Chapter two Basic Theory of Chaos and Fractals

2.1 Chaos2.2 Bifurcation and Ways to Generate Chaos 2.3 Criteria and Guidelines to Study Chaos 2.4 Bifurcation

2.1Chaos2.1.1The Features of ChaosChaos has definite movement characteristic: non-periodic and orderly, had discovered three roads to the chaos, the Feigenbaum universal constant, the boundedness and have the strong sensitivity to the initial valueThe main features of Chaos 1Sensitive initial state 2Elongates and folds 3Have rich hiberarchy and self-similarity structure 4The existence of chaotic attractor in nonlinear dissipative systems

2.1.2Definition of ChaosThe chaos system takes time as the independent variable of the differential equation, and the goal is to predict the final condition of the solution in the remote future or in the remote past. Let the system is based on the form of discrete characters, that is:

F is a mapping from set M to M, then we want to know the final states of the sequence with n enlarging

So we call the set

is the forward orbit of x, express as

If is the homographism, we may also define the set of point as the entire orbit of x the afterward orbit of x is defined as

Definition2.1 if but for the natural number k which is less than n ,then say is a n periodic point of when is a n periodic point of then

and only has n different elements

Definition 2.2 when is a n periodic point of f say

is n periodic orbit of f

Definition 2.3 Let (X, ) is a compact metric space, f: X X is a continuous mapping, we say f is chaotic at X, if: (1) f has sensitive dependence on initial value, (2) f is transmission topology in X, (3) the periodic points of f is dense in X.

The word Chaos was first proposed by T. Y. Li and J. A. Yorke. in 1975, they delivered a speech entitled Period Three Implies Chaos , in The American Mathematical Monthly, and gave a mathematical definition of chaos, now known as the Li-Yorke definition or Li-Yorke theorem

consider a continuous, single-parameter mapping which map the interval [a, b] to itself , ,can also be written in the form of point mapping ,Definition2.4 continuous or point mapping is chaoticif1exist periodic points of all periods 2exist uncountable subset

S dont have periodic points ,cause , , , , , , p is a periodic point

According to the Li-Yorke definition, a chaotic system should have three properties: 1 the existence of periodic orbits of all bands; (2) there is an uncountable set, this set contains only chaotic orbits, and neither any two tracks tend to far away or do not close off, but two states alternately take place, at the same time any of the orbits do not tend to any periodic orbits, namely the set do not have asymptotical periodic orbits; (3) chaotic orbit is highly unstable.

Obviously the periodic orbit and the chaotic motion have close relation, manifested in two aspects First, study the steady state of motion in the parameter space , the system often must experience a series of cyclical system first in the parameter change process, then enters the chaotic state. This constitutes the so-called "road leading to chaos."Second, a chaotic attractor with a infinite number of unstable periodic orbits;, or there are many long or short fragments in a chaotic orbit , they are very close to this or that unstable periodic orbits.

2.1.3Strange attractorThere is another category of system, in its movement, the phase space volume contract to the attractor with its dimension is lower than the original dimension of phase space , this kind of system is the dissipation system. In dissipative systems, there are some equilibrium point (fixed point) or sub-space,along with time increase, the orbit or movement approaches to it, it is the attractor.

2.1.3Strange attractorIn phase space, the dissipative system may have many attractors. Usually dissipation system's simple attractors are fixed point, limit cycle and torus. Simple attractor is influenced by the system parameters. With parameters changing, chaos will appear in the dissipative movement , then the attractor becomes strange attractor.The chaotic motion is characterized by the strange attractor is the unique nature of dissipative systems.

The four attractors of dissipative system are often explained as follow:(1) Fixed-point attractor or Trivial attractorsit is a zero-dimensional attractors and a point in phase space is, it shows that the system is in balance movement . (2) Limit cycle: a one-dimensional attractor and a closed curve around the equilibrium points in phase space, it corresponds to periodic motion. (3) Quasi-periodic attractors: two-dimensional torus in phase space, which is similar to the surface of the donut, the track around the surface of the torus in the state space

(4) Strange Attractors: also called "Random attractors," "chaotic attractor." It is a infinitely collection of points in the phase space, these points correspond to the chaotic state of the system. Definition 2.5 Due to the phase space volume of dissipative systems is contraction, the steady-state movement of n-dimensional dissipative systems will be located in a "surface" (hyper surface), which is less than n-dimension, roughly to say that this surface is the attractor .Definition 2.6 First ,it should be an attractor, that is, there is a set U, makes (1)U is a neighborhood of A (2) For each initial point ,when t > 0should have ;when that is A is the attractor , in addition

