Kaotik Davranış Kriterleri Olarak Fraktal Boyut Değişimi Ve Dinamik Sistemlere Uygulanması

Abstract

Chaos theory has opened new horizons in science and is already considered by many as öne of the most important discovery in the twentieth century after relativity and quantum mechanics. Deterministic chaos denotes the irregular ör chaotic motion that is generated by nonlinear systems whose dynamical laws uniquely determine the time evolution of the system state from previous history. in recent years, it has become clear that chaotic phenomena are abundant in nature and have far-reaching conse- quences in many branches of science, and especially in dynamic meteorology. According to dynamical system theoıy time evolution of a phenomenon can be given by its trajectories in the phase space. Coordüıates of this space are spanned by the variables which are necessary to specify the time evolution of the system. Any trajectory in this space represents the time evolution of a given set of initial conditions. in some systems, trajectories initiated from different phase space points eventually converge and stay on a distinct pattern. This kind of a pattern attracting ali trajectories is called "attractor". Systems that develop deterministically have low dimensional attractors such as point, limit cycle and tour. These types of attractors can be characterised by an integer dimension. An important property of these attractors is that ali trajectories converging on it stay at a constant distance from each other and this property imposes long term prediction of the system under study. it has been shown for many dynamical systems that their attractors are not topological. These attractors are called "fractal set" and can be characterised by a noninteger dimension. in literatüre, these kind of attractors are referred to as "strange (ör chaotic) attractors". The most important characteristic of the chaotic attractors is their exponential divergence of nearby trajectories. This implies that system is sensitive to initial conditions and consequently long term predictions become impossible. By using a time series of a single state variable, it is possible to reconstruct xi phase space. Before applying reconstruction procedure we need some informa- tion (i.e. dimension) related with the attractor. The new phase space can be formed by adding an extra independent coordinate until no more information about the process is gained. Öne of the independent coordinates mentioned above is taken as the time series itself. The remaining coordinates are formed by its (m-1) derivatives ör, in the discrete case, the (m-1) lagged time series shifted by (w-l) multiples of the correlation time T, at which correlations between coordinates become zero. There are some criteria for chaotic motion, as follovvs: i) Time series behave erratically, ü) Autocorrelation function decays exponentially, iii) Power spectrum has broad band noise at low frequencies, iv) Poincare map fills some region of space completely and irregularly, v) At least öne of the Lyapunov exponents is positive, vi) Dimension of attracting set is fractal. The first three criteria are functions of time t, delay time r and frequency w. Any time series gives only a simple idea about the underlying system. it is not possible to distinguish two time series sampled out from deterministic chaotic ör stochastic processes. The autocorrelation analysis is valid under two important assumptions which are that: - the probability density function (PDF) of the given data is normal, - the autocorrelation function is a means of measuring the linear dependence between two time series which originate from the same series with the delay time T. it is a well known fact that PDF of the most data encountered in applications are not normal, in such cases, autocorrelation analysis will give results different from the expected values. Although the power spectrum is a reliable method for periodicity, it is not effective for determining whether the chaotic time series is sampled out from a deterministic system ör not. Broad band noise in power spectra can arise from stochastic ör deterministic processes, but the decay in the spectral power at high frequencies is different for the two cases. Poincare map transforms a continuous system into a discrete system. If a phase space is spanned by the m state variables, we can cut this space by an (m-1) dimensional space called hyperplane. For phase space with dimension m>3, it xii is not possible to examine the Poincare section at least visually. in addition, Poincare sections of forced systems are very similiar to each other. For example, Poincare map of the chaotic and nonchaotic behaviour of quasi-period- ically forced van der Pol equation can not be distinguished easily. Only the last two criteria give numerical results, so they are regarded as more objective tools than the others. Lyapunov exponents (^.) measure the rate of exponential divergence ör convergence of nearby trajectories in the phase space. in other