Lyapunov Characteristic Exponents

It can be shown that A both exists and is finite. Moreover, we can always find a set of n tangent-space basis vectors, c (i = 1,... n), such that Ax = Sxi,..., Sx ) — "The divergence (or contraction) along a given basis direction, e, is then measured by the j Lyapunov characteristic exponent, A. These n (possibly... 

Lyapunov Dimension An interesting attempt to link a purely static property of an attractor, - as embodied by its fractal dimension, Dy - to a dynamic property, as expressed by its set of Lyapunov characteristic exponents, Xi, was, first made by Kaplan and Yorke in 1979 [kaplan79]. Defining the Lyapunov dimension, Dp, to be... 

G. Benettin, L. Galgani, and J-M. Strelcyn. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems a method for computing all of them. Meccanica, 15 9-20, 1980. 

This value is a measure of the mean exponential rate of divergence (convergence) of two initially very close trajectories, i.e. when (xq,0) 0. The values from (59) are the so called Lyapunov characteristic exponents, which can be ordered by size ... 

The nature of the intramolecular motion may also be identified by studying the way the separation of two trajectories evolves with time [353]. If the motion is regular (quasi-periodic) the separation is linear with time, but exponential if the motion is irregular (chaotic). If the separation is exponential, the rate of the separation — called the Lyapunov characteristic exponent — provides qualitative information concerning the IVR rate for the chaotic trajectories. This type of analysis has been reported, for example, for NO2 [271] and the Cl CHsBr complex [354]. 

Since the very beginning non graphical methods for the detection of chaos where implemented, taking into account the exponential divergence of nearby chaotic orbits (Froeschle 1970,b). The introduction, in a comprehensive way, of the Lyapunov Characteristic Exponents (LCEs hereafter) and a method for computing all of them (Benettin et al. 1980) made a major breakthrough for the characterization of chaos. Actually, the largest Lyapunov Exponent was already computed earlier (Froeschle 1970,b) but was called indicator of stochasticity since the works of the Russian mathematician where not known by the author (Froeschle 1984). 

Froeschle, C. (1984). The Lyapunov characteristic exponents and applications. Jomal de Mec. theor. et apll., Numero special 101-132. 

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. 

If the first non-zero Lyapunov value is positive and if all non-critical characteristic exponents (71,...,7n) lie to the left of the imaginary axis in the complex plane, then the equilibrium state is a complex saddle-focus, as shown in Fig. 9.3.2(b). Its stable manifold is and the unstable manifold coincides with the center manifold W, The trajectories lying neither in nor pass nearby the equilibrium state. 

We have seen in Sec. 10.4 that in the case of weak resonance cj = 2nM/N N > the stability of the critical fixed point is, in general, determined by the sign of the first non-zero Lyapunov value. The same situation applies to the critical case of an equilibrium state with a purely imaginary pair of characteristic exponents. However, there is an essential distinction, namely, for a resonant fixed point only a finite number which does not exceed N—3)/2 of the Lyapunov values is defined. The question of the structure of a small neighborhood of the fixed point in the case where all Lyapunov values vanish is difficult, so we do not study it here. Instead, we consider two examples. 

For cases having an extra degeneracy (for example an equilibrium state with zero characteristic exponent and zero first Lyapunov value) the boundary of the stability region may lose smoothness at the point There may also exist situations where the boimdary is smooth but bifurcations in different nearby one-parameter families are different (i.e. there does not exist a versal one-parameter family, for example, such as the case of an equilibrium state with a pair of purely imaginary exponents and zero first Lyapunov value). In such cases the procedure is as follows. Consider a surface 971 of a smaller dimension (less than (p — 1)) which passes through the point and is a part of the stability boundary, selected by some additional conditions in the above examples the condition is that the first Lyapunov value be zero. If (fc — 1) additional conditions are imposed, then the surface 971 will be (P fc)-dimensional and it is defined by a system of the form... 

For regular motion, T> t) grows only linearly with time, so that the exponents are all zero. On the other hand, because chaotic flows are characterized by exponential divergences of initial nearby trajectories, a characteristic signature of such flows is the existence of at least one positive Lyapunov exponent. 

First, in order to simplify the description of the dynamics we separate the whole system, locally in the phase space, into two parts based on a gap in characteristic time scales. This is done using the concept of normally hyperbolic invariant manifolds (NHIMs) [4-8]. Here, the characteristic time scales are estimated as the inverses of the absolute values of the local Lyapunov exponents [5,6]. Then, the Fenichel normal form offers a simplified description of the local dynamics near a NHIM [7]. 

In general Ai may depend on the initial condition xo- But for ergodic systems it has the same value for almost all initial positions (Eck-mann and Ruelle, 1985), i.e. everywhere except in a set of measure zero. The characteristic signature of chaotic advection is that at least one of the Lyapunov exponents is positive, representing exponential growth of the distance separating the two particles. 

For calculating the dimension of the set of spatial locations which at each time are occupied by trajectories with non-typical values of the asymptotic Lyapunov exponent we cover these locations with objects of size which can be estimated from the dynamics Since the proportion of fluid elements experiencing a finite-time average stretching A / A°° decays as exp(—G(A)f), and in an incompressible flow the area of fluid element remains constant (we refer to the two-dimensional situation for simplicity), the total area covered by these fluid elements decreases also as A (t) exp(—G(X)t). This area is stretched by the chaotic dynamics locally characterized by A, so that its characteristic width shrinks as w (t) exp(—At). The number of boxes of size l = W needed to cover such set of fluid elements can be estimated as... 

In addition to the Lyapunov exponent, that is a measure of chaotic advection inside the mixing zone, the transport in open flows has another characteristic timescale associated to the escape rate of fluid elements. This can be defined from the distribution of escape times r, that for large values (in hyperbolic systems) has an exponential form... 

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... 

The transition between the two regimes takes place when the characteristic timescale of chaotic mixing (e.g. the inverse Lyapunov exponent) is comparable to the characteristic decorrelation time of the of the local oscillations, that was found to be significantly larger than the oscillation period of the reaction. 

Theoretical models suggest that the transition from a coil-stretch transition within random flows depends on the principle Lyapunov exponent 7.1 and the longest relaxation time for the molecule Tr. To stretch the molecules, 7.iXr must be >1 [31,32]. The Lyapunov exponent can be approximated by taking the fluctuating velocity u, divided by the characteristic length scale, which in turbulence, is determined by the eddies at the viscous scale i.e., X = m7t [31]. 

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