(3) when have sensitivity of x0 (when initial value of error is infinitely small, its phases error is in exponential growth along with t), that is, A is a strange attractor (4)For there is ,let and strange attractors should not be divided into two. Strange attractor has the following important features: (1) have a very sensitive dependence on the initial states. (2) its power spectrum is a wide spectrum. (3) the existence of horseshoe system. (4) it has a very peculiar form of topology and geometry

2.2Bifurcation and the Way to Generate Chaos 2.2.1 Bifurcation TheoryChaotic motion occurs after a series of solution mutations. The parameter values of a mutation is known as the bifurcation point. The main contents of bifurcation theory is to study the numbers of the solutions of non-linear equations how to have mutation in the change process of parameters.

When the control parameters change to a critical value , the dynamic behavior of the system have qualitative change. this phenomenon is called Bifurcation, it is a characteristic inherent of non-linear system. The most common Bifurcation are: fork-type bifurcation or symmetric saddle node bifurcation, cut bifurcation or saddle node bifurcation, transcritical bifurcation, hysteresis bifurcation, Hopf bifurcation, period-doubling bifurcation, homoclinic and heteroclinic bifurcation.

2.2.2The roads lead to Chaos2.2.2.1 Period-doubling bifurcation lead to chaos It is also called Feigenbaum Way. This way is one kind of regular state of motion For example, steady-state solution or the periodic solutionscan transit gradually to the chaotic motion condition by continuous doubling period .Such as:

Logistic mapping leads chaos by period-doubling bifurcation, as following diagram

Feigenbaum noted that the Logistic map bifurcation point of the parameter values (m = 1, 2, 3, ...) to form infinite sequences, and a limit value = 3.569945672 Simultaneously, Feigenbaum found the Logistic mapping system with the emergence of period-doubling bifurcation, but also has other complex dynamic behavior. Logistic mapping, described below. 1Chaos and strange attractors when = , the infinite cycle has infinitely number of solutions, mapping into the chaos, as a whole it is stable and it also is a strange attractor. The mapping, because it has two basic properties: 1stretching and folding 2Irreversible2inverse cascade When = 4, the cycle for any integer solution exists, and are unstable.

Therefore, in [0,1] can be listed on an unlimited number of unstable periodic points, after the removal of these points [0,1] are still not out of an unlimited number of points, the points will form a strange attractor.3periodic windows When ,the Logistic map exists the solutions of cycle 3. In 1975, Li and Yorke had put forward the "cycle of 3 means that the chaos" of the thesis.4Universal Sequence also known as MSS Sequence or kneading sequence. It said that in the single-peak mapping that the location of the window of their cycle with a certain order.

2.2.2.2Paroxysmal to chaosParoxysmal chaos refers to when the system from an order into chaosIn the non-linear non-equilibrium conditions, changes in certain parameters when it reaches a certain critical threshold, the system sometimes acts of the time cycle (orderly), sometimes chaotic, oscillating between the two.

2.2.2.2Paroxysmal to chaosAt the earliest , Paroxysmal chaos can be found in the Lorenz model, but more detailed studies are in some non-linear mapping .The chaos generated by Paroxysmal chaos and generated by period doubling bifurcation is twinning phenomena.Those who observed period-doubling bifurcation of the system, in principle, can be found in the phenomenon of intermittency chaos

2.2.2.3Hopf bifurcation to chaosIs a rule of motion after at most 3 times Hopf bifurcation can be transformed into a state of chaotic motion. Specifically,The transform to chaos can be expressed as Fixpointlimit cycleTwo-dimensional toruschaos, Each time a bifurcation can be seen as a Hopf bifurcation, bifurcating a new non-divided frequency.

2.3 Criteria and guidelines for the study of chaos2.3.1Poincare section method French mathematician Poincare provided us an efficient method for researching the track of complex multi-variable dynamic system, that is Poincare section method. In the multi-dimensional phase space appropriately select a cross-section, which may either be flat or curved. Then consider the change of a series of intercept points of the continuous dynamics and this cross-section. In this way, we could get rid of the track of phase space, and draw the intercept points of the Poincare section with the help of computer. Information about movement characteristics could be get from them.

