Self-similarity principle and iterative generation principle are important principles of fractal theory. It means that fractal is invariant under the usual geometric transformation, that is, scale independence. Self-similarity is based on symmetry of different scales, that is, recursion. The self-similarity in fractal can be the same or similar in statistical sense. The standard self-similar fractal is a mathematical abstraction, which iteratively generates infinitely fine structures, such as Koch snowflake curve and Sierpinski carpet curve. There are only a few such regular fractal, and most of them are statistical random fractal.
Fractal dimension, as a quantitative characterization and basic parameter of fractal, is another important principle of fractal theory. Fractal dimension, also known as fractal dimension or fractional dimension, is usually expressed by fractions or numbers with decimal points. For a long time, people are used to defining points as zero dimension, straight lines as one dimension, planes as two dimensions and spaces as three dimensions. Einstein introduced the time dimension into the theory of relativity, thus forming a four-dimensional space-time. Considering a problem in many aspects, we can build a high-dimensional space, but all of them are integer dimensions. Mathematically, the geometric objects in Euclidean space are constantly stretched, compressed and twisted, and the dimension remains unchanged, which is the topological dimension. However, this traditional view of dimension has been challenged. Mandelbrot once described the dimension of a rope ball: observing the rope ball from a distance can be regarded as a point (zero dimension); Seen from a close distance, it is full of a spherical space (three dimensions); Closer, you will see the rope (one dimension); Microscopically, the rope becomes a three-dimensional column, which can be decomposed into one-dimensional fibers. So, what about the intermediate state between these observation points?
Obviously, there is no exact boundary between a rope ball and a three-dimensional object. Mathematician Hausdorf put forward the concept of continuous space in 19 19, that is, the dimension of space can change continuously, and it can be an integer or a fraction, which is called Hausdorff dimension. Written as Df, the general expression is: K=LDf, also known as K=( 1/L)-Df. Take the logarithm and get Df=lnK/lnL, where l is the multiple of an object expanding in each independent direction and k is the multiple of the original object. Obviously, Df is generally a fraction. Therefore, Mandelbrot also defines fractal as a set with Hausdorff dimension greater than or equal to topological dimension. Why can't the British coastline be accurately measured? Because Euclid's one-dimensional measure is inconsistent with the dimension of coastline. According to Mandelbrot's calculation, the dimension of British coastline is 1.26. Using fractal dimension, the length of coastline can be determined.
Fractal theory is the frontier and important branch of nonlinear science, and it is also a new interdisciplinary subject. As a methodology and epistemology, its enlightenment is various: first, the similarity between fractal whole and local morphology inspires people to know the whole through cognitive part and the infinite from finite; Secondly, fractal reveals a new form and new order between whole and part, order and disorder, complexity and simplicity; Thirdly, fractal reveals the picture of universal connection and unity in the world from a specific level.
=============================================
Fractal theory and its development course
Fractal theory called natural geometry is a new branch of modern mathematics, but its essence is a new world view and methodology. It is combined with the chaos theory of dynamic system and complements each other. It recognizes that under certain conditions, some parts of the world may. In the process, in a certain aspect (shape, structure, information, function, time, energy, etc. ), it shows the similarity with the whole, and it admits that the change of spatial dimension can be both discrete and continuous, thus expanding the horizon.
The concept of fractal geometry was first put forward by the French American mathematician Mandelbrot in 1975, but the earliest work can be traced back to 1875. German mathematician K.Weierestrass constructed a continuous but differentiable function. G Cantor, the founder of set theory, is a German mathematician.
1890, Italian mathematician g Peano constructed a curve to fill the space.
1904, Swedish mathematician H.von Koch designed a curve similar to snowflakes and island edges.
19 15 years, the Polish mathematician W.Sierpinski designed geometric figures similar to carpets and sponges. These are counterexamples to solve the problems of analysis and topology, but they are the source of fractal geometry.
19 10, the German mathematician F.Hausdorff began to study the properties and quantity of singular sets and put forward the concept of fractal dimension.
