurcation diagram for the same a -window and compare/discuss both plots. So if we plot on an X Y coordinate system the depth of the bulge, X, on the X axis, and the Lyapunov exponent, at first flow is laminar and the Y value, ie the Lyapunov exponent stays 0. 3236 using the Matlab LET tool-box. Lyapunov exponents. Hybrid Toolbox Author: Alberto Bemporad The Hybrid Toolbox is a Matlab/Simulink toolbox for modeling and simulating hybrid dynamical systems, for designing and simulating model predictive controllers for linear and for hybrid systems subject to constraints, and for generating equivalent piecewise linear control laws that can be directly embedded as C-code in real-time applications. Bachelorproject 2016 K. Python; cbnfreitas / lyapunov_exponent_map_and_ode Star 4 Code Issues Pull requests Lyapunov exponent of maps and ODE in Python 3, example with Henon Map and Lorenz System. m Lyapunov exponents for logistic map. 1 (1993): 117-134. m - algorithm lorenz_ext - Lorenz system extended rhs run_lyap - example of calling and result visualization Govorukhin V. 7 Lyapunov Exponents for Rotating Rayleigh-B´enard Convection 143. VOICEBOX: Speech Processing Toolbox for MATLAB Introduction. Use MatLab or Octave to compute the eigenvalues of your matrix. See the complete profile on LinkedIn and discover Yury’s connections and jobs at similar companies. This code corresponds to the formulas in Cochran and Ellner (1992). ATM S 591:Source Code. "On the Numerical Computation of the Lyapunov Spectrum: A Stopping Rule,"CCSE Technical report-CCSE-016 (1994). In this work, an alternative approach involving computation of the FTLE on-the-ﬂy during direct numerical simulation of the. Sandri in order to determine the maximum Lyapunov exponent as well as all the Lyapunov exponents. First it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. For a discrete system x n+1 =f(x n) and for an orbit starting with x 0, the Lyapunov exponent can be. The Lyapunov exponent was found to be sensitive to changes in evolution time but not sensitive to the cutoff frequency of the low pass filter when it was above 3. 1: Lyapunov exponents of a smooth map f on R with x 0 as initial point be defined as λ (x 0) = [ ln ( f (x ) ln ( f (x ) ln ( f (x ) ln ( f (x ) ] n 1 Lim 0 2 n 1 provided the limit exists. 577 0577 0577]. com and play around with the different constants to see what each does to the graph. Specify the number of dimensions of phase space vectors as a scalar or vector from the MATLAB workspace. When for at least one fixed point in a cycle both the Lyapunov exponents are zero we prove that the entire projective bundle is a single chain recurrent component. References:  Sandri, M. We can solve for this exponent, asymptotically, by ˇln(jx n+1 y n+1j=jx n y nj) for two points x n;y nwhere are close to each other on the trajectory. The reason for taking the logarithm and dividing by the magnitude of the integration time T is the parallel that we then establish with autonomous systems. The 2 Lyapunov exponents for this ﬂlter are ¡ﬂﬁf, corresponding to a complex conjugate pair of eigenvalues. A PowerBASIC translation is also available along with an executable version. Lyapunov exponent calcullation for ODE-system. For the proposed system, the first two conditions are satisfied obviously. If you have time series data, you can use this code. LyE calculations include 3 steps: Time lag (τ) calculation via Average Mutual Information Embedding dimension (m) calculation via Global False Nearest Neighbor Distance calculation τand m transform the force profiles into state space. " Physica D: Nonlinear Phenomena 65. For different perturbations of initial conditions we have different Lyapunov exponents often called as a Lyapunov spectrum. Now I need to plot lyapunov versus parameter, ''say a''. Theoretical studies of Oseledec  and numerical algorithm of Benettin et al. Skip to content. This "sensitivity to initial conditions" is generally taken as the definition of chaos. I hope you are able to start up MATLAB and actually run expshow while you are reading this blog. Before computing The Largest Lyapunov Exponent, you must find the minimum embedding dimension(m), time delay(tao) and mean period parameters. If the linear range of curve is determined,code can calculate the largest lyapunov exponent. That’s the maximum limit of our foreknowledge!. 3236 using the Matlab LET tool-box. The simulation of the small data method to calculate the largest Lyapunov exponent. Therefore to solve a higher order ODE, the ODE has to be ﬁrst converted to a set of ﬁrst order ODE’s. four Lyapunov exponents is less than zero. -Hai-Feng Liu, Zheng-Hua Dai, Wei-Feng Li, Xin Gong, Zun-Hong Yu(2005) Noise robust estimates of the largest Lyapunov exponent,Physics Letters A 341, 119Ò127 -Sprott,J. The Estimate Lyapunov Exponent task lets you interactively estimate the Lyapunov exponent of a uniformly sampled signal. Analysis of the maximum finite time Lyapunov exponent in time domain using data from torso stability tests @inproceedings{Dong2015AnalysisOT, title={Analysis of the maximum finite time Lyapunov exponent in time domain using data from torso stability tests}, author={Chaoke Dong}, year={2015} }. Combined with one null exponent along the flow and one negative exponent to ensure the boundness of the solution, the minimal dimension for a (continuous) hyperchaotic system is 4. Collins, and Carlo J. 3, we have the Lyapunov exponent for the logistic map: Eq. The Lyapunov Exponent Test and the 0 1 Test for Chaos compared Kristel Lok S2393263 First supervisor: A. Finite-time Lyapunov exponents, and in some cases vectors, have also been used to analyze ﬂuids , ,  and atmospheric circulation . The table below summarizes our results. This blog post states that:. MATDS (Govorukhin VN, 2010) is MATLAB -based software for investigation of dynamic systems; it displays bifurcations and Lyapunov exponents, has few system analysis methods and its graphical interface is a bit complex to use. is a delay differential equation including a parameter K. 7 Lyapunov Exponents for Rotating Rayleigh-B´enard Convection 143. In particular, the characteristic quantities of chaos have been obtained for the hard-ball ﬂuids thanks to the methods of statistical mechanics [3–5]. The Lyapunov spectra of system (2)witht = 1000, starting from (1. 