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Dynamics of the generalized M set on escape-line diagram
Applied Mathematics and Computation 206 (2008) 474–484
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Dynamics of the generalized M set on escape-line diagram Xingyuan Wang *, Xu Zhang, Yuanyuan Sun, Fanping Li School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116023, China
a r t i c l e
i n f o
Keywords: Bifurcation diagram Escape-line algorithm Generalized M set Period Fractal
a b s t r a c t The basic contour diagrams of the M set are briefly reviewed. With the method combining escape-line diagram of the generalized M sets and bifurcation diagram of a family of onedimensional maps, the dynamics of a family of one-dimensional maps are studied and the graphic method to determine the period of midgets corresponding to the map is given. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Since Poincaré carried out the first mathematical study in chaos in 1890, nonlinear dynamical systems have been used in more and more areas [1–8]. Among the nonlinear dynamical systems, the family of one-dimensional multiple map x pðx; uÞ depending on one parameter has been deeply studied [9]. Bifurcation diagrams are usually chosen as a normal tool to study the dynamics of a family of one-dimensional maps. Fig. 1a gives the bifurcation diagram of one-dimensional quadratic map. We can also study the dynamics of a one-dimensional quadratic map by means of the Mandelbrot set like of its complex form [10]. In this case, the neighbor of the real axis exhibits graceful graphics [11]. In recent years, researchers have studied deeply on the one-dimensional quadratic maps of the M set [10,12]. In this paper, the real part of the complex map z za þ cða > 1Þ, which is used to construct the generalized M set [13,14], is studied, and the dynamics of a family of one-dimensional maps are analysed.
2. Theory and method There are many different methods to draw the M sets [15]. In this paper the escape-line method is chosen to draw the M set and the generalized M set since the method can offer us the more clear pictures of midgets and hyperbolic components. Actually, both the escape-line method [16–19] and the equipotential line method [9] are contour line methods. They are defined by different escape radius of escape time algorithm [20]. For the complex map z za þ cða > 1Þ, we get equipotential line pictures when r > 21=ða1Þ (r is the escape radius), while we get escape-line pictures when r ¼ 21=ða1Þ . Romera studied the characteristics of the M set escape-line [11]. In this paper the escape-line of the generalized M set is studied. In the escape-line pictures, every line is called an escape-line and is associated with a number, just as the equipotential line associates with potential in electric field. The numbers of the escape-lines indicate the iteration times by which a point needs to escape from the center of a circle with radius r = 21/(a1). Fig. 1b and c gives the equitentional line picture and the escape-line picture of the M set respectively. The escape-line equation is as follows:
n f ð0Þ ¼ r: c
* Corresponding author. E-mail address: [email protected] (X. Wang). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.09.035
ð1Þ
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Fig. 1. Bifurcation diagram of one-dimensional quadratic map and contour lines pictures of the corresponding M set.
From r = 21(a1), when a = 2, r = 2.0; when a = 4, r 1.2599. From Eq. (1), for n = 0,1,2,. . ., we have different escape-lines which are labeled by the number according to it. The escape-lines in Fig. 1c are marked with its corresponding iteration times n. The set of c-values for which fcn ð0Þ converges to an n-cycle is an open set, and the connected components are called hyperbolic component [16,19]. A midget [11] is a tiny copy of the M set; and, in the case of the one-dimensional quadratic map, is the intersection of the midget of the M set and the x-axis. The definition holds for the generalized M set, too. The main aim of the paper is to make certain the period of the hyperbolic components and midgets. When we say ‘‘the period of a midget”, we mean ‘‘the period of the main cardioid of the midget” [11]. 3. Experiments and results 3.1. When a is positive and even Fig. 2 is the bifurcation diagram of one-dimensional multiple map when a = 2,4,6. From Fig. 2 we can see that the mapping system first appears one-period, then bifurcates to two-period, four-period. . . and goes to the chaotic status finally with the decrease of the parameter. The property also holds when. a are other positive even numbers. That is to say, chaotic patterns of one-dimensional map arise from period-doubling bifurcation when .a is a positive real number. In this paper, our main work is to discuss how to determine the period of the point in chaotic region and we could get more information from the escape-line picture of the M set. Fig. 3 shows the escape-line pictures of the generalized M set corresponding to the three cases of Fig. 2. We shall analyse the neighbor of the real axis of the generalized M set in order to determine the period of the midgets of the main antenna. Fig. 4 gives the amplifying parts of the rectangle area in chaotic region in Fig. 3 and the iteration numbers are set to N = 4000. The ranges of the real axis in Fig. 4a, b and c are 2.0071 < c < 1.3986, 1.2612 < c < 1.1625 and 1.1505 < c < 1.1086, respectively. Each escape-line in Fig. 4 is labeled by the corresponding number. We can conclude from Fig. 4 that a family of one-dimensional maps whose index a is a positive even number have the same characteristics in chaotic regions. With the increase of the value of the even numbera, its period becomes chaotic
Fig. 2. The bifurcation diagrams of one-dimensional positive even maps.
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Fig. 3. The escape-line pictures of the generalized M set when a is positive and even.
