Finite-time Lyapunov exponents

Accordingly, the finite-time Lyapunov exponent a may be defined via... 

To characterize these invariant structures and the changes of reaction coordinates, the concept of finite-time Lyapunov exponents can be useful [44]. The original definition of the Lyapunov exponents needs ergodicity (see, e.g.. Ref. 45) to make sure that the time average of the exponents converges. However, for chaotic itinerancy, the exponents would not converge. Moreover, the finite-time Lyapunov exponents can be more useful to detect whether... 

Development of the ES FR was motivated by the seminal work by Evans, Cohen and Morriss. This paper focussed on constant energy dynamics and, based on results for the 2 dimensional Lorentz model, proposed that the measure of a trajectory was related to the exponential of the sum of the positive finite-time Lyapunov exponents of that trajectory. In this way they obtained a relationship for... 

FINITE-TIME LYAPUNOV EXPONENTS IN MANY-DIMENSIONAL DYNAMICAL SYSTEMS... 

The aim of this chapter is twofold. One is to give a new method for computing finite-time Lyapunov exponents and vectors in many-dimensional dynamical systems, and the other is to discuss the Lyapunov instability of a cj)4 model with this method. 

When a dynamical system is nonhyperbolic, there exist time intervals where part of the finite-time Lyapunov exponents accumulate around zero. Hence the spectra of the exponents are (quasi-)degenerate. These degenerate spectra impede our ability to obtain accurate numerical values of finite-time Lyapunov exponents using the existing numerical methods, namely, the QR method and the SVD method [9,17] ... 

The QR methods, based on the matrix factorization of QR decomposition [18], are effective and thus are widely used algorithms for computing the Lyapunov exponents [1,19-22]. However, for the, finite-time Lyapunov exponents, these methods introduce errors that decrease only algebraically... 

Now we introduce the finite-time Lyapunov exponents and corresponding vectors, utilizing the SVD of the stability matrix. The SVD of any n x n matrix, say A, is a matrix factorization [18]... 

From this decomposition, the finite-time Lyapunov exponents in the time interval from f,- to zy are given by... 

The exact exponents Xj are directly computed by diagonalizing the symmetric matrix MTM with high-precision computation to evade its roundoff error. Figure 2 plots the error in the smallest exponents, (t, 0) — ki(t, 0), against t. We can see that until t 30 the error rapidly decreases. After the initial dropping stage, it decreases slowly as 1/f (inset). This quite slow convergence shows that A,9R is not a sufficiently accurate approximation of the finite-time Lyapunov exponent. Note here that behavior of Aj is similar to that of iL-... 

Now we present our novel method for computing accurate values of finite-time Lyapunov exponents and vectors, by correcting the finite-time error in [24], To this end, we construct a sequence of refinements U 1], d, r, and Vik] (k = 0,1,2,...) satisfying... 

Figure 5 shows that the leading exponents are positive and thus the 4>4 MTRS is chaotic. In addition, they varies depending on the time intervals We see relatively small instability regions, for example, around the intervals 9000-12,000 and 70,000-75,000. The variations of finite-time Lyapunov exponents have been related to the alternations between qualitatively different motions, such as (a) chaotic and quasi-regular, laminar motions in two-dimensional systems [11] and (b) random and cluster motions in high-dimensional systems [12], and they have been utilized for detecting these ordered motions. 

Let us confirm the correspondence between the relative magnitude of instability and the orders of motions with this < )4 MTRS. Figure 6 shows stroboscopic data of j = 0 mode variables (a0, Po) with sampling period l/fi>o-The left panel presents the data in the time interval 70,000 < t < 75,500, where the leading finite-time Lyapunov exponent is relatively small. A coherent, quasi-periodic motion is clearly seen in these variables. Since almost all energy concentrates in the zeroth mode, the other mode variables j / 0 are hardly excited in this coherent motion. In contrast, the right panel in Fig. 6 presents the data in the time interval 20,000 < t < 25,500, where the leading exponent is... 

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