249984719222, ω = 0. This book shows how to code different algorithms for drawing Lyapunov exponent of real quadratic map in PDL, IDL, MATLAB, Octave, C and FORTRAN77 by. Following posts gives a very basic example to hel user use the lyapunov function in Matlab. pdf), Text File (. The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. Lyapunov Exponents, Stochastic Diﬀerential Equations, QR-decomposition, numer-ical methods. Hybrid Toolbox Author: Alberto Bemporad The Hybrid Toolbox is a Matlab/Simulink toolbox for modeling and simulating hybrid dynamical systems, for designing and simulating model predictive controllers for linear and for hybrid systems subject to constraints, and for generating equivalent piecewise linear control laws that can be directly embedded as C-code in real-time applications. Lyapunov Exponent Lyapunov exponent represents the features of a chaotic system and can largely express the overall performance of chaotic maps. like to thank Dr. A key point of this research was to understand how the Lyapunov exponent changed with time as it approached a critical event. The task automatically generates MATLAB ® code for your live script. ro NIKOLAY KUZNETSOV Dept. 285-317, 1985. (3) exists for all t>0 and satis es the exponential decay estimate jv(t)j Ce tjv(0)j for all t 0. Lok S2393263 ais the max-imum rate of population growth and K represents a sort of "ideal" population or "carrying capacity", which is basically the maximum size of the population. 1, 0, 0, -1. Maximal Lyapunov exponent with the Kantz algorithm: None in GNU Octave: C: wrapped in C++/mfile/octfile code: lyap_r: Maximal Lyapunov exponent with the Rosenstein algorithm: None in GNU Octave: C: wrapped in C++/mfile/octfile code: lyap_spec: Full spectrum of Lyapunov exponents: None in GNU Octave: C: wrapped in C++/mfile/octfile code: lzo-gm. For this, one can plot ln d(k) vs k and apply a linear fit. Math 319 or 320, or consent of instructor. View Yury Adamov’s profile on LinkedIn, the world's largest professional community. The maximal Lyapunov exponent can be calculated in any dimension by monitoring the separation distance in phase space used to write the Matlab code. Source code can be seen below or downloaded, as can the Java executable (Lyapunov. Stochastic systems biology: a paper by Golightly and Wilkinson. The chaotic state is detected most often, using an index called the Lyapunov characteristic exponent (,t). Now I need to plot lyapunov versus parameter, ''say a''. K, can anybody help me out?. Figure 2: A comparison of resampled Monte{Carlo (J. values and eigenvectors  and ﬁnite-time Lyapunov exponents and vectors . GaussLegendre, etc. This version is tuned for the Mackey–Glass equation, but modification should be straightforward. In a code block in your script, type a relevant keyword, such as Lyapunov or Lyapunov exponent. University of Auckland. Set the matrix (must be square) and append the identity matrix of the same dimension to it. Roughly speaking the (maximal) Lyapunov exponent is the time constant, lambda, in the expression for the distance between two nearby orbits, exp (lambda * t). Information on toolbox-specific routines can be obtained by the MATLAB help command (e. The alogrithm employed in this m-file for determining Lyapunov exponents was proposed in A. Mathematics Department. Figure 2: Lyapunov exponent of the heap model of Fig. In a code block in your script, type a relevant keyword, such as Lyapunov or Lyapunov exponent. m Orbit diagram for logistic map logistic_lyapunov. My problem is that I don't know first Lyapunov exponent is average of all the lyapunovs when k = 1 or average of all the lyapunovs for the first time that data[i] - data[j] < eps? Is this right implementation for Lyapunov exponent? And this is the Numerical Calculation of Lyapunov Exponent. I am trying to find the maximal lyapunov exponent of a system described by 3 differential equations and one of the diff eq. Similar to the asymptotic approximation of amplitude and period, expressions are derived for the non‐zero Lyapunov exponent for both small and large parameter values. Reset your password. which is likely to be observable and relevant in any physical system. Terms offered: Fall 2009, Spring 2003, Spring 2000 Table-driven and retargetable code generators. It is based on the comparison of Taylor series for the Lyapunov Exponent definition and the Taylor series for the. The non-linear dynamic examples studied here are the Lorenz and the Rosler systems as well as the non-isothermal chemical system of Tomlin and Scott. Maxime Fortier-Bourque. You can choose and change arbitrary the number of iteration. It estimates the maximal Lyapunov exponent from a time series (more information available from the TISEAN documentation located here). Lyapunov Exponents.