Fig. 4. Partial amplification picture of the rectangles in Fig. 3: (a) amplification picture of the rectangle region in Fig. 3a; (b) amplification picture of the rectangle region in Fig. 3b; (c) amplification picture of the rectangle region in Fig. 3c.
and the projections around the x-axis neighbors along the negative direction become smaller. In order to determine the periods of the midgets in chaotic region, we give the descriptions of filaments [11]. We could see that there are some symmetrical filaments besides the midgets. These filaments attract most of the escape-lines from infinitude. When a = 2 we call the 21, 22, 23,. . ., 2n bigger symmetric filaments of a midget the 1-filaments, 2-filaments, the 3-filaments, the n-filaments; when a = 4 we call the 41, 42, 43,. . .,4n bigger symmetric filaments of a midget 1-filaments, 2-filaments, 3-filaments and n-filaments. We can deduce by analogy that when a = m (m is an even number) n-filaments are the mn bigger symmetric filaments of this midget. As is shown in Fig. 4a, each 1-filaments of the 3-period midgets (the center position isc3 = 1.754. . .) attracts all the escape-lines with number of 3,4,. . .,1; each 1-filaments of the 5-period midgets (the center position is c5 = 1.625. . .) attracts all the escape-lines with number of 5,6,. . .,1. We can conclude that all the midgets in the chaotic region have the same regularities. In Fig. 4b and c the 1-filament of each midget attract escape-lines with numbers 3,4,. . .,1 and 5,6,. . .,1, respectively. We could get a rule to determine graphically the period of a midget: for one-dimensional positive even maps, the 1-filaments of a midget whose period is p attract all the escape-lines with numbers p; p þ 1; . . . ; 1. That is to say the period of a midget is equivalent to the least escape-line number attracted by its 1-filaments. The rule holds when used to determine the period of a hyperbolic component. In order to explain which escape-lines are finally attracted by the 1-filaments, we choose distances rate between two escape-lines in the direction of the 1-filaments [11]. In Fig. 4a, we extend the method in [11] to other conditions when a are other positive even numbers. But in Fig. 4b and c the distances rate between two escape-lines in the direction of the 1-filaments varies apparently. Fig. 5 is the magnification of the generalized midgets with period 5 in Fig. 4b and c. Let dn be the distance between escape-lines numbered n and n + 1 in the direction of 1-filament, and denote the distance rate to be qn = dn/dn+1. In Fig. 5a and b we can easily see that the distance between the escape-lines 4 and 5 becomes larger than the distance between the escape-lines 5 and 6. Apparently the distance rate value q4 is an abrupt point. That is to say the escape-line labeled by 5 is the last escape-line attracted by the 1-filaments of the midget. In Fig. 6, the escape-line picture of generalized midget in Fig. 3 is locally magnified. The ranges of the x-axis neighbors in Fig. 6a, b and c are 1.8845499 < c < 1.8704139, 1.19527417 < c < 1.19358735 and 1.148097 < c < 1.148084, respectively. Each escape-line in Fig. 6 is labeled by the corresponding number. For a family of one-dimensional maps when a is even, the last escape-line attracted by 1-filaments can be calculated by the above mathematical method, and thus determine its period. But in the process of graphic analysis, we hope to decide the period of a midget macroscopically. Now we give a simple method: suppose a midget whose period is known and determine
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Fig. 5. Partial amplification picture of Fig. 4b and c.
Fig. 6. Partial amplification escape-line picture of the generalized M set when a is positive and even.
the periods of the other midgets in the neighbors of the midget. In Fig. 6, the midgets with period 9, 11 and 11 are given. According to the above rule, the bigger midget in each figure attracts the last escape-lines 9, 11 and 11. We can label the last escape-line number attracted by the 1-filaments and count the other escape-lines numbers. With this method we can directly work out the periods of midgets in the neighbors of those midgets whose periods are known. The above method to determine the period of a midget is under the condition that some other midgets’ periods are known. Next we discuss how to determine the period of a midget without other midgets’ information. Fig. 7 gives a midget whose period is 10 and center point is c10 = 1.53624327. . . when a = 2. In Fig. 7a, there are 21 1filaments which attract escape-lines with number of 10,11,. . .,1; amplifying Fig. 7a we get Fig. 7b, and we can see 22 2-filaments which attract escape-lines with number of 20,21,. . .,1; amplifying Fig. 7b we get Fig. 7c, and we can see 23 3-filaments which attract escape-lines 30,31,. . .,1. When we amplify continually we find the escape-line numbers attracted by midget with period p are np with the increase of n-filament in the form of 2n. It can be deduced by analogy when a is other
Fig. 7. Partial amplification picture of the M set: (a) amplification picture of a period-10 midget; (b) partial amplification picture of (a); (c) partial amplification picture of (b).
