Chapter 3

Chaos and Fractals in the Two-dimensional Non-linear

Mapping

CONTENT

3.1 Chaos in the one-dimensional Logistic mapping

3.2 Quantitative observation of the LMGS chaotic attractor’s characters

3.3 Computer simulation of three-dimensional renderings of strange attractors

3.4 Chaos and Fractals in the two-dimensional Logistic mapping

3.5 Strange attractors and Fractal in the general two-dimensional quadratic mapping

3.1 Chaos in the one-dimensional Logistic mapping

Based on Eqs. (2.3), can give another form of Logistic mapping:

3.1 The bifurcation process is shown in Fi

g. 3.1 (not pro-rata but schematic) :

Discussion of xn’s behavior when n and with the changes of parameter

When 0 0.25 ， = 0 is a equilibrium point. If 1 0.75 ， = 1 is a equilibrium point If 1 2 0.85056…… ， 2-cycle solution appe

ars. If 2 3 0.88602…… ， 4-cycle solution appe

ars. 。 If continues increasing, followed by the emergenc

e of a equilibrium periodic solutions, that is 8-cycle solution, 16-cycle solution and 32-cycle solution... ....

The above sequence of bifurcation value 1 ， 2 ， 3

，……， a limit of solution 0.89248……

When reached the value, the equilibrium solution of the system is a “ 2k-cycle solution”. So we can see that when the parameter’s value increases, the system will experience period doubling continually;

When the value of exceeds , the system will enter into the chaotic area. That is, when ∈ , the solution sequence of the system will be "attracted" to the periodic solution (including the special case of equilibrium state).

In addition to the usual two attractors like the stable equilibrium state and periodic solutions, when ∈ 1 of these are usually outside of the system when, the equilibrium state or the limit of the solution sequence will appear strange attractors, which is chaos in physical phenomenon.

It shows that: during the process of a series of bifurcation that leading Logistic map to chaos, it behaves the characteristics of Self-similarity and invariance under scaling, both in parameter space and phase space. and the. The evolution like this is generic in the non-linear system.

Therefore, M.J. Feigenbaum used the renormalization group method, after observe and research carefully, he found that when the system experienced period-doubling bifurcation to enter into the chaotic state, the quantity relation is regular, that is, Feigenbaum constant and and .

The first universal constant is the limit of the interval during the process of converge of bifurcation value n sequence.

He also pointed out that in the x-axis the distance between the periodic solutions is reducing by the factor = 2.5029078751..., which is the second universal constant.

The two constants are mutual to all period-doubling bifurcation which reflect the regularity of the process which leading period-doubling bifurcation to chaos.

From the above analysis, we can give the important universal characteristic between period-doubling bifurcation and chaos :

Regardless of the different continuous flow (the solution of differential equations) or the different discrete mapping, they have the same structure of period-doubling bifurcation, and at last they will both lead to chaos. Convergence of period-doubling bifurcation and the changes during the process to chaos both obey to a certain law, and both presented by the same universal constants , and .

Chaotic motion has self-similar structure on different levels, and the bifurcation structure of the sub-window (secondary and tertiary, ...) in each cycle of chaos strap obey the same law as the first bifurcation structure , and has the same universal constant.

3.2Quantitative observation of LMGS attractor’s chaotic characteristic

3.2.1 Method of constructing LMGS attractor

3.2.2 Results of Simulating LMGS attractor

3.2.3 Conclusion

3.2.1 Method of constructing LMGS attractor

Definition 3.1 LMGS defined as follows:

(3.2)

The subset of LMGS in 2-dimensional space can be expressed as:2DLMGS. 2DLMGS is defined as follows:

(3.3)

The operators in the above definition are the four basic operations: add, subtract, multiply and division. But we believe that the operator can also be other forms, such as power, extraction, etc., so that the LMGS defined above can be extended, and the extended LMGS can be used to protract the more beautiful graphics.

