Lyapunov exponent, largest

The largest Lyapunov exponents for the relativistic hydrogenlike atom in a uniform magnetic field... 

Highly unstable systems lead to two positive Lyapunov exponents that show the hyperchaotic behavior [116]. Now, Eq. (3) is numerically examined with damping constants 71=72 = 0.01. In Fig. 9a we see only the two largest Lyapunov exponents of all the spectrum versus the modulation parameter O. The... 

Figure 9. The two largest Lyapunov exponents (a) and the bifurcation diagram (the maxima of yi) (b) versus the modulation parameter Q. Parameters are/o — 1. — y2 — 0.01 and the initial...

The two largest Lyapunov exponents versus a duration of pulse 7] are presented in Fig. 18a for the cases of BCL. There are a two regions of hyperchaos. A Typical hyperchaotic phase portrait is presented in Fig. 18b. 

Figure 18. (a) The two largest Lyapunov exponents X and %2 versus the pulse duration T, for... 

Changes in the largest Lyapunov exponent have also been used to signal phase changes, however work in this area has mostly been at the fluid dynamics level, using integral equations. 

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). 

Figure 4 Fast Fourier transform of g(t) = (t)Q(I(t), tp(t)), where (7(t),

The length of the vectors M(V grows or decreases exponentially in time as exp(AH). After long time the sum (2.72) will be dominated by the term corresponding to the largest Lyapunov exponent. Thus, for almost all initial directions of the separation vector the distance... 

Lyapunov exponents for different parameter values. Figure 6.12(b) shows the parameter values for which the largest Lyapunov exponent is positive,... 

Sacker bifurcation curve emerging from the Zero-Hopf point, (b) Parameter values, for which an attractor with positive largest Lyapunov exponent exists. 

Fig. 15.9. Dynamics of the foodweb model (15.8) in the phase coherent regime as a function of the control parameter 6. (a) Bifurcation diagram, plotted are the maxima of w, (b) largest Lyapunov exponent A (c) mean frequency cj (solid line). Further indicated is the approximation wo b) = Vb (dotted line).

As described in a previous section, the Lyapunov exponents are a generalized measure of the growth or decay of perturbations that might be applied to a given dynamical state they are identical to the stability eigenvalues for a steady state and the Floquet exponents for a limit cycle. For aperiodic motion at least one of the Lyapunov exponents will be positive, so it is generally sufficient to calculate just the largest Lyapunov exponent. 

For shear rates between 0 and 0.8 the results appear robust and reproducible. There is a steady change in the largest exponent with shear rate with A ox = 0 from 7 = 0 to 0.18, and then returning to 0 just before 0.8. Between 0.18 and 0.35 the largest Lyapunov exponent is very close to 0. There are small windows near 7 = 0.34,0.42 and 0.52 where the largest exponent appears to be zero, but there is no obvious change in the trajectory at these places. 

C Complex complicated motion of the alignment tensor. This includes periodic orbits composed of sequences of KT and KW motion with multiple periodicity as well as aperiodic, erratic orbits. The largest Lyapunov exponent for the latter orbits is positive, i.e., these orbits are chaotic. 

A strange (or chaotic) attractor is by definition an attractor for which the largest Lyapunov exponent is positive. Then trajectories starting from nearby points will separate exponentially fast as time evolves. Therefore, all information about the initial conditions is rapidly lost, since any uncertainty, no matter how small, will be magnified until it becomes as large as the attractor thus there is sensitive dependence on initial conditions (RUELLE [38]). Long term predictions about the state of the system are impossible. 

The largest Lyapunov exponent for a system described by a one-dimensional map (as in Fig. 2) can be computed from the map ... 

Fig. 3. A schematic diagram illustrating the procedure developed by Wolf and Swift for computing the largest Lyapunov exponent from experimental data (see text and refs. [36,37]). 

The largest Lyapunov exponent can be computed from a map or directly from measurements of the divergence rate of nearby trajectories. The two approaches yield the same positive value for the exponent for a given set of data hence the data are described by a strange attractor. 

This model, introduced a few years ago by Rossler, is among the simplest ones exhibiting chaos. We chose it because the largest Lyapunov exponent has already been computed by the Santa-Cruz group [ 12] over a wide range of the control parameter c (for... 

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