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even number. Such as a = 4, as shown in Fig. 8a, a given generalized midget whose center point is c5 = 1.19981625. . . and period is 5 has 41 1-filaments which attract escape-lines with numbers 5,6,. . .,1; we amplify Fig. 8a and get Fig. 8b, we can see 42 2-filaments which attract escape-lines with numbers from 10,11,. . .,1; amplifying the rectangle area in Fig. 8b and get Fig. 8c (1.197689 < c < 1.19415), we can see the rule holds. From the above discussion, we can get the following conclusion for a family of complex maps z za + c when a is even: the n-filaments of each midget whose period is p attract the escape-lines with numbers np,np + 1,. . .,1. Finally give the rule to determine the period of a midget: one midget’s period is the last escape-line number attracted by its 1-filaments or the count of different escape-lines between the n-filaments and the (n+1)-filaments. Using the rule we can determine the period of an isolated and period-unknown midget. For example, we amplify the chosen real axis neighbors of generalized midgets in Fig. 6a and b and get Fig. 9a and b. The range of real axis neighbors in Fig. 9a is 1.8784018 < c < 1.8783675, and the range of real axis neighbors in Fig. 9b is 1.1949677 < c < 1.19491549. We can find the numbers of different escape-lines between the escape-line with number ‘a’ and the escape-line with number ‘b’ in Fig. 9a and b are 9 and 13, respectively. That is to say the periods of the generalized midget are 9 and 13, respectively. We can get more information in Fig. 9a: the escape-line with number ‘a’ is attracted by 3-filaments, so the value of ‘a’ should be 3 9 = 27 and the value of ‘b’ is 4 9 = 36. By analogy, in Fig. 9b, the value of ‘a’ is 2 13 = 26 and the number of escapeline ‘b’ is 3 13 = 39. In this way, we have a rule to determine the period of an isolated midget simply and effectively. 3.2. When a is positive and odd We could see from the bifurcation diagrams of a family of one-dimensional maps when a = 3,5,7 in Fig. 10. It shows that the bifurcation diagram present linear relation when parameter. l is bigger than a certain value. There is similar property when a chooses the other odd numbers. That is to say, the period of one-dimensional maps is 1 period invariably when a is a positive odd number. And the radius of the one single period increases with the increase of the value of a. Fig. 11 shows the escape-line diagram of the generalized Mandelbrot set when a is positive and odd, corresponding to Fig. 10. We can see clearly that all the escape-lines for a family of one-dimensional maps when a is odd converge to a1 vertexes. Here we mainly work on the real axis neighbors of a family of one-dimensional maps complex form. We find that these pictures are symmetric cardioids about y-axis and there are no midgets in the projection area in x-axis neighbors (Fig. 12). It means
Fig. 8. Partial amplification picture of the generalized M set when a = 4 (a) amplification picture of the generalized midget with period 5; (b) amplification picture of the rectangle region in Fig. 8a; (c) amplification picture of the rectangle region in Fig. 8b.
Fig. 9. Partial amplification picture of Fig. 6a and b.
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Fig. 10. Bifurcation diagram of a family of one-dimensional maps when a is positive and odd (No bifurcation is present because in the three cases only the period 1 is present in the real map, as will be seen in Fig. 11).
Fig. 11. Escape-line picture of the generalized M set when a is positive and odd.
Fig. 12. Partial amplification picture of rectangles in Fig. 11 (a) amplification picture of the rectangle region in Fig. 11a; (b) amplification picture of the rectangle region in Fig. 11b; (c) amplification picture of the rectangle region in Fig. 11c.
that the period of a family of one-dimensional maps keep 1 single period invariable when a is and odd number. Fig. 12 gives the amplification picture of the rectangle areas in Fig. 11. We can see the variations of the escape-lines when period is fixed, which is good for comparing with the study before. 3.3. When a is a positive decimal number De Moivre’s theorem is used to calculate the value
za ¼ jza j½cosðahÞ þ i sinðahÞ:
ð2Þ
This involves the choice of the principal value of the phase angle h. We have chosen four types as follows: [0,2p), [p,p), [3p/2,p/2) and [p/2,3p/2). When a is a positive integer, the use of Eq. (2) will not affect because
cosðahÞ ¼ cosðah þ 2pÞ; sinðahÞ ¼ sinðah þ 2pÞ:
ð3Þ
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But when a is a decimal number, Eq. (3) is not valid. So the different choice of the principal value for h will give rise to different evolutions of the generalized Mandelbrot sets. Two cases are studied under these four conditions: one is that a varies from even to odd, the other is that a varies from odd to even. In the following part, we illustrated the condition when the value of a approaches 2. When 1 < a < 2, with the increase of the value of a, the periodic behaviors of a family of one-dimensional maps vary from the single period of the odd maps to the period-doubling bifurcation of the even maps. The closer the value of a approaches to 2, the clearer the phenomenon becomes. In Fig. 13, the number of points corresponding to chaotic region of the perioddoubling bifurcation diagram when a = 2 increases with the increase of the value of a. That is because the number of formed midgets corresponding to the generalized M set in the x-axis increases. We know that the period of the points in chaotic region is also the period of the midgets. Only found the midget can we determine the period by the method mentioned above. Let us illustrate the formation feature of one-dimensional maps by the generalized M set when a = 1.999. Figs. 14–17a are the partial amplification picture in the x-axis neighbors of the generalized M set when period-doubling bifurcation of a family of one-dimensional maps appears, the angle ranges are (p,p], (p/2,3p/2], (3p/2,p/2] and (0,2p], respectively. Figs. 14–17b are magnifications of the rectangle regions in Figs. 14–17a, where the x-axis is denote by a thin wire. In Fig. 14a, we can see a midget with center pointc = 1.75. . ., which also can be seen from Figs. 15–17a. Corresponding to the center point there are some periodic points appearing in the bifurcation diagram in Fig. 13a, but both sides of periodic points are blank bands, which indicates that the fractal picture of the generalized M set has no figures of midgets, shown in Fig. 14b. Although there are many midgets as shown in Figs. 15–17b, there is only one midget with center point c = 1.75. . . lies in x-axis, and the other midget lie below or above x-axis. For these reasons, only the periodic point has the periodic behavior. According to the above method, the period of the midget is 3, that is to say the number of the last escape-line attracted by 1-filaments is 3. When 2 < a < 3, the periodic behaviors of a family of one-dimensional maps vary from the period-doubling bifurcation of the even maps to the single period of the odd maps with the increase of the value of a. Fig. 18 shows the bifurcation diagram of a family of one-dimensional maps: the farther the value of a away from 2, the smaller the range of the period-doubling bifurcation becomes; the number of points becomes less when compared with the case of a = 2. That is because the number of midgets formed in the x-axis of the generalized M set is decreasing. Next let us illustrate the formation feature when a = 2.0001.