3.2.2 Results of Simulating LMGS attractor

Figure 3.2 Shows 2DLMGS which are generated by Logistic mapping. In fig. 3.2, 3.2a, 3.2c, 3.2e, 3.2g, 3.2i, 3.2k, 3.2m, 3.2o are graphics and 3.2b, 3.2 d, 3.2f, 3.2h, 3.2j, 3.2l, 3.2n, 3.2p are attractors. The point coordinates of fig. 3.2a and 3.2b is:

when the operator is chosen to be and, minus, multiply and division, we will get the graphics and attractors as figure 3.2c and 3.2d 、 3.2e and 3.2f 、 3.2g and 3.2h 、 3.2i and 3.2j, respectively. The point coordinates of fig. 3.2k and 3.2l is:

The point coordinates of fig. 3.2m and 3.2n is:

The point coordinates of fig. 3.2o and 3.2p is:

(a) (b)

(c)

We can see from fig. 3.2, if the operator is chosen to be relevant, then the attractor and the graphics are very similar, such as the point coordinates of figure 3.2c and 3.2d is:

The point coordinates of figure 3.2k and 3.2l is:

Figure 3.2c and 3.2k is similar ， so is figure 3.2d and 3.2l.

In order to clarify the impact of the initial point, and to discuss the relation between the graphics and the attractor. We studied when the operator is chosen to be division, what changes will be find on 2-D LMGS graphics with the increase number of iteration.

Figure 3.3 shows the graphics generated by n iterations of Logistic map, when increased from 0 to 1.0(n has been written on the bottom left corner of each graph). In this process, the total iteration times N = 1000. According to the algorithm, we can see that when increases, the end point of the last iteration can be seen as the initial point of the next iteration. We can see from Figure 3.3, with the increase of iteration times, graphics growing gradually, finally, the edge of the graphics is similar to its corresponding attractor [Figure 3.2j].

In phase space, chaotic systems is described by the irregular trajectory of strange attractors, and to score the behavior of strange attractors quantitatively, Lyapunov exponent and the fractal dimension is proposed.

Lyapunov exponent describes the sensitivity to the initial value of chaotic system.

Fractal dimension describes freedom information of the chaotic systems.

They describe chaotic systems on different aspects.

Chaotic systems have at least one positive Lyapunov exponent and a non-integer fractal dimension .

According to the observed one-dimensional phase space reconstruction theory proposed by Packard in 1980, seen the abscissa attractor time series as data to observe system behavior, which is obtained by iterating 150,000 times using Logistic mapping by 2-D LMGS definition. Lyapunov exponent and the fractal dimension of the 2-D LMGS attractor is calculated.

According to the method to calculate the largest Lyapunov exponent 1 from time series, which is proposed by Wolf, select N=50000, m is beginning from 2,the time delay and Step is determined by testing different values repeatedly. Scalmx is set to be 0.02 and Scalmn is 1E6. Table 3.1 shows 1 of the 2DLMGS attractor presented in fig. 3.2.

By the method of calculating correlation dimension D2 proposed by Grassberger and Procaccia, select 10,000 data points. Figure 3.4 shows the curve of the 2-D- LMGS attractor presented in figure 3.2. The calculation results of D2 is shown in table 3.2.

3.2.3 Conclusion

In this section, the of definition of LMGS proposed by Sun Hai Jian et al. is expanded.

Analyzed the relation between Graphics and attractors.

Studied nonlinear mapping such as Logistic mapping.

3.3 　 computer simulation of three-dimensional Strange attractors renderings

3.3.1 Method

3.3.2 Results

3.3.3 Conclusion

3.3.1 Method Choose 3-demension nonlinear mapping as:

The initial point is (0,0,0) ， After five million times iteration, so the point falls into the viewing area. To translation and rotation the trajectories of the dynamic systems expressed in eq. (3.4) , we can use the coordinate transformation as follows :

In which, yt and zt is the three-dimensional coordinates which are not converted to explicit coordinates yet, parameters mx, my and mz represent the positions of the observer, q and j is the rotation angle and tilt angle, respectively. Trajectories of the system is mapped from the three-dimensional space to two-dimensional screen (the direction of x-axis).

In order to display the internal structure of three-dimensional strange attractor, we can use the method of drawing perspective .

The trajectory of 3-dimensional strange attractor has no intersection point in phase space, but its projection on two-dimensional plane can intersect. Therefore, when we drawing perspective, the color of pixel points is: add the times of trajectory in the phase space projected onto the pixel point in eq. (3.4):

We divided basic color into seven groups: blue, green, black, red, magenta, brown and gray, given the incremental steps of 32 to each color so that the color degrees varied between 0 ~ 63. However, the above method will result in a high-value-based color picture, we can hardly see any fine structure characteristics.

Hence ， suppose

Where t is the critical value. At the same time, using histogram equalization, and to the pixels spread wrongly, using dispersion error to spread them to neighboring pixels to be adjusted. So that we can obtain the renderings with fine structure.

3.3.2 Results The dynamics behavior of nonlinear mapping

is determined by the control parameters a, b, c, d and e. In order to inspect the behavior of the system in control parameter space more roundly, we studied the evolution of system behavior with the changes of the control parameters.

Figure 3.5 shows the strange attractor of non-linear mapping, In which the initial point is selected as(x0, y0, z0) = (0, 0, 0), when the initial point changes, there is little significant changes in Fig.3.5, indicating that strange attractors have the stability of on the whole.

At the same time, we also observed that the strange attractor is not a continuous entity, but there is large number of inanition structures, apart from some large inanition , there are different levels of small inanition, which makes the strange attractors with infinite levels of self-similar geometric structure.

To observe Figure 3.6a, 3.6b and 3.6c, when the control parameters changes continuously, the overall structure of graphics changed suddenly. Figure 3.6a and 3.6c show a number of discrete points, indicating that they are non-chaotic attractor, so the system output is predictable; and Figure 3.6b shows the strange attractor, and the system is chaotic.

3.3.3 Conclusion From the above discussion, we found that the

strange attractor has the following main features:

A nonlinear system, with the process of time, its orbit in the phase space will be contracted to the attractor stably.

When the system is chaotic, there is two types of motion: from the whole, the system is moving around some big inanition, this movement has an average cycle . But in fact the system is moving around numerous large or small inanition at the same time, this movement is naturally random (but not completely random.

Strange attractors often hides in the back of chaotic phenomena, which is a class of infinite levels of nested self-similar geometry.

3.4 　 Chaos and Fractals in 2-Dimensional Logistic Map 3.4.1 Study of Chaos

3.4.2 Study of Fractal

3.4.3 Conclusion

3.4.1 Study of Chaos

3.4.1.1 Simple coupling

3.4.1.2 Quadratic coupling

3.4.1.1 Simple coupling The two-dimensional Logistic map with simple coupl

ing

whose dynamic behavior is decided by the control parameters 1 、 2 and . In order to study the behavior of the system in the control parameter space more complete, we use the phase diagram, reconstruction phase diagram, bifurcation diagram and Lyapunov exponent map, respectively, and studied the evolution of system behavior when the control parameters changes along some trajectory in parameter space.

The first trajectory is obtained when 120.60,0.90 、 =0.1 ， and the initial point is selected as x0,y00.10,0.11. When =0.60, the system tends to a stable fixed point in phase plane; When is up to 0.69 ， there are two stable fixed points in phase plane; When is up to 0.787, the two fixed point become unstable, the new equilibrium state become two limit cycles around the two fixed points before fig. 3.7a ， Fig. 3.7a shows that the two stable limit cycle is a closed cycle, adjacent trajectories is convergent to it; When =0.80, the two limit cycles increasing and deformed; fig. 3.7b ； When increasing from 0.81 to 0.815, further bifurcation occurs and strange attractors can be found in phase plane fig. 3.7c; When continue increasing to 0.84, there is 4 periodic solution fig. 3.7d; Finally when =0.89 the system is chaotic. Fig.3.8 are two-dimensional Logistic map strange attractors, which are obtained by the phase-space reconstruction method .

3.4.1.2 Quadratic coupling Two-dimensional Logistic map with symme

tric quadratic coupling

Whose dynamic behavior is decided by the control

parameters 1 、 2 and . When 120.7, 0.10,0.64, we studied the evolution of system behavior with the use of phase diagram, bifurcation diagram and Lyapunov exponent maps,respectively. show as fig. 3.11 and fig. 3.12 。

When =0.10 ， the system tends to a stable fixed point in phase plane fig. 3.12b;When =0.184 ， there are two stable fixed points in phase plane fig. 3.12b; When up to 0.425, the two fixed point become unstable, the new equilibrium state become two limit cycles around the two fixed points before fig. 3.11a, this process is called Hopf bifurcation, fig. 3.11a shows that the two stable limit cycle is a closed cycle, adjacent trajectories is convergent to it ; When =0.445, the limit cycle appears again fig. 3.11b; When up to 0.49, we can see from fig. 3.12a that the system becomes chaotic; But when continue increasing to 0.525, there are two limit cycles again in phase plane, and the size is bigger and they have two “corner” fig. 3.11c ， which shows that the system has returned to periodic motion.

We can see from fig. 3.12a and fig. 3.12b, there is cycle window, which confirmed the above conclusions. when continue to increase, the density of points on the trajectory increased too, and the trajectory has complex distortion, When =0.550, see from fig. 3.12a, appeared in the phase plane is strange attractor fig. 3.11d,the system’s behavior is chaotic; With the increasing of , the two separate parts of the strange attractor become larger and shaped and near to each other fig. 3.11e, the separate parts connected each other finally fig. 3.11f ～ 3.11h. We can see from fig. 3.11d ～ fig. 3.11h,that the strange attractor has the complexity of movement behavior described before.

Figure 3.12a shows how the Lyapunov exponent changing, the two-dimensional Logistic map with quadratic coupling, when the parameter changed. We can see that, when < 0.550is periodic motion; and when > 0.550, mostly, it is corresponds to chaotic motion. But in the chaotic zone ( > 0.815), there are a few narrow location, which corresponds to the window of different cycles the chaotic zone. The points are the bifurcation point.

From the analysis above, we can see that the evolution process is based on the cycle behavior and chaotic phenomena, in the intermittent of alternating it broke out and routed to chaos, that is, through the Pomeau-Manneville to chaos, and the intermittent is relate to the Hopf bifurcation .

3.4.2 Study of Fractal

3.4.2.1 Simple coupling

3.4.2.2 Quadratic coupling

3.4.2.1 Simple coupling In this paper, we used the escape time algorithm, c

hoose escape radius R=1000 、 Escape time limit N = 256, and in accordance with the different escape time given different colors to each point, and constructed the colored fractal images of the two-dimensional Logistic mapping with simple coupling (as shown in Figure 3.13). Black represents the limited domain of attraction. It is equivalent to generalized Julia set of the two-dimensional Logistic mapping with simple coupling when complex constant c = 0.

As shown in Figure 3.31,

Outside the black limited attracted domain of the fractal images,there is the domain full of twists and turns surrounding with the various types of ribbon color ,we call them the equipotential zone .

when 1=2 [as shown in Figure 3.13a].,the limited attracted domain of the fractal images is connected and axisymmetric on a straight line. Due to the limited of the computer precision, observed at this time the border of the limited domain of attraction do not seem to have a fractal structure, but increase the magnification, we will see the structure of fractal[as shown in Figure 3.13b].As constructed the Julia set by the complex mapping, when c=0 ,Julia set is the unit circle with no border of fractal. When c0 Julia set has a delicate border on the fractal structure, though in the case of low magnification sophisticated ,the fractal structure can not be seen. When 12 [as shown in Figure 3.31c], the limited attracted domain of the fractal images are no longer connected and Symmetrical.

in accordance with the local amplification of the fractal figures 3.13b and 3.13d , we can see that the junction and the equipotential line of the limited domain of attraction is smooth and has a fractal nature.

3.4.2.2 Quadratic coupling The method of the construction using

the fractal map of the two-dimensional Logistic mapping with a symmetric coupling of the second (as shown in Figure 3.14) is the same to the method of the construction using the fractal map of the two-dimensional Logistic mapping with a symmetric coupling of the first.

From Figure 3.14 we can be observed the same conclusion with the fractal map of the two-dimensional Logistic mapping with a symmetric coupling of the second: for example,

when 1=2 [as the Figure 3.14a] ,the fractal images is axisymmetric on a straight line.

when 12 [as the Figure 3.14c] ,the fractal images is not axisymmetric.