words, Lyapunov exponents are used to determine whether the system is sensitive to initial conditions ör not. in three dimensional phase space, the largest Lyapunov exponent is negative for a point attractor, zero for a limit cycle ( periodic âttractor) and n-torus as well as positive for chaotic attractor. For system where the equations of motion are explicitly known and linearized equations exist, there are straightfonvard techniques for computing a complete Lyapunov spectrum. in case of experimental data, it is stili difficult to measure the Lyapunov exponents. in Euclidean space, dimension is the minimum number of coordinates needed to specify a point uniquely. in contrast, the dimension of a dynamical system is the number of state variables needed to specify the dynamics of the system. Ali of these dimensions are integer so they can not characterise a chaotic attractor. Hovvever, almost ali chaotic attractors have fractal (noninteger) dimension. There are fifferent dimension definitions such as capacity, information, correlation dimension ete. Two main uses of these dimensions are: - To distinguish chaotic and nonchaotic attractors, - To determine the minimum number of state variables that are needed to specify the dynamics of the underlying system. If the points on the attractor are almost uniformly distributed in the phase space then the difference betvveen dimensions are insignificant. On the other hand, some subsets of the attracting set are visited more frequently than the others, then a nonuniform distribution of points will be obtained. Although the capacity dimension is not sensitive to frequent visit of some locations by the trajectory in the phase space, it is strongly dependent on the geometry of the attractor. Contributions of the frequently and rarely visited volume elements to the capacity dimension remain the same. On the other hand, the correlation dimension is sensitive to the relative frequency of the visit. in brief, due to this sensitivity to the dynamics of the process, it is convenient to use the correlation dimension, which ıs given by where />. is the relative frequency of the typical trajectory that enters the ith xüi AT(e) **EP? (1) j r. i-1 * ' da » Lım-- 8-0 Lne volume element. To obtain correlation dimension by using eq. (1) öne must estimate p. values especially through the box counting tecnique. As this techniqe is not practical in application, a fast algorithm was proposed by Grassberger-Procaccia (GPA). Consider the set {X{, /=1,...^V} of points on the attractor, i.e. X.=X(t+ij} with a fixed time increment T between the measure- ments. in this algorithm "correlation integral" is given by C(e)=£im -L£ £ 0(e- |*rx |)j (2) Af- N* i-1 >1 W J where 6 (y) is the Heaviside step function defined as.».U Î3 in order to obtain correlation dimension from eq. (1), it is necessary to know the relationship between S(Pj)2 and C(e). it is easily shown that such a relationship expressed as W(e) C^^p,2 (4) 1=1 Finally, combination of eqs. (1) and (4) leads to ia, Um ±m m e-o Ln e As the value of e decreases, clearly the number of paired correlations will decrease as well, and hence, C(e) « ed° (6) If we use sufficient number of noise free data points, plot of in C(e) versus /n(e) will give a linear part called "scaling region". Slope of this region is the estimated value of the correlation dimension dG (v is also frequently used notation). A noninteger value of v indicates chaotic behaviour; the underlying dynamics is deterministic and shows a sensitive depedence on the initial xiv conditions. There are some problems in calculating the correlation dimension. Noisy and limited number of data, difficulties in determining the scaling region exactly, errors arising from computer experiments etc. cause biased estimation of v. Above all, it looks unrealistic that a single number can fully describe the complex structure of a strange attractor. This implies that the value of fractal dimension must be handled so that it carries more information about the dynamical system under study. In applications, GPA is the most frequently used algorithm to estimate the fractal dimension. Distance matrix used to calculate the correlation integral is given by o d" (?) Due to the symmetry, it is sufficient to consider only upper triangle matrix so the number of pairs with distance less than e, say N(e), is to be normalised with N(N-l)/2 instead of N2. It is well known in the study of lacunarity, if e approches to zero then the expected form of the correlation integral becomes C(e)=$(e)ev (8) where the function <£(e) reflects the lacunarity of the set. The structure of <&(e) is generated by sparse or empty regions in the object. Distribution of these empty regions can be constant, periodic or aperiodic. Because sets show different lacunarity, one can expect that contribution of each row/band of the distance matrix to the correlation integral will be different. Calculation of the correlation integral at each row/band can be called "Row and Band Scanning Approach (RSA/BSA)".