Fig. 13. Bifurcation diagram of a family of one-dimensional maps.
Fig. 14. Partial amplification of the generalized M set when a = 1.999, h 2 (p,p].
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Fig. 15. Partial amplification of the generalized M set when a = 1.999, h 2 (p/2,3p/2].
Fig. 16. Partial amplification of the generalized M set when a = 1.999, h 2 (3p/2,p/2].
Figs. 19–22a are the partial amplifications in the x-axis neighbors of the generalized M set when period-doubling bifurcation of a family of one-dimensional maps appears, whose angle ranges are (p,p], (p/2,3p/2], (3p/2,p/2] and (0,2p], respectively. Figs. 19–22b are gotten by amplifying the rectangle regions of Figs. 19–22a, where the x-axis is denoted by a thin wire. Fig. 19b shows that there are two helixes composed by many midgets, but there is no projection on the x-axis. We can see that there are many figures composed by midgets similar to the x-axis neighbors corresponding to the general-
Fig. 17. Partial amplification of the generalized M set when a = 1.999, h 2 (0,2p].
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Fig. 18. Bifurcation diagram of a family of one-dimensional maps.
Fig. 19. Partial amplification picture of the generalized M set when a = 2.0001, h 2 (p,p].
Fig. 20. Partial amplification picture of the generalized M set when a = 2.0001, h 2 (p/2,3p/2].
ized M set when a = 2. These midgets do not have the periodic behavior because they lie above or below the x-axis. This is in coincident with the phenomenon in Fig. 15 that there are narrow blank band in the area of l 2 ½1:798; 1:790 and before it. For the other midget period determination in Fig. 15b is the same as when 1 < a < 2. According to the above research, we can get the periodic regularity of the a family of one-dimensional maps when a is a positive decimal number: When a varies from odd number to even number, the periods of the one-dimensional maps varies from the single period of odd maps to the period-doubling bifurcations of the periods of even maps; When a varies from even number to odd number, the periods of the one-dimensional maps varies from the period-doubling bifurcations of even maps to the single period of the periods of odd maps. There are midgets in x-axis neighbor on the real axis corresponding to the
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Fig. 21. Partial amplification picture of the generalized M set when a = 2.0001, h 2 (3p/2,p/2].
Fig. 22. Partial amplification picture of the generalized M set when a = 2.0001, h 2 (0,2p].
generalized M set on the positions of periodic points in bifurcation diagrams. The periods of these midgets can be determined according to the one-dimensional positive even maps. 4. Conclusion In this paper we first use the bifurcation diagrams to check the periods of a family of one-dimensional maps and find out the periodic regularities of them. Then we draw the generalized M set using the escape-line method. According to the cases of the escape-lines in the real axis, we get the midgets’ periods corresponding to the points in the real axis in the bifurcation diagram. We can determine the period of any midget using the method by amplifying the midgets so long as what we need is within computer’s ability. Acknowledgements This research is supported by the National Natural Science Foundation of China (No: 60573172), the Superior University doctor subject special scientific research foundation of China (No: 20070141014) and the National Natural Science Foundation of Liaoning province (No: 20082165). References [1] D. Ruelle, F. Takens, On the nature of turbulence, Commun. Math. Phys. 20 (1971) 167–192. [2] B.L. Hao, Bifurcation, chaos, strange attractor, turbulence and all that: on intrinsic stochasticity in deterministic systems, Progr. Phys. 3 (1983) 329–416 (in Chinese). [3] D. Avnir, The Fractal Approach to Heterogeneous Chemistry: Surface, Colloids, Polymers, Wiley, New York, 1989. [4] M. Ghil, R. Benzi, G. Parisi, Turbulence and predictability of geophysical flows and climate dynamics, in: Proceedings of the International School of Physics, NorthHolland, Amsterdam, 1985, pp. 60–78.