2.3.2Phase space reconstructionReconstruction of time series: when time series is extended to three-dimensional or higher-dimensional phase space, chaotic information of time series could fully come to light. Packard et al proposed the reconstruction of an "equivalent" phase space from a one-dimensional variable that can be observed.Reconstruct m-dimensional phase space from a certain time series that can be observed, and get a group of phase space vector

where t is the time delay, ,d is the number of variables of the system. M is less than Nand have the same order of magnitude with N.

The key of reconstruction phase space is the selection of embedding space dimension m and time delay t.1. The selection of embedding space dimension mPackard and F.Takens proposed the time delay method to construct phase space. It is based on embedded topology theory:Basic idea: If a one-dimensional curve is limited to a two-dimensional surfacethis curve will intersect on the reconstruction normal surface. Small deformation in the curve wont cause the disappearance of the points of intersection. On the contrary, if put a curve in the three-dimensional space ,all the points intersect with themselves can be eliminated through a small deformation. Thus, these intersection points can be viewed as occasional. Conclusion promotion: two objects A and B, their dimensions are respectively

They are in the d-dimensional space. We define a codimension to denote the dimension of

The codimension has the nature of addition

so (2.5)The deterministic of formula (2.5) could beindicate by several examples, for example, put twocurves in a surface, and usually there would be some overlapping, here , .According to (2.5) we get But in three-dimensional space, they wont intersect, because , .

In the three-dimensional space, the curve of will intersect with the curve of ,because (2.5) can be used for the same objects ,This object should stretch freely in the d-dimensionspace, that is, any part of A wont touch any other part of A when it is stretching, because .This is not a good marker method. Let the left and of (2.5) be , then the embedding spacedimension is (2.6)The result of (2.6) is usually bigger than actualneed. To ensure the correct restoration of theattractor, the embedding space dimension must beat least twice of the dimension of the attractor.

The select of the value of m should be determined througha number of attempts. When m is too large, all the values of m corresponding to non-scale reign is the same. In the calculation of the movement characteristics ofchaos, the least dimension of the embedded space isdetermined by what physical quantity we are going to extract from the time series. Roux et al. discussed the selection of m and t. They believe that in most cases, m could be a little smaller thanthe value determined by the inequality m 2d+1. Wolf et al. got the same conclusion when they were studying the calculation of Lyapunov exponential. The majority view is that the value of m should bedetermined through a number of calculation. Roux et al.proposed that let m add 1 every time, until no additionalstructure appear in the phase diagram.

2. The selection of time delay t1986, Fraser et al. pointed out that the method ofautocorrelation function only measure the linear relation ofvariables. In order to measure the general dependence relation between two variables, we should select the delay twhen the mutual information function between the twocomponents appears the first minimum. Mutual information is defined as follows:

are two series. In the surface of S-Q, with count boxes approach we could get probability distribution and joint probability distribution The concept that using logarithm denote the information:When we get an element from a set with n elements, (2.7)

Shannon considered the case that there are n subset in S,Apply formula (2.7) for every subset : , ,and average the information for each subset, then get theinformation of S: (2.8)

Then we get joint information:

From and we get:

and are defined as follow, and that is mutual information :

In the actual applications, the selection of the best delay time t needs repeated attempts.

2.3.3 Power spectrum analysis Welch proposed the average periodogram method to calculate the power spectrum estimate value of scalar signal :Suppose the power spectrum of sequence is divide the sequence into K overlapping segment with length of L, we can obtain the amended periodic map of the estimated value. In the realization process, paragraph overlapping sequences from a sample point, the total number of all sequences is The ith section of value is defined as

is data window function of L points ( for example: rectangular window function, Hamming window function)

After window processing, Discrete Fourier Transform of M point of sequence is:

is calculated with the FFT algorithm

2.3.4 Correlation dimension Strange attractor has the fractal structure. Fractal dimension can divide the chaotic extent of attractors further. there are several definitions of Fractal dimension, the correlation dimension as the measured parameters of chaotic behavior has been widely used. based on the experimental data in the phase space reconstruction , calculate the relevant points of formula (2.4) (2.11)

Any distance less than a given positive vector r, called the associated vector, here H is the Heaviside function

If the value of r is appropriate, as the increase of r , there is a rapid increase in an exponential times, the correlation dimension is defined as :

In the calculation, with the changes of embeddingdimension d ,in log-log graph curve beamthe slope of the straight lines parallel with each other is the correlation dimension D2.

