The Lyapunov exponent

The non-dimensional parameter a = UT/L of the sine-flow that controls the extent of chaotic advection can be interpreted as a ratio of two characteristic timescales. One of them is the typical advection time over the characteristic lengthscale of the velocity field L/U. This is a property of the instantaneous velocity field and would be the same for a steady flow. Therefore it can not characterize the dynamics of chaotic mixing. For a time-periodic velocity field another timescale is the period of the flow. In the case of an aperiodic time-dependent flow an analogous timescale can be defined as 

A more exact quantitative characterization of the chaotic advection can be given by considering the relative dispersion of fluid particles. Let us consider two fluid elements moving on trajectories r(t) and r (t). When the distance between them is small compared to the characteristic lengthscale of the velocity field (L) the velocity difference can be approximated by Taylor expansion and the separation S(t) = r (t) — r(t) satisfies the equation 

The Oseledec theorem (Eckmann and Ruelle, 1985) implies that under rather general conditions the eigenvectors of M)r(xo)Mt(x0) converge, as t — oo, to a set of Lyapunov vectors v. Since MfMt is symmetric and positive definite the Lyapunov vectors form an orthonormal set and the eigenvalues A are all positive. If the initial separation is chosen to be oriented along one of the Lyapunov vectors, So = with 5o C L, the distance between the particles at a later time t is 

The linear approximation for the velocity differences in (2.67) is only valid while 5(t) Lso this will only hold for arbitrary long time if we first take the limit 5o — 0. Then the set of Lyapunov exponents can be defined as the growth rate of the inter-particle distance as 

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. 

---

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

Since the speed of information propagation is, as we shall see in chapter 4, related to the Lyapunov exponent for the CA evolution, and is a direct measure of the sensitivity to initial conditions, it should not be surprising to learn that various rules can also be distinguished by the degree of predictability for the outcome of... 

It has been suggested that the signature, S = ( sign (Ai), sign (A2),. .., sign (A )), of the Lyapunov exponents may be used to provide a qualitative characterization of of attractors for dissipative flows. Among the possibilities for n = 4, for example, we have the following ... 

Except for simple cases, it is generally a nontrivial task to compute the Lyapunov exponents of a flow. In trying to estimate A(x(0)) in equation 4.59, for example, the exponentially increasing norm, V t), may lead to computer overflow problems. 

As defined above, the Lyapunov exponents effectively determine the degree of chaos that exists in a dynamical system by measuring the rate of the exponential divergence of initially closely neighboring trajectories. An alternative, and, from the point of view of CA theory, perhaps more fundamental interpretation of their numeric content, is an information-theoretic one. It is, in fact, not hard to see that Lyapunov exponents are very closely related to the rate of information loss in a dynamical system (this point will be made more precise during our discussion of entropy in the next section). 

For example, the time average definition of the Lyapunov exponent for one-dimensional maps, A = lim v->oo (which is often difficult to calculate in prac-... 

We stress that the chaos identified here is not merely a formal result - even deep in the quantum regime, the Lyapunov exponent can be obtained from measurements on a real system. Quantum predictions of this type can be tested in the near future, e.g., in cavity QED and nanomechanics experiments (H. Mabuch et.al., 2002 2004). Experimentally, one would use the known measurement record to integrate the SME this provides the time evolution of the mean value of the position. From this fiducial trajectory, given the knowledge of the system Hamiltonian, the Lyapunov exponent can be obtained by following the procedure described above. It is important to keep in mind that these results form only a starting point for the further study of nonlinear quantum dynamics and its theoretical and experimental ramifications. 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

We note that the integral over the energetically allowed phase space— that is, the classical level density (97)—was found in Fig. 20 to be in excellent agreement with the quantum-mechanical level density. This finding indicates that there is a valid correspondence between the quantum-mechanical two-state system and its classical mapping representation. A similar conclusion was drawn in a recent smdy of a mapped two-state problem, which focused on the Lyapunov exponents and the energy level statistics of the system [124, 235]. 

Consequently, we can ensure that the simulation process is correct. Note that if there is at least one positive Lyapunov exponent, trajectories obtained from two very close initial conditions diverge, and when all Lyapunov exponents are negative the same trajectories converge. A practical procedure to numerically determine the Lyapunov exponents is given in the Appendix. 

From the study presented in this chapter, it has been demonstrated that a CSTR in which an exothermic first order irreversible reaction takes place, can work with steady-state, self-oscillating or chaotic dynamic. By using dimensionless variables, and taking into account an external periodic disturbance in the inlet stream temperature and coolant flow rate, it has been shown that chaotic dynamic may appear. This behavior has been analyzed from the Lyapunov exponents and the power spectrum. 

The Lyapunov exponents provide a computable measure of the sensitivity to initial conditions, i.e. characterize the mean exponential rate of divergence of two nearby trajectories if there is at least one positive Lyapunov exponent, or convergence when all Lyapunov exponents are negative. The Lyapunov exponents are defined for autonomous dynamical systems and can be described by ... 

It is possible to generalize the previous concept to describe the mean rate of exponential growth (decrease) of a m-dimensional volume in the tangent space of R , where m < n. The Lyapunov exponent of order m is defined as... 

The calculation of the Lyapunov exponents of order m has been carried out as follows. The following notation is useful ... 

Then, the Lyapunov exponent of order m is calculated from ... 

Taking into account Eq.(63)the value of the Lyapunov exponent associated to the direction m is deduced as follows ... 

Fig. 9. Maximum Lyapunov exponents along the orbits in the space phase for scale maximum Sm = 1-5 (compare with Figure 8). Even for experiment E2.b (which has the smallest attractor) the maximum Lyapunov exponent holds positive. The average values of the Lyapunov exponents where computed by neglecting the transitory effects (AVWT means Average Value without Transitory effects).

Figure 3. Spectrum of Lyapunov exponents of a dynamical system of 33 hard spheres of unit diameter and mass at unit temperature and density 0.001. The positive Lyapunov exponents are superposed to minus the negative ones showing that the Lyapunov exponents come in pairs L, —L, as expected in Hamiltonian systems. Eight Lyapunov exponents vanish because the system has four conserved quantities, namely, energy and the three components of momentum and because of the pairing rule. The total number of Lyapunov exponents is equal to 6 x 33 = 198.

The Lyapunov exponents and the Kolmogorov-Sinai entropy per unit time concern the short time scale of the kinetics of collisions taking place in the fluid. The longer time scales of the hydrodynamics are instead characterized by the decay of the statistical averages or the time correlation functions of the... 

This condition can be rewritten in terms of Ruelle s function defined as the generating function of the Lyapunov exponents and their statistical moments ... 

In systems with two degrees of freedom such as the two-dimensional Lorentz gases, there is a single positive Lyapunov exponent X and the partial Hausdorff dimension of the set of nonescaping trajectories can be estimated by the ratio of the Kolmogorov-Sinai entropy to the Lyapunov exponent [ 1, 38]... 

SO that the escape rate can be directly related to the fractal dimension and the Lyapunov exponent ... 

The difference between both entropies per unit time is minus the sum of all the Lyapunov exponents which is the rate of contraction of the phase-space volumes under the effects of the nonHamiltonian forces ... 

The stability matrix carries the necessary information related to the vicinity of the trajectory and provides an efficient numerical procedure for computing the response function. It plays an important role in the field of classical chaos the sign of its eigenvalues (related to the Lyapunov exponents) controls the chaotic nature of the system. Interference effects in classical response functions have a different origin than their quantum counterparts. For each initial phase-space point we need to launch two trajectories with very close initial conditions. [For 5(n) we need n trajectories.] The nonlinear response is obtained by adding the contributions of these trajectories and letting them interfere. 

At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. 

Differences between the lifetimes obtained from equilibrium point quantization and periodic-orbit quantization appear as the bifurcation is approached. The lifetimes are underestimated by equilibrium point quantization but overestimated by periodic-orbit quantization. The reason for the upward deviation in the case of periodic-orbit quantization is that the Lyapunov exponent vanishes as the bifurcation is approached. The quantum eigenfunctions, however, are not characterized by the local linearized dynamics but extend over larger distances that are of more unstable character. 

Figure 13. Scattering resonances of a two-degree-of-freedom collinear model of the dissociation of Hgl2 [10], The filled dots are obtained by wavepacket propagation, the crosses by equilibrium point quantization with (2.8), the dotted circles from periodic-orbit quantization with (4.16). The solid lines are the curves corresponding to the Lyapunov exponents, Im E = -(A/2)Xp(Re ), of the fundamental periodic orbits p = 0, 1,2. The dashed line is the spectral gap, Im = ftP(l/2 Ref). The long-short dashed line is the curve corresponding to the escape rate, Im E = (h/2)P( Re ).

---