484 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
X. Wang et al. / Applied Mathematics and Computation 206 (2008) 474–484 L.F. Olsen, H. Degn, Chaos in biological systems, Q. Rev. Biophys. 18 (1985) 165–225. A. Babloyantz, J.M. Salazar, C. Nicolis, Evidence of chaotic dynamics of brain activity during the sleep cycle, Phys. Lett. A. 111 (1985) 152–156. B. West, Fractal Physiology and Chaos in Medicine, World Scientific Publishing Company, Singapore, 1990. A.R. Smith, Plants, fractals and formal languages, Comput. Graph. 18 (1984) 1–10. X.Y. Wang, The Fractal Mechanism of the Generalized M-J Set, Dalian University of Technology Press, Dalian, 2002. G. Pastor, M. Romera, An approach to the ordering of one-dimensional quadratic map, Chaos Solition Fract. 7 (1996) 565–584. M. Romera, G. Pastor, F. Montoya, Graphic tools to analyse one-dimensional quadratic maps, Comput. Graph. 20 (1996) 333–339. G. Pastor, M. Romera, G. Alvarez, et al, How to work with one-dimensional quadratic maps, Chaos Solition Fract. 18 (2003) 899–915. B.B. Mandelbrot, On the quadratic mapping z ! z2 l for complex l and z: the fractal structure of its M set and scaling, Phys. D 7 (1983) 224–239. G. Pastor, M. Romera, G. Alvarez, et al, Chaotic bands in the Mandelbrot set, Comput. Graph. 28 (2004) 779–784. H.O. Peitgen, D. Saupe, The Science of Fractal Images, Springer Verlag, Berlin, 1988. G. Alvarez, M. Romera, G. Pastor, et al, Determination of Mandelbrot set’s hyperbolic component centres, Chaos Solition Fract. 9 (1998) 1997–2005. K.W. Philip, Field lines in the Mandelbrot set, Comput. Graph. 16 (1992) 443–447. H.O. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals, Springer Verlag, New York, 1992. M. Frame, A.G.D. Philip, A. Robucci, A new scaling along the spike of the Mandelbrot set, Comput. Graph. 16 (1992) 223–234. B.W. Sun, Fractal Algorithm and Program Design, Science Press, Beijing, 2004.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Dynamics of the generalized M set on escape-line diagram Xingyuan Wang *, Xu Zhang, Yuanyuan Sun, Fanping Li School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116023, China
a r t i c l e
i n f o
Keywords: Bifurcation diagram Escape-line algorithm Generalized M set Period Fractal
a b s t r a c t The basic contour diagrams of the M set are briefly reviewed. With the method combining escape-line diagram of the generalized M sets and bifurcation diagram of a family of onedimensional maps, the dynamics of a family of one-dimensional maps are studied and the graphic method to determine the period of midgets corresponding to the map is given. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Since Poincaré carried out the first mathematical study in chaos in 1890, nonlinear dynamical systems have been used in more and more areas [1–8]. Among the nonlinear dynamical systems, the family of one-dimensional multiple map x pðx; uÞ depending on one parameter has been deeply studied [9]. Bifurcation diagrams are usually chosen as a normal tool to study the dynamics of a family of one-dimensional maps. Fig. 1a gives the bifurcation diagram of one-dimensional quadratic map. We can also study the dynamics of a one-dimensional quadratic map by means of the Mandelbrot set like of its complex form [10]. In this case, the neighbor of the real axis exhibits graceful graphics [11]. In recent years, researchers have studied deeply on the one-dimensional quadratic maps of the M set [10,12]. In this paper, the real part of the complex map z za þ cða > 1Þ, which is used to construct the generalized M set [13,14], is studied, and the dynamics of a family of one-dimensional maps are analysed.
2. Theory and method There are many different methods to draw the M sets [15]. In this paper the escape-line method is chosen to draw the M set and the generalized M set since the method can offer us the more clear pictures of midgets and hyperbolic components. Actually, both the escape-line method [16–19] and the equipotential line method [9] are contour line methods. They are defined by different escape radius of escape time algorithm [20]. For the complex map z za þ cða > 1Þ, we get equipotential line pictures when r > 21=ða1Þ (r is the escape radius), while we get escape-line pictures when r ¼ 21=ða1Þ . Romera studied the characteristics of the M set escape-line [11]. In this paper the escape-line of the generalized M set is studied. In the escape-line pictures, every line is called an escape-line and is associated with a number, just as the equipotential line associates with potential in electric field. The numbers of the escape-lines indicate the iteration times by which a point needs to escape from the center of a circle with radius r = 21/(a1). Fig. 1b and c gives the equitentional line picture and the escape-line picture of the M set respectively. The escape-line equation is as follows:
n f ð0Þ ¼ r: c
* Corresponding author. E-mail address: [email protected] (X. Wang). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.09.035
ð1Þ
X. Wang et al. / Applied Mathematics and Computation 206 (2008) 474–484
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Fig. 1. Bifurcation diagram of one-dimensional quadratic map and contour lines pictures of the corresponding M set.
From r = 21(a1), when a = 2, r = 2.0; when a = 4, r 1.2599. From Eq. (1), for n = 0,1,2,. . ., we have different escape-lines which are labeled by the number according to it. The escape-lines in Fig. 1c are marked with its corresponding iteration times n. The set of c-values for which fcn ð0Þ converges to an n-cycle is an open set, and the connected components are called hyperbolic component [16,19]. A midget [11] is a tiny copy of the M set; and, in the case of the one-dimensional quadratic map, is the intersection of the midget of the M set and the x-axis. The definition holds for the generalized M set, too. The main aim of the paper is to make certain the period of the hyperbolic components and midgets. When we say ‘‘the period of a midget”, we mean ‘‘the period of the main cardioid of the midget” [11]. 3. Experiments and results 3.1. When a is positive and even Fig. 2 is the bifurcation diagram of one-dimensional multiple map when a = 2,4,6. From Fig. 2 we can see that the mapping system first appears one-period, then bifurcates to two-period, four-period. . . and goes to the chaotic status finally with the decrease of the parameter. The property also holds when. a are other positive even numbers. That is to say, chaotic patterns of one-dimensional map arise from period-doubling bifurcation when .a is a positive real number. In this paper, our main work is to discuss how to determine the period of the point in chaotic region and we could get more information from the escape-line picture of the M set. Fig. 3 shows the escape-line pictures of the generalized M set corresponding to the three cases of Fig. 2. We shall analyse the neighbor of the real axis of the generalized M set in order to determine the period of the midgets of the main antenna. Fig. 4 gives the amplifying parts of the rectangle area in chaotic region in Fig. 3 and the iteration numbers are set to N = 4000. The ranges of the real axis in Fig. 4a, b and c are 2.0071 < c < 1.3986, 1.2612 < c < 1.1625 and 1.1505 < c < 1.1086, respectively. Each escape-line in Fig. 4 is labeled by the corresponding number. We can conclude from Fig. 4 that a family of one-dimensional maps whose index a is a positive even number have the same characteristics in chaotic regions. With the increase of the value of the even numbera, its period becomes chaotic
Fig. 2. The bifurcation diagrams of one-dimensional positive even maps.