In 1928, G.Bouligand applied Minkowski's ability to non-integer dimensions, so that spirals can be well classified.
Box dimension was introduced by Pontryagin in 1932.
In 1934, A.S.Besicovitch gave a deeper insight into the properties of Hausdorff measure and the fractal dimension of singular sets. He made important contributions to the research field of Hausdorff measure and its geometry, and thus put forward the concept of Hausdorff-Besicovitch dimension. Since then, the research work in this field has not attracted more attention, and the pioneer's work has only been circulated in the textbook of analysis and topology as a counterexample.
two
During the period of 1960, when studying the long-term behavior of cotton price change, Mandelbrot found the symmetry between large scale and small scale. In the same year, when studying the transmission error of signals, it was found that the error transmission and error-free transmission were arranged according to Cantor set in time. In the mathematical analysis of the Nile water level and the British coastline, a similar law was found. He summed up the symmetry of many phenomena in nature from the perspective of scale transformation. He called this kind of set self-similar, and its strict definition can be given by similarity mapping. He thinks that Euclidean measure can't describe the essence of this kind of set, and instead studies dimension, and finds that dimension is invariant under scale transformation, and advocates describing this kind of set with dimension.
From 65438 to 0975, Mandelbrot published the first book of fractal geometry, Fractal: Shape, Opportunity and Dimension, in French. This book has been published in English again. It concentrates Mandelbrot's main thoughts on fractal geometry before 1975. It defines fractal as a set whose Hausdorff dimension is strictly greater than its topological dimension, and summarizes the method of calculating experimental dimension according to self-similarity. Because the similarity dimension is only meaningful for small sets with strict self-similarity, Hausdorff dimension is extensive, but it is difficult to get it by calculation in many cases, which limits the application of fractal geometry.
1982, Mandelbrot's new book Fractal Geometry of Nature was published. Fractal is defined as a set similar to the whole to some extent, and the box dimension is discussed again. It is easier to calculate than Hausdorff dimension, but the box dimension of dense countable set is equal to the spatial dimension of the set. In order to avoid this defect, Tricot (1982) introduces the filling dimension,
In 1983, P.Grassberger and I. Proch proposed an algorithm to directly calculate the attractor dimension of dynamical systems according to the observed time data series.
1985, Mandelbrot proposed and studied the self-affine sets widely existing in nature, including self-similar sets, which can be strictly defined by affine mapping. In 1982, F.M.Dekking studied recursive sets. This kind of fractal set is generated by iterative process and embedding method, which has a wide range, but it is very difficult to study the dimension. Degin obtains the upper bound of the dimension. 1989, Zhong et al. solved the Deking conjecture and determined the dimensions of a large class of recursive sets.
With the development of fractal theory and the gradual improvement of dimension calculation method, after 1982, fractal theory has been gradually applied to many fields and become more and more extensive. It is still an arduous task to establish a simple and universal dimension calculation method to meet the needs of application development.
Fractal in nature is closely related to probability statistics and stochastic processes. Adding randomness to the deterministic classical fractal set will produce random fractals such as random Cantor set and random Koch curve. 1968, when studying the stochastic process of Brownian motion, Mandelbrot extended it to fractional Brownian motion related to fractal. In 1974, he proposed a fractal seepage model. In 1988, j.T.Chayes gives a detailed mathematical analysis. From 65438 to 0984, U.Zahle obtained a very interesting fractal structure by random deletion. Random fractal can describe and simulate natural phenomena more truly.
three
Fractal set in dynamical system is the most active and fascinating research field in fractal geometry in recent years. The strange attractor of dynamic system is usually a fractal set generated by the iteration of nonlinear function and nonlinear differential equation. 1963, meteorologist E.N.Lorenz discovered the first strange attractor named after him while studying the convection motion of fluid, which is a typical fractal set.