In accordance with local enlargement of the figure 3.14b , we can see the limited domain of attraction and the junction with equipotential line is smooth and has a fractal nature.

The proof of the symmetry refer to the theorem3.1.

Fig.3.14 the fractal map of the two-dimensional Logistic mapping with a symmetric coupling of the second

3.4.3 Summary (1) The chaotic motion of dissipative systems,

there appears to be the opposite of two processes.

On the one hand, the dissipative action lead to the orbital contraction. On the other hand, have separate tracks (divergence). As a result of contraction is determined by the equation of its own (the existence of dissipation term), it is the whole phase space, and it tends to track the distance (contraction) to limited (attractors);

The phase-space divergence of the specific nature of near-term, it is the local nature of divergence, it is close to the track to separate mutually exclusive. In this way, all the track on the final (excluding the beginning of the transient process time) on the phase space within the limited side and separated from the side to separate and fold, fold back and forth numerous times before the formation of complex chaos, as shown in Figure 3.7 and Figure 3.11 ,the two-dimensional Logistic map attractor is this case. For this reason, the attractor of chaotic motion must have a complex infinite levels of a self-similar structure.

Because of the features above, it is said that such a chaotic state with infinite levels of self-similar structure of attractors call the strange attractor, to distinguish between fixed points and limit cycles. To distinguish with the past completely random or disorder chaotic concept (such as the molecular kinetic theory of chaos in the concept or the word dictionary on the interpretation of chaos) , people call the chaotic which is talking about is a deterministic chaos.

(2)Chaos is received by the determinism of a stochastic equation with no cycle time course. Fractal is a structure or graphic in the space, the course of their growth is also to comply with a number of decisions on the equation, but also to comply with the randomness of its own law, above is the difference between the chaos and the fractal.

Chaos and Fractals, however there are also some common .

They are the results of non-equilibrium process described by nonlinear equations. Chaotic motion is closely related to the stochastic fluctuation of the initial state and fractal’s structure of the specific shape or its close ties with the random nature is closely related to the fluctuation of the initial state too ; Chaotic motion’s strange attractors and fractal structure have self-similarity. In fact, the chaotic motion is the random self-similarity with different time scales.

(3) Study of the unsteady state of motion in the parameter space , the system often have to change in the parameters of the process to going through a series of first-cycle system, and then into the chaos. This constitutes is the so-called "road leading to chaos," mastering these cycle-oriented system of law can help to understand the ultimate nature of the chaotic state. Two-dimensional Logistic map, when the control parameters change along a different path in the parameter space, can be through with Pomeau-Manneville way, Ruelle –Takens-Newhouse and Feigenbaum way towards chaos. Control parameters in a larger space in the region, the road leading to chaos have relationship with the Hopf bifurcation, and in these channels can observe the phase-locked and quasi-periodic motion.

(4) In conclusion, though the studying of chaos, people has greatly expanded the outlook and actived the thinking. Over the past, some mechanical equations which are considered to be determined and reversible have the randomness and non-reversible inherent.

Determined equation can be drawn on the results of uncertainty, which broke the gap between systems of the two sets of random and determine .

This gives a great impact on traditional science, traditional science has been transformed in a sense , this will promote the further development of other disciplines.

3.5 General strange attractors and fractal in the two-dimensional quadratic mapping.

3.5.1 General two-dimensional quadratic mapping of Chaos and Fractals

3.5.2 General two-dimensional quadratic mapping of fractal

3.5.3 Conclusion

3.5.1 General two-dimensional quadratic mapping of Chaos and Fractals

3.5.1.1 Theory and Method

3.5.1.2 Results and Analysis

3.5.1.1 Theory and Method Analysis of complex nonlinear systems can be

observed in orbit in phase space. Nonlinear systems with time will tend to the evolution of dimension of phase space than the original limit of low collection - attractor.

Attractors usually have a simple fixed point, limit cycle and torus. The simple attractor have the impact of controlled parameters, with the changes in control parameters, the simple attractor can be gradually developed into a strange attractor, at this time the system is chaotic.