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Fig. 3. The escape-line pictures of the generalized M set when a is positive and even.
Fig. 4. Partial amplification picture of the rectangles in Fig. 3: (a) amplification picture of the rectangle region in Fig. 3a; (b) amplification picture of the rectangle region in Fig. 3b; (c) amplification picture of the rectangle region in Fig. 3c.
and the projections around the x-axis neighbors along the negative direction become smaller. In order to determine the periods of the midgets in chaotic region, we give the descriptions of filaments [11]. We could see that there are some symmetrical filaments besides the midgets. These filaments attract most of the escape-lines from infinitude. When a = 2 we call the 21, 22, 23,. . ., 2n bigger symmetric filaments of a midget the 1-filaments, 2-filaments, the 3-filaments, the n-filaments; when a = 4 we call the 41, 42, 43,. . .,4n bigger symmetric filaments of a midget 1-filaments, 2-filaments, 3-filaments and n-filaments. We can deduce by analogy that when a = m (m is an even number) n-filaments are the mn bigger symmetric filaments of this midget. As is shown in Fig. 4a, each 1-filaments of the 3-period midgets (the center position isc3 = 1.754. . .) attracts all the escape-lines with number of 3,4,. . .,1; each 1-filaments of the 5-period midgets (the center position is c5 = 1.625. . .) attracts all the escape-lines with number of 5,6,. . .,1. We can conclude that all the midgets in the chaotic region have the same regularities. In Fig. 4b and c the 1-filament of each midget attract escape-lines with numbers 3,4,. . .,1 and 5,6,. . .,1, respectively. We could get a rule to determine graphically the period of a midget: for one-dimensional positive even maps, the 1-filaments of a midget whose period is p attract all the escape-lines with numbers p; p þ 1; . . . ; 1. That is to say the period of a midget is equivalent to the least escape-line number attracted by its 1-filaments. The rule holds when used to determine the period of a hyperbolic component. In order to explain which escape-lines are finally attracted by the 1-filaments, we choose distances rate between two escape-lines in the direction of the 1-filaments [11]. In Fig. 4a, we extend the method in [11] to other conditions when a are other positive even numbers. But in Fig. 4b and c the distances rate between two escape-lines in the direction of the 1-filaments varies apparently. Fig. 5 is the magnification of the generalized midgets with period 5 in Fig. 4b and c. Let dn be the distance between escape-lines numbered n and n + 1 in the direction of 1-filament, and denote the distance rate to be qn = dn/dn+1. In Fig. 5a and b we can easily see that the distance between the escape-lines 4 and 5 becomes larger than the distance between the escape-lines 5 and 6. Apparently the distance rate value q4 is an abrupt point. That is to say the escape-line labeled by 5 is the last escape-line attracted by the 1-filaments of the midget. In Fig. 6, the escape-line picture of generalized midget in Fig. 3 is locally magnified. The ranges of the x-axis neighbors in Fig. 6a, b and c are 1.8845499 < c < 1.8704139, 1.19527417 < c < 1.19358735 and 1.148097 < c < 1.148084, respectively. Each escape-line in Fig. 6 is labeled by the corresponding number. For a family of one-dimensional maps when a is even, the last escape-line attracted by 1-filaments can be calculated by the above mathematical method, and thus determine its period. But in the process of graphic analysis, we hope to decide the period of a midget macroscopically. Now we give a simple method: suppose a midget whose period is known and determine
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Fig. 5. Partial amplification picture of Fig. 4b and c.
Fig. 6. Partial amplification escape-line picture of the generalized M set when a is positive and even.
the periods of the other midgets in the neighbors of the midget. In Fig. 6, the midgets with period 9, 11 and 11 are given. According to the above rule, the bigger midget in each figure attracts the last escape-lines 9, 11 and 11. We can label the last escape-line number attracted by the 1-filaments and count the other escape-lines numbers. With this method we can directly work out the periods of midgets in the neighbors of those midgets whose periods are known. The above method to determine the period of a midget is under the condition that some other midgets’ periods are known. Next we discuss how to determine the period of a midget without other midgets’ information. Fig. 7 gives a midget whose period is 10 and center point is c10 = 1.53624327. . . when a = 2. In Fig. 7a, there are 21 1filaments which attract escape-lines with number of 10,11,. . .,1; amplifying Fig. 7a we get Fig. 7b, and we can see 22 2-filaments which attract escape-lines with number of 20,21,. . .,1; amplifying Fig. 7b we get Fig. 7c, and we can see 23 3-filaments which attract escape-lines 30,31,. . .,1. When we amplify continually we find the escape-line numbers attracted by midget with period p are np with the increase of n-filament in the form of 2n. It can be deduced by analogy when a is other
Fig. 7. Partial amplification picture of the M set: (a) amplification picture of a period-10 midget; (b) partial amplification picture of (a); (c) partial amplification picture of (b).
