Lyapunov dimension

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... 

Nevertheless, the determination of the fractal dimension from a data set thought to be chaotic is often of interest. A number of different dimensions exist in the literature, including the Hausdorff dimension, the information dimension, the correlation dimension, and the Lyapunov dimension. Which of these is the true fractal dimension Of the ones in this list, the information dimension, Di, has the most basic and fundamental definition, so we often think of it as the true fractal dimension. Because the information dimension is impractical to calculate directly, however, most investigators have taken to finding the correlation dimension, Dq, as an estimate of the fractal dimension. Grassberger and Procaccia published a straightforward and widely used algorithm for the calculation of the correlation dimension. On the other hand, the Lyapunov... 

An example of a calculation of the Lyapunov exponents and dimension, for a simple four-variable model of the peroxidase-oxidase reaction will help to clarify these general definitions. The following material is adapted from the presentation in Ref. 94. As described earlier, the Lyapunov dimension and the correlation dimension, D, serve as upper and lower bounds, respectively, to the fractal dimension of the strange attractor. The simple four-variable model is similar to the Degn—Olsen-Ferram (DOP) model discussed in a previous section but was suggested by L. F. Olsen a few years after the DOP model was introduced. It remains the simplest model the peroxidase-oxidase reaction which is consistent with the most experimental observations about this reaction. The rate equations for this model are ... 

Figure 8, Example of the phase-space analysis of water dripping intervals from the HeWs Half Acre infiltration test (Test 8, Dripping Point 6) (a) time-series of dripping water intervals, (b) autocorrelation function, (c) mutual information junction used to determine the time delay, (d) false nearest neighbors used to determine the global embedding dimension, (e) local embedding dimension, and (f) Lyapunov exponents and Lyapunov dimension.

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. 

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 basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

Nonlinear analysis requires the use of new techniques such as embedding of data, calculating correlation dimensions, Lyapunov exponents, eigenvalues of singular-valued matrices, and drawing trajectories in phase space. There are many excellent reviews and books that introduce the subject matter of nonlinear dynamics and chaos [515,596-599]. 

Suppose we have a saddle with index 1. Then, a NHIM of 2N — 2 dimension exists above it in the phase space, with two directions that are normal to it. Along these normal directions, with negative and positive Lyapunov exponents, 2N — 1)-dimensional stable and unstable manifolds exist, respectively. The normal directions of the saddle correspond to the degree of freedom that is the reaction coordinate near the saddle, and they describe how the reaction proceeds locally near the NHIM. 

We conclude this section by pointing out an important relationship between the decay rate 7, the Lyapunov exponent A and the fractal dimension d of a one-dimensional self-similar fractal. In the context of fractals, the Lyapunov exponent is the rate of stretching given by... 

The proportion of fluid elements experiencing a particular anomalous value of the Lyapunov exponent A / A°° decreases in time as exp(—G(X)t). In the infinite-time limit, in agreement with the Os-eledec theorem, they are limited to regions of zero measure that occupy zero volume (or area in two dimensions), but with a complicated geometrical structure of fractal character, to which one can associate a non-integer fractional dimension. Despite their rarity, we will see that the presence of these sets of untypical Lyapunov exponents may have consequences on measurable quantities. Thus we proceed to provide some characterization for their geometry. 

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... 

Therefore, in the t —> oo limit, fluid elements experiencing a certain particular anomalous value of the Lyapunov exponent form a fractal set with dimension... 

The chaotic saddle and its manifolds are also sets of zero measure with fractal structure. The set of points, seen in Fig. 2.13 corresponding to inflow coordinates with very large, singular, escape times, typically form also a fractal set determined by the intersection of the saddle s stable manifold and the line containing the initial conditions. There is a connection between the dimension of the chaotic saddle and the dimensions of its manifolds. The trajectories on the chaotic saddle have a set of Lyapunov exponents whose number is equal to the dimension of the full space, d. The sum of the Lyapunov exponents is zero due to incompressibility and chaotic dynamics implies... 

The dynamical and geometrical properties of the advection in open flows are related through the generalization of the Kaplan-Yorke formula that gives the fractal dimension of the manifolds as a function of Lyapunov exponents and escape rate. In the case of... 

In the previous sections we considered flows with a smooth spatial structure in which the relative dispersion of fluid trajectories is exponential in time and can be characterized by a single timescale, the inverse of the Lyapunov exponent. This is also valid for two-dimensional turbulent flows that have a smooth velocity field in the small-scale enstrophy cascade range (Bennett, 1984). A similar behavior occurs in any dimension at scales below the Kolmogorov scale (the so-called Batchelor or viscous-convective range, see below). In the inertial range of fully developed three-dimensional turbulence, however, the velocity field has a broad range of timescales and they all contribute to the relative dispersion of particle trajectories and affect the transport properties of the flow. 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

---