These methods apply to the calculation of the fractal mass dimension of an attractor, i. In the case of a low-dimensional phase space, a visual inspection of the plot of the projection of the orbit on the coordinate planes indicates whether this condition is fulfilled. Uniformity of the sampling over the set is an assumption that remains to be tested cf. Mayer-Kress for a discussion of the practical difficulties encountered with the estimates of attractor dimensions; the number of sampling points needed for an evaluation of the dimension increases exponentially with the dimension to be estimated.
Correlation Dimension If a set A given in the sampled form 1.
Voss Perdang V, number of points of the cloud inside the ball of radius R centred at the reference point Xj. Therefore, by Eqs. Let T now tend to infinity in 1. If a scaling relation 1. The occurrence of a plateau in this plot gives the value of the correlation dimension Fe, if it exists. Note that the correlation dimension is not an intrinsic parameter of the geometry of a set: it characterises a distribution of a sampling points over the set. Accordingly, an estimate of the correlation dimension of an m-manifold does not necessarily coincide with m; there is no guarantee for the correlation dimension to preserve the simple properties 1.
Generalised Dimensions The correlation dimension is one representative of a continuous class of dimensionalities attached to distributions of sampling points over geometrical sets A, known as generalised dimensions Hentschel and Procaccia , Grassberger ; see also Rasband Pawelzik and Schuster indicate that the latter are recovered from a correlation type integral 1.
The derivative of 1. Schlogl , so that the Renyi information is a non-increasing function of q d. Beck for a closer discussion of the inequalities obeyed by the Renyi dimensions. Therefore the generalised dimension is a decreasing function of q 1.
[Fractal art in separative sciences].
We mention a final fractal dimension specifically attached with attractors of dynamical systems Grassberger , Rasband The latter measure the local convergence [divergence] of the departures ox t from the trajectory x t the component ox; along the eigenvector of L; evolves as '" e L ;1 , the occurrence of positive Lyapunov exponents characterising the 'sensitive dependence on initial conditions' of chaotic behaviour. Since expression 1. This parameter is known as the Lyapunovor Kaplan- Yorke dimension.
Lyapunov dimensions have not been estimated in an astrophysical context. It has recently been argued that they are reliably obtained from observational time series d. Briggs Multifractal Measures: Spectrum of Singularities While a dispersion of the generalised dimensions indicates that the sampling of the set is not uniform, the knowledge of the spectrum of dimensions F q does not directly provide an information on the statistics of the local fractal mass dimensions associated with the distribution of points. We introduce therefore another characterisation which refers directly to the statistics of the local fractal mass dimensions.
This information is actually contained in the generalised dimensions, as will be shown in this section. Since F q is a decreasing function of q, this potential has a single extremum. The function I a , on the other hand, just like the Legendre transforms of the internal energy in thermodynamics, preserves the original information on the probability distribution, and has a single extremum in a: it has a parabolic shape of maximum given by the grid dimension cf.
The width of the parabola gives an indication of the scatter of the dimensions.
The function I a is the multifracta] spectrum, or spectrum 01 singularities, since it characterises the distribution of the points over the cells. A large [small] exponent ai thus characterises a low [high] concentration of points.
Hunting the Hidden Dimension
The width of the interval of exponents [am, aM] is a measure of the lack of homogeneity of the distribution. We shall be interested in the statistical distribution of the exponents over the interval [am, aM], rather than in their individual values. In terms of this distribution, the summation over the cells in the defining relation of the Renyi information Eq, 1. Definition 1. Halsey et al.
Of course, a similar information is obtained by computing directly the local fractal mass dimensions FM X Eq. Arneodo et al. Attractors With chaotic dynamics as a potential framework for interpreting the complex observational time series of various astrophysical phenomena light curves or velocity curves of Variable Stars, etc , the problem of reconstructing the attractor from a given time series has received considerable attention in the literature.
Casdagli Favard Takens for a rigorous proof. Whitney's modified theorem thus supplies the lowest dimensional RN capable of carrying any set A of fractal dimension FM. The estimate of the fractal dimension of the image A' in the trial phase space RN must not exceed the value FM given by 1.
So far virtually all authors who have analysed astronomical time series have applied the method of the correlation dimension Fe 1. Such an identification is subject to several criticisms. In the first place, the fractal dimensions are metric properties, and therefore they are not topological invariants: if A and A' are homeomorphic, then their fractal dimensions are generally not equal.
Secondly, as has been shown by numerical experiments by Cannizzo et al.
fensterstudio.ru/components/tegepyx/pox-programa-espia-para.php Geometry and Dynamics of Fractal Sets 17 which is strongly affected by a nonlinear change of variables. Thirdly, the correlation dimension is more sensitive to the distribution of the sampling points than for instance the mass dimension. These observations signal that estimates of the dimension of the true attractor drawn from time series via the single determination of Fe of a trial attractor remain problematic. Additional tests, such as the nonintersection test a , should be carried out. Moreover, the singularity spectrum f a should be determined, or a direct estimate of the statistical distribution of the local mass dimensions should be made.
In the case of stellar pulsations, Kovacs and Buchler have estimated Fe from the time series of the stellar radius R t of their full hydrodynamic models of Population II Cepheids; for all models tested the dimension lies in the range 1. One may surmise that in the case of theoretical models the uncertainties of the method are minimised the radius is a good phase space variable sampling the attractor approximately uniformly, so that Fe remains close to F M. Moreover these estimates agree with the dimensions of the attractors of elementary onezone models developed by various authors cf.
Perdang In the observational context, a recent analysis of the visual lightcurve of R Scuti by Kollath has led to a correlation dimension between 4 and 5 significantly higher than the theoretical model attractor dimensions. Besides the difficulties already mentioned, estimates of such a high dimension are intrinsically unreliable as a consequence of the limited number of data points. A search for a dimension of the attractor of white dwarf oscillations was made by Auvergne and Baglin , and for the variability of Mira, R Leonis and V Bootis by Cannizzo et al.
In the case of observational signals resulting from astrophysical mechanisms which are only partially understood and which involve more complex physics than the dynamical oscillations of classical stellar variables, several authors have attempted to estimate attractor dimensions. Such dimensions provide constraints on the dynamics responsible for the signal effective number of degrees of freedom.
The correlation integral for the outbursts of the Cataclysmic Variable SS Cygni reveals no stable dimension Cannizzo and Goodings Construction of Hierarchical Sets. Associated Fractal Dimensions. We analyse here the geometric construction of a versatile class of model fractals, namely the deterministic and statistical hierarchical sets by means of an algebraic construction based on an iterative transformation which prepares the dynamical models to be discussed below.
For a pure geometric construction see Mandelbrot 18 Jean M. Perdang , ; cf. A minor modification of our iterative transformation changes the latter into Barnsley's iterative function system IFS. The action on B of an individual map Tj produces a set 2. Since each component map Tj is a contraction, the iterations 2. Geometrically the definition of H as an invariant of the iterative transformation T implies that this structu.
Therefore, if M H is the PM-measure of the set H, then 7 Recipes for generating fractal sets are found in Voss , and Saupe based on fractional Brownian motions and related techniques , and in Barnsley , a, b IFS. Note that in the presence of selfintersections of the resulting set H, Eq. The general visual aspect of a limit set depends on the precise transformation T. Accordingly, the dimension F. Instead of a single transformation T take a family T s depending on discrete or continuous parameters s, with associated probability density p s , the scaling parameter r and the number of individual maps P being kept constant.
Then we define a statistical hierarchical set as the limit set of iteration 2.
The mass dimension of the set is again given by Eq. For if Tf denotes the map Tj with the translation terms discarded, then it follows from r k The asymptotic set H is the counterpart of an attractor for the transformation T. We carry out the following discussion for an embedding space R2 only. It is then convenient to choose as the basic set the unit line segment 8 1 on the x axis, centred at the origin. After K 20 Jean M. Perdang iterations Eq. Assuming that condition 2. Under condition 2.
The corresponding limit set 2. The limit set is a hierarchical tree. As independent dimensions we choose the global similarity dimension of this hierarchy Eq. The limit set 2. We have defining the global dimension 2. U cePe ' 2. The global similarity dimension of the hierarchy describes the border of the islands plus coastline island chain dimension ; the special dimension is again a filament dimension, similarity dimension of the filament, and of the 22 Jean M.
Perdang boundary of any individual island of the chain. The partial dimensions are island subchain dimensions which correspond to the subchains obtained by suppressing all polygons but one, c Pi in the first iterate. Any symbol of length P can therefore be expanded as a product of P symbols of length 1. We use the following terminology Temperley With a geometric point in the plane associate a sign i; to a line segment of unit length joining two points symbolised by i and j there corresponds the pair symbol i, j. The union of two geometric sets of associated symbols E' and E" corresponds to the symbolic product.
With these conventions, the geometrical structure of any first iterate B1 is encoded in a symbol E of the above form. We shall use the signs o and 1 for the invariant points of the two affine maps T 1 and T 2 Eq.