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even number. Such as a = 4, as shown in Fig. 8a, a given generalized midget whose center point is c5 = 1.19981625. . . and period is 5 has 41 1-filaments which attract escape-lines with numbers 5,6,. . .,1; we amplify Fig. 8a and get Fig. 8b, we can see 42 2-filaments which attract escape-lines with numbers from 10,11,. . .,1; amplifying the rectangle area in Fig. 8b and get Fig. 8c (1.197689 < c < 1.19415), we can see the rule holds. From the above discussion, we can get the following conclusion for a family of complex maps z za + c when a is even: the n-filaments of each midget whose period is p attract the escape-lines with numbers np,np + 1,. . .,1. Finally give the rule to determine the period of a midget: one midget’s period is the last escape-line number attracted by its 1-filaments or the count of different escape-lines between the n-filaments and the (n+1)-filaments. Using the rule we can determine the period of an isolated and period-unknown midget. For example, we amplify the chosen real axis neighbors of generalized midgets in Fig. 6a and b and get Fig. 9a and b. The range of real axis neighbors in Fig. 9a is 1.8784018 < c < 1.8783675, and the range of real axis neighbors in Fig. 9b is 1.1949677 < c < 1.19491549. We can find the numbers of different escape-lines between the escape-line with number ‘a’ and the escape-line with number ‘b’ in Fig. 9a and b are 9 and 13, respectively. That is to say the periods of the generalized midget are 9 and 13, respectively. We can get more information in Fig. 9a: the escape-line with number ‘a’ is attracted by 3-filaments, so the value of ‘a’ should be 3 9 = 27 and the value of ‘b’ is 4 9 = 36. By analogy, in Fig. 9b, the value of ‘a’ is 2 13 = 26 and the number of escapeline ‘b’ is 3 13 = 39. In this way, we have a rule to determine the period of an isolated midget simply and effectively. 3.2. When a is positive and odd We could see from the bifurcation diagrams of a family of one-dimensional maps when a = 3,5,7 in Fig. 10. It shows that the bifurcation diagram present linear relation when parameter. l is bigger than a certain value. There is similar property when a chooses the other odd numbers. That is to say, the period of one-dimensional maps is 1 period invariably when a is a positive odd number. And the radius of the one single period increases with the increase of the value of a. Fig. 11 shows the escape-line diagram of the generalized Mandelbrot set when a is positive and odd, corresponding to Fig. 10. We can see clearly that all the escape-lines for a family of one-dimensional maps when a is odd converge to a1 vertexes. Here we mainly work on the real axis neighbors of a family of one-dimensional maps complex form. We find that these pictures are symmetric cardioids about y-axis and there are no midgets in the projection area in x-axis neighbors (Fig. 12). It means
Fig. 8. Partial amplification picture of the generalized M set when a = 4 (a) amplification picture of the generalized midget with period 5; (b) amplification picture of the rectangle region in Fig. 8a; (c) amplification picture of the rectangle region in Fig. 8b.
Fig. 9. Partial amplification picture of Fig. 6a and b.
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Fig. 10. Bifurcation diagram of a family of one-dimensional maps when a is positive and odd (No bifurcation is present because in the three cases only the period 1 is present in the real map, as will be seen in Fig. 11).
Fig. 11. Escape-line picture of the generalized M set when a is positive and odd.
Fig. 12. Partial amplification picture of rectangles in Fig. 11 (a) amplification picture of the rectangle region in Fig. 11a; (b) amplification picture of the rectangle region in Fig. 11b; (c) amplification picture of the rectangle region in Fig. 11c.
that the period of a family of one-dimensional maps keep 1 single period invariable when a is and odd number. Fig. 12 gives the amplification picture of the rectangle areas in Fig. 11. We can see the variations of the escape-lines when period is fixed, which is good for comparing with the study before. 3.3. When a is a positive decimal number De Moivre’s theorem is used to calculate the value
za ¼ jza j½cosðahÞ þ i sinðahÞ:
ð2Þ
This involves the choice of the principal value of the phase angle h. We have chosen four types as follows: [0,2p), [p,p), [3p/2,p/2) and [p/2,3p/2). When a is a positive integer, the use of Eq. (2) will not affect because
cosðahÞ ¼ cosðah þ 2pÞ; sinðahÞ ¼ sinðah þ 2pÞ:
ð3Þ
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But when a is a decimal number, Eq. (3) is not valid. So the different choice of the principal value for h will give rise to different evolutions of the generalized Mandelbrot sets. Two cases are studied under these four conditions: one is that a varies from even to odd, the other is that a varies from odd to even. In the following part, we illustrated the condition when the value of a approaches 2. When 1 < a < 2, with the increase of the value of a, the periodic behaviors of a family of one-dimensional maps vary from the single period of the odd maps to the period-doubling bifurcation of the even maps. The closer the value of a approaches to 2, the clearer the phenomenon becomes. In Fig. 13, the number of points corresponding to chaotic region of the perioddoubling bifurcation diagram when a = 2 increases with the increase of the value of a. That is because the number of formed midgets corresponding to the generalized M set in the x-axis increases. We know that the period of the points in chaotic region is also the period of the midgets. Only found the midget can we determine the period by the method mentioned above. Let us illustrate the formation feature of one-dimensional maps by the generalized M set when a = 1.999. Figs. 14–17a are the partial amplification picture in the x-axis neighbors of the generalized M set when period-doubling bifurcation of a family of one-dimensional maps appears, the angle ranges are (p,p], (p/2,3p/2], (3p/2,p/2] and (0,2p], respectively. Figs. 14–17b are magnifications of the rectangle regions in Figs. 14–17a, where the x-axis is denote by a thin wire. In Fig. 14a, we can see a midget with center pointc = 1.75. . ., which also can be seen from Figs. 15–17a. Corresponding to the center point there are some periodic points appearing in the bifurcation diagram in Fig. 13a, but both sides of periodic points are blank bands, which indicates that the fractal picture of the generalized M set has no figures of midgets, shown in Fig. 14b. Although there are many midgets as shown in Figs. 15–17b, there is only one midget with center point c = 1.75. . . lies in x-axis, and the other midget lie below or above x-axis. For these reasons, only the periodic point has the periodic behavior. According to the above method, the period of the midget is 3, that is to say the number of the last escape-line attracted by 1-filaments is 3. When 2 < a < 3, the periodic behaviors of a family of one-dimensional maps vary from the period-doubling bifurcation of the even maps to the single period of the odd maps with the increase of the value of a. Fig. 18 shows the bifurcation diagram of a family of one-dimensional maps: the farther the value of a away from 2, the smaller the range of the period-doubling bifurcation becomes; the number of points becomes less when compared with the case of a = 2. That is because the number of midgets formed in the x-axis of the generalized M set is decreasing. Next let us illustrate the formation feature when a = 2.0001.