Can be expressed as. If it is a fixed point, then we give the following fixed point equation

Can be obtained from equation (3.12)

3.5.1.2Results and Analysis General two-dimensional quadratic mapping of th

e dynamic behavior is determined by the control parameters.

In order to observe the behavior of the system for a more complete study in parameter space, the author first studied the selected control parameter space along the rail line for a few changes in the evolution of system behavior.

Figure 3.15 for the initial check point (x0,y0)0.05,0.05 , the control parameters remain unchanged by changes in the general two-dimensional quadratic map and the Lyapunov index bifurcation diagram Fig. It has more representative.

Eckmann had a variety of non-linear system bifurcation process research, identifies three ways to chaos: Feigenbaum way ； Ruelle-Takens-Newhouse way ； Pomeau-Manneville way.

The author found that the control parameters vary, but generally can be two-dimensional quadratic map by Feigenbaum way, Ruelle-Takens-Newhouse and Pomeau-Manneville way to chaos, and in the control parameter space of a larger region, the road leading to chaos with the Hopf bifurcation, and in these channels can be observed the phase-locked and quasi-periodic motion. Figure 3.15 is a typical representative of the way.

3.5.2 General two-dimensional quadratic fractal mapping Fractal geometry is mentioned by the famous

mathematician Mandelbrot in the 70's first, and it gives a description of the proposed morphological characteristics with random and infinite details of a natural phenomenon of the new model which is more accurately in the geometric.

The author uses the escape time algorithm [21], select the escape radius R = 1000, escape time limit N = 256, and according to the escape of the various points at different times, draw a general two-dimensional quadratic fractal color mapping.

As figure 3.17 shown , black for the limited domain of attraction. It is equivalent to constant for the 0:00 complex the general two-dimensional mapping of the generalized quadratic Julia sets.

3.5.3 conclusion 1. This section analyzes the general two-dimension

al quadratic mapping of the stability of the fixed points. The general two-dimensional quadratic map bifurcation occurred in the first equation of the border are given in the parameter space.

2. When the control parameters in the parameter space along a different path changes, using the calculation methods, this section reveals that in general two-dimensional quadratic map can through Pomeau-Manneville way, Ruelle Takens Newhouse and Feigenbaum way toward chaos, and in the control parameters in the larger region of space, the road leading to chaos have the relationship with the Hopf bifurcation.

Criteria for the use of Lyapunov index, in this section , structure the strange attractor map of the general two-dimensional quadratic.

We found that the second general two-dimensional mapping of strange attractors in the phase space of a limited region.Get instability stucture from the infinitely points in an assembly, and its geometry has a certain degree of self-similarity, which because it is a general two-dimensional quadratic map as 3.11 caused by the repeated iteration of the "genetic" effect. the structural features of the strange attractor can interpreted as follows:

Strange attractors contained in the complex geometry is difficult to study using conventional Euclidean geometry .such things as the complexity of research methods - Fractal ,on naturally be applied to strange attractors to the study.

The fractal dimension of strange attractors set of geometric features and the strange attractor in the track of the evolution over time for quantitative description of this,

we according to the speculation of Kaplan and Yorke. Use Lyapunov index to get the fractal dimension obtained by the strange attractor. At the same time, though the studies of the second general two-dimensional fractal images mapping have shown that different control parameters and fractal images different from each other, and their border is fractal. For general two-dimensional quadratic map of the roads leading to chaos and further theoretical and experimental work to be researched.

Summarize the discussion above, we can get a number of conclusions on the discrete two-dimensional mapping

(1) Two-dimensional discrete fixed point mapping and classification of the stability of differential equations with two variables singular point (steady state) is similar.

(2) Discrete two-dimensional nonlinear mapping may also appear as a variety of continuous flow in the bifurcation.

(3) One-dimensional nonlinear mapping is not reversible, it is always on behalf of non-conservative process. However, both two-dimensional nonlinear mapping can be dissipative, or they may be conservative, by the Jacobian determinant of the decision value.

(4) Dissipative two-dimensional mapping system through a large number of iterations may be trying to cut back on some one-dimensional curve to a certain extent chaotic orbits (attractors), such a chaotic orbit has infinite levels of self-similar .