2.3.5 Lyapunov Index2.3.5.1 Definition of Lyapunov index For strange attractors ,Lyapunov index can express the sensitive dependence of initial conditions. For the continuous dynamic system in n-dimensional phase space, study the long evolution of an infinitesimal n-dimensional ball. due to the local deformation characteristics of flow, it will become a n-dimensional spherical ellipsoid. The ith Lyapunov index is defined by ellipsoid axis length pi (t) as: (2.12)

(2.12) Indicates that the size of Lyapunov index shows average convergence or divergence index rate in similar phase space orbit .

Lyapunov index is a very general characteristic value, it define each type of attractors . For n-dimensional phase space, there are n real indexes, they are also known as spectrum, ordered in accordance with their sizes. Generally :

For the case of strange attractors, the largest Lyapunov index is positive (and also at least one Lyapunov index is negative), and the greater the Lyapunov index , the system is more chaotic; and vice versa

For one-dimensional (single variable) case, the attractor may be only fixed point (a stable steady state). Lyapunov index is negative. For two-dimensional case, attractors are fixed points or limit cycles. for fixed points for limit cycles In the three-dimensional cases fixed points limit cycles two-dimensional torus

unstable limit cycles unstable two-dimensional torus strange attractor

In continuous four-dimensional dissipative systems, there are three different types of strange attractors

2.3.5.2 Kaplan-Yorke GuessThe relationship between Lyapunov index and fractal dimensionsort the index Lyapunov-based, start from the largest (at least one of the index is greater than zero), then add up the following index.Suppose when adds sum is positivebut when the next one sum becomes negativeVery naturally think that the range of attractor dimension between k and k +1.

Determined score part of dimension by linear interpolation, Kaplan and Yorke [60] has been speculated this relationship :

Here d is Fractal dimension k is the largest value of K makes

In the two-dimensional circumstances, reduced to the type:

2.3.5.3 Differential equations method to calculate the Lyapunov index

Definition 2.7 Space-based differential equation:

f is continuously differentiable on the map on . Suppose is Jacobi Matrix of fnamely

let

Put n complex modulus of eigenvalue, and order them according to size:

then, the Lyapunov index of f is defined as

2.3.5.4 Differential equations method to calculate the largest Lyapunov indexIn 1976Benettin proposed Differential equations method to calculate the largest Lyapunov index The method as follows In the phase space which is confirmed by the given differential equations, choose two initiate points closely and ,the space between them is and is very smallin a small time interval to integrate this differential equationsuse transform get

The space between two points is Then choose a new point its position is in the line of and and make For and

use transform againcan get and and Repeat this processas shown in fig. 2.3then can be calculated by the following equation

Here n is the times of integral so n must be very largesuch as but must be very smallsuch as so long as is not too largethe result is independent of size of

Figure 2.3 Calculate the evolution process of Lyapunov index in differential equations

2.3.5.5 Experimental data to calculate Lyapunov index1The length evolution law In 1985Wolf et al. in summing up the results of previous studies, proposed one way to calculate the largest non-negative Lyapunov index based on the experimental data - the length evolution lawCalculated as follows: from the time series of experimental data , constructed m-dimensional space using time-delay method, every point in space is given by { , , }. First, find out the nearest points from the initial point { , , },and express the distance between these two points as . To the moment has evolved into , then look for a new data point as the following two principles :

after evolution it has a small distance from the reference point ; and with a very small angle between and . repeat the process as fig.2.4 shownexhausted until all of the data points. then is

N is the total evolution numbers of length element. When tend to a stable value, the calculation to be successful.

Figure 2.4 the evolution and replace process to calculate the Lyapunov index in length evolution method

2Area evolution lawMethod as followsFor considering the attractors of (+, 0, -) spectra in the near the plane local structure (Ie, require ), in the reconstructed attractors identify three neighbor points and , is any point in the attractor, for instance check the first point corresponding to the time Point and are to use exhaustive or other methods to find the nearest neighbor points of , then forward the development of time-series to get the new location of the three points in time as areas of the two triangles are defines as and

Maintain the point as a triangle vertex, and search for its new nearest neighbor point and .The area of triangle is .Clearly the amount reflect local expansion and contraction characteristics of attractor. Repeat this step until all the data are used, the estimated value of the sum of two largest Lyapunov index is

Where N is the total substitution steps, is the kth step substitution time. when tend to a stable value, the calculation to be successful.

Fig. 2.5 the evolution and replace process to calculate the Lyapunov index in area evolution method

2.3.6 Metric entropyAnother statistical nature of dynamic systems is entropy, There is a certain relationship between it and Lyapunov index and the Hausdorff dimension .as a measure of chaos, the most common entropy are topological entropy and metric entropy (or measure theoretic entropy)Metric entropy is an extension from the thermodynamic entropy which we are familiar with. In statistical thermodynamics, entropy

is the probability in state i. Entropy S is a measure of the degree of system disorder. According to S, the metric entropy K can be introduced

K-entropy is defined as follows: consider the orbit of dynamical systems on strange attractor . For d-dimensional phase space is divided into the boxes with size , the system state can be observed in time interval of time . suppose is joint probability that in box in box , in boxAccording to Shannon Formula

K-entropy is defined as the average loss rate of information

the limit illustrate K has nothing to do with selection of division. For discrete time step mapping , can be omitted.

K is closely related to positive Lyapunov index. For finite-dimensional differential mapping (2.18)For all the positive Lyapunov index ,gives the upper limit of K-entropy. In practice , the equation (2.18) tend to set up, it became the so-called Pesin equality

2.4 Fractal The term fractal is proposed by Mandelbrot to describe the complex structure of all scales of irregular and broken shapes . He has published three books in French and English in 1975,1977 and 1982, especially the booksFractals:From,Chance,and DimensionThe Fractal Geometry of Nature introduced many people into the fractal fields.

Definition 2.8 If a set in Euclidean space, its Hausdorff dimension is larger than its topological dimension constantly , that is, the set is called fractal sets, take fractals for shortDefinition 2.9 An integral part in some way similar to the whole body is called fractal. British mathematician Falconer in his book Fractal Geometry Mathematical Foundations and Applications proposed that the definition of fractal should be given in a similar way as the definition of "life" in biologists, that is, not to seek the precise definition of fractal, but search for the characteristics of the fractal.

In general, the set F is called fractal, that is, it has the following typical properties: 1 F has a fine structure, that is, any small-scale details 2 F is irregular, which can not be described in traditional geometric language 3 F usually have some form of self-similarity ,may be similar or statistics. 4 F in one way or another, the defined "fractal dimension" is usually greater than its topological dimension. 5 interestingly in most cases, F can be defined in a very simple way , you may get by iteration

2.4.1 the relationship between fractal and chaos In the non-linear science, fractal and chaos have different origins, but they are the non-equilibrium process and results which are described by nonlinear equations , indicating that they share a common mathematics ancestor - Dynamical Systems, strange attractors is the fractal sets, or chaos is fractal in time , and fractal is chaos in space.

2.4.2 the escaping time algorithm to construct Fractal structureDynamical systems: definition 2.10 the dynamical system on metric space (X,) is a transformation f: XX . the orbit of point x in X is the sequence . Set (X,) is a given metric space, (F(X),h) represent the corresponding non-empty compact subset space (that is, fractal space) with Hausdorff distance. so the certainty fractal set A is the fixed point set of the contraction mapping and collapse transform on (F(X),h), and X,f (f is transformation on (X,)) construct the dynamical fractal system on (F(X),h).

Formula 2.1 suppose (Y,) is metric space XY is non-empty contact subset of YAnd supposefXY is continuous ,and have f(X)Xthen (1) by AF(X) define a transform WF(X)F(X) (2)W have fixed point AF(X)it is determined by the following formula

if f also meet suppose UX is the open subset of metirc space (X,)then f(U) the open subset of metirc space (f(X),).then (3)W is a continuous transform from metric space (F(X),h) to itself.

The fractal set A can be expressed as:

That is A is constituted of the points whose orbit are not away from Xit is the remain set of points whose orbit escape from Aso the escaping time algorithm to construct Fractal set is given as follows:(1)Given dynamical system X, f the view window W and the escaping-radius R and escaping-time limit N; (2) Define the escaping-time function

(3) Calculate with the point in the view window(4)if then (fractal set)if then

2.4.3 Julia setsDefinition 2.11:If is the polynomial whose order is larger than 1, represents the set of the points whose orbits do not tend to infinite points, i.e., , then the sets are called filled Julia sets corresponding to f. The boundary of is called the Julia sets of f, which denoted by , i. e.,

2.4.4 Mandelbrot setquadratic function ,for each , is a dynamical system which is dependent on two parameters. All the possible values of parameters is called parameter space.Definition 2.12 the Mandelbrot set according to dynamical system is

Definition 2.5 a family of dynamical system (cC) corresponding to the Julia set is connectiveif and only if

fig. 2.6 classical Mandelbrot set

Figure 2.6 show the M set which is made by escaping-time algorithm.M set have very complex structureIt has some apparent features.The elaborate structure shows that M set is connective setand has been proved by Hubbard and Douady mathematically .