Fig. 13. Bifurcation diagram of a family of one-dimensional maps.
Fig. 14. Partial amplification of the generalized M set when a = 1.999, h 2 (p,p].
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Fig. 15. Partial amplification of the generalized M set when a = 1.999, h 2 (p/2,3p/2].
Fig. 16. Partial amplification of the generalized M set when a = 1.999, h 2 (3p/2,p/2].
Figs. 19–22a are the partial amplifications in the x-axis neighbors of the generalized M set when period-doubling bifurcation of a family of one-dimensional maps appears, whose angle ranges are (p,p], (p/2,3p/2], (3p/2,p/2] and (0,2p], respectively. Figs. 19–22b are gotten by amplifying the rectangle regions of Figs. 19–22a, where the x-axis is denoted by a thin wire. Fig. 19b shows that there are two helixes composed by many midgets, but there is no projection on the x-axis. We can see that there are many figures composed by midgets similar to the x-axis neighbors corresponding to the general-
Fig. 17. Partial amplification of the generalized M set when a = 1.999, h 2 (0,2p].
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Fig. 18. Bifurcation diagram of a family of one-dimensional maps.
Fig. 19. Partial amplification picture of the generalized M set when a = 2.0001, h 2 (p,p].
Fig. 20. Partial amplification picture of the generalized M set when a = 2.0001, h 2 (p/2,3p/2].
ized M set when a = 2. These midgets do not have the periodic behavior because they lie above or below the x-axis. This is in coincident with the phenomenon in Fig. 15 that there are narrow blank band in the area of l 2 ½1:798; 1:790 and before it. For the other midget period determination in Fig. 15b is the same as when 1 < a < 2. According to the above research, we can get the periodic regularity of the a family of one-dimensional maps when a is a positive decimal number: When a varies from odd number to even number, the periods of the one-dimensional maps varies from the single period of odd maps to the period-doubling bifurcations of the periods of even maps; When a varies from even number to odd number, the periods of the one-dimensional maps varies from the period-doubling bifurcations of even maps to the single period of the periods of odd maps. There are midgets in x-axis neighbor on the real axis corresponding to the
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Fig. 21. Partial amplification picture of the generalized M set when a = 2.0001, h 2 (3p/2,p/2].
Fig. 22. Partial amplification picture of the generalized M set when a = 2.0001, h 2 (0,2p].
generalized M set on the positions of periodic points in bifurcation diagrams. The periods of these midgets can be determined according to the one-dimensional positive even maps. 4. Conclusion In this paper we first use the bifurcation diagrams to check the periods of a family of one-dimensional maps and find out the periodic regularities of them. Then we draw the generalized M set using the escape-line method. According to the cases of the escape-lines in the real axis, we get the midgets’ periods corresponding to the points in the real axis in the bifurcation diagram. We can determine the period of any midget using the method by amplifying the midgets so long as what we need is within computer’s ability. Acknowledgements This research is supported by the National Natural Science Foundation of China (No: 60573172), the Superior University doctor subject special scientific research foundation of China (No: 20070141014) and the National Natural Science Foundation of Liaoning province (No: 20082165). References [1] D. Ruelle, F. Takens, On the nature of turbulence, Commun. Math. Phys. 20 (1971) 167–192. [2] B.L. Hao, Bifurcation, chaos, strange attractor, turbulence and all that: on intrinsic stochasticity in deterministic systems, Progr. Phys. 3 (1983) 329–416 (in Chinese). [3] D. Avnir, The Fractal Approach to Heterogeneous Chemistry: Surface, Colloids, Polymers, Wiley, New York, 1989. [4] M. Ghil, R. Benzi, G. Parisi, Turbulence and predictability of geophysical flows and climate dynamics, in: Proceedings of the International School of Physics, NorthHolland, Amsterdam, 1985, pp. 60–78.
484 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
X. Wang et al. / Applied Mathematics and Computation 206 (2008) 474–484 L.F. Olsen, H. Degn, Chaos in biological systems, Q. