Finite-time Lyapunov exponents systems

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] ... 

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. 

The question of what controls the asymptotic decay rate and how is it related to characteristic properties of the velocity field has been an area of active research recently, and uncovered the existence of two possible mechanisms leading to different estimates of the decay rate. Each of these can be dominant depending on the particular system. One theoretical approach focuses on the small scale structure of the concentration field, and relates it to the Lagrangian stretching histories encountered along the trajectories of the fluid parcels. This leads to an estimate of the decay rate based on the distribution of finite-time Lyapunov exponents of the chaotic advection. Details of this type of description can be found in Antonsen et al. (1996) Balkovsky and Fouxon (1999) Thiffeault (2008). Here we give a simplified version of this approach in term of the filament model based... 

Here we also include the contribution of Okushima, in which the concept of the Lyapunov exponents is extended to orbits of finite duration. The mathematical definition of the Lyapunov exponents requires ergodicity to ensure convergence of the definition. On the other hand, various attempts have been made to extend this concept to finite time and space, to make it applicable to nonergodic systems. Okushima s idea is one of them, and it will find applications in nonstationary reaction processes. 

Let us address these in turn, without being entirely formal. The sensitive dependence on initial conditions can be taken to mean that if a pair of initial points of phase space is given which are separated by any finite amount, no matter how small, then the gap between these solutions grows rapidly (typically exponentially fast) in time. A problem with this concept is that we often think of molecular systems as having an evolution that is bounded by some sort of domain restriction or a property of the energy function the exponential growth for a finite perturbation can therefore only be valid until the separation approaches the limits of the accessible region of phase space. In order to be able to make sense of the calculation of an exponential rate in the asymptotic t oo) sense, we need to consider infinitesimal perturbations of the initial conditions, and this can be made precise by consideration of the Lyapunov characteristic exponents mentioned at the end of this chapter. 

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