1976, French astronomer M.Henon got the enon attractor when considering the standard quadratic mapping iterative system. It has some self-similarity and fractal properties. 1986, Lawwill transformed Smale's horseshoe map into Lawwill map, and the limit of unstable manifold under its iteration was integrated into a typical strange attractor, and its cross section with the horizontal line was a Cantor set. In 1985, C. Greppo and others constructed a two-dimensional iterative function system, and its adsorption bound was Wilstrass function, and the box dimension was obtained. In 1985, S.M.MacDonald and Greppo obtained three types of fractal adsorption:
(1) locally disconnected fractal set;
(2) locally connected fractal quasicircle;
(3) It is neither locally connected nor quasi-circular. The former two have quasi-self similarity.
Another kind of fractal set in dynamic system comes from the iteration of analytic mapping on complex plane. This research was initiated by G.Julia and P.Fatou on1918-1919. They found that the iteration of analytic mapping divides the complex plane into two parts, one is the normal map, and the other is the Julia set (J set). They don't have a computer when dealing with this problem, and rely entirely on their own inherent imagination, so their intellectual achievements are limited. In the following 50 years, little progress was made in this field.
With the use of computers to do experiments, this research topic has gained vitality again. 1980, Mandelbrot drew the first picture of Mandelbrot's special collection (M Collection) named after him by computer. 1982 A.Douady constructs a quadratic complex map fc with parameters, and its Julia set J(fc) presents various fractal images with the change of parameter c, such as the famous Youdidier and Saint Kyle attractor. In the same year, D.Ruelle obtained the relationship between J- set and mapping coefficient, and solved the problem of calculating Hausdorff dimension of hit set of analytic mapping. L.Garnett obtained the numerical solution of Hausdorff dimension of J(fc) set. In 1983, M.Widom further generalized some results. The research of whole function iteration began with the normal graph 1926. 198 1 year, M.Misiuterwicz proved that the J set of exponential mapping is a complex plane, which solved the problems raised by normal graphs and aroused great interest of researchers. It is found that there is a difference between the J set of transcendental whole function and rational mapping J. In 1984, R.L.Devanney proved that the J (Eλ) set of exponential mapping eλ is a Cantor bundle or a complex plane, and J(fc) is a Cantor dust or a connected set.
The point c on the complex plane that makes J(fc) a connected set constitutes a Mandelbrot special set. According to H.Jurgens and H-O.Peitgen, the properties of M sets have always been and will continue to be a huge problem in mathematical research. Through the combination of mathematical theory and computer graphics experiments, and the basic research work carried out by H.Hubbard and others in this field, great progress has been made in solving this problem and people's understanding of M sets has been deepened. In 1982, Dodi and Hubbard proved that M- sets are connected and simply connected, and people speculated that M- sets are locally connected. At present, every computer diagram has confirmed this conjecture, but no one has been able to prove it yet. It is not clear whether m is arc connected. The dimension of M-set boundary is also one of the problems worth studying.
M-set not only divides J-sets into connected and disconnected categories, but also acts as a graph table of infinite J-sets, that is, the graph around point C of M-set is an integral part of J-set related to point C. However, the mathematical secret of this discovery has not yet been determined. Tan Lei (1985) proved that there is similarity between adjacent M sets and related J sets of each Mihewitz point. Eugene et al. obtained fractal images similar to natural morphology in the study of m-set electrostatic potential. At present, many researchers, including Eugene, are devoted to exploring M episodes with the help of computer activity videos. The research work of other fractal sets is making progress. In 1990, Dwayne observed through numerical experiments that the complex graph of M set consists of many stable regions of periodic orbits with different periods. In 199 1, Huang Yongnian proved this fact by his algebraic analysis method, and studied the global analytic characteristics of M sets and their generalized periodic orbits.
Basle (B.M.Barnsley) and S. Demko (1985) introduced the iterative function system. Many fractal sets, such as J sets, are attractive sets of some iterative functions, and fractal sets generated by other methods can also be approximated by iterative function systems. 1988, Lawwill found that the Pythagorean tree flower is a J set of an iterative function system through numerical research. Basle et al. studied the iterative dynamical system of functional system with parameters in 1985, and obtained the connectivity difference between m sets d, d and m. Under the iteration of a linear mapping system, a famous fractal curve-Gemini curve can be generated. 1986, Shui Gu and others studied its dynamic system.
The Hausdorff dimension dH of fractal set in general dynamic system is difficult to be obtained by theoretical method or calculation method. For fractal sets with overlapping structure, T.Bedford et al. gave an effective algorithm in 1986, but these results are difficult to apply to fractal sets generated by general nonlinear mapping iterative dynamic systems, and the conclusion and algorithm of Hausdorff dimension dH actually do not exist. Kaplan (j.L.Kaplan) and York (J.A. York) introduced the Lyapunov dimension dL in 1979, and speculated that dL=dH. 198 1 year Lelapier proves that dH≤dL. Yang (L.S.Young) 1982 proved that dH=dL in two dimensions. A.K.Agarwal et al. illustrated in 1986 that Kaplan-York conjecture does not hold in high-dimensional cases. This conjecture attempts to infer geometric structure from dynamic characteristics, and its inverse problem is to infer chaotic mechanics from attractor dimension, which is worth studying. But at present, there is little work in this field, and it is mainly limited to computer research. In addition, the fractal dimension of parametric dynamic system in chaotic critical state or sudden change needs further study.
Multifractal is another important fractal set related to the strange attractor of dynamical systems, and its concept was first put forward by Mandelbrot and A.Renyi. In 1983, J.D.Farmer and others defined the generalized dimension of multifractal. In 1988, T.Bohr and others introduced topological entropy into the dynamic description and thermodynamic analogy of multifractals. In 1988, Arnedo and others applied wavelet transform to multifractal research. J.Feder, T.Tel and others have studied multifractal subsets and scale indices. Eminem Tricca studied the inverse problem of multifractal, put forward the generalized partition function, gave the generalized transcendental dimension, and revised the previous dimension. J.Lee and others discovered the phase transition of multifractal thermodynamic form. In 1990, C.Beck obtained the upper and lower bounds and limits of generalized dimensions, and studied the uniformity measure of multifractals. Mandelbrot studied random multifractals and negative fractal dimensions. Covic introduced a binary iterative system in 199 1, and derived the dimension, entropy and Lyapunov exponent by using the maximum eigenvalue and Gibbs potential, which provided a general scheme for the classification of multifractal phase transitions. General scheme of multifractal phase transition classification. Although many methods have been put forward to deal with multifractals, from a mathematical point of view, these methods are not strict enough, and some problems are difficult to deal with mathematically.
four
Fractal theory has only been developed for more than ten years, and it is in the ascendant. Many theories need further study. It is worth noting that in recent years, the application and development of fractal theory far exceeds the development of theory, which puts forward newer and higher requirements for fractal mathematical theory. The establishment, improvement and perfection of various fractal dimension calculation methods and experimental methods make their theory simple and easy to operate, which is a common concern of scientists who apply fractal. In theoretical research, the theoretical calculation and estimation of dimensions, fractal reconstruction (that is, finding a dynamic system so that its attractive set is a given fractal set), the properties, dynamic characteristics and dimensions of J sets and M sets and their extended forms will become very active research fields for mathematicians. The perfection and rigor of multifractal theory and how to solve practical problems with these theories may arouse scientists' extensive interest, and dynamic characteristics, phase transition and wavelet transform may become several hot spots.
In philosophy, people are interested in the universality of self-similarity, the simplicity and complexity of M set and J set, the unity of complex number and real number, the relationship between multifractal phase transition and catastrophe theory, the characterization of self-organized criticality (SOC) and the transformation of various contradictions in fractal system. It can be predicted that a discussion on the philosophy of fractal science will be held in China soon.
======================================
Research on fractal theory and wave theory/forex/forexinfo/2005091612161974322.shtml.
Fractal theory and its application in chemical industry/resource/article/jylw/3/306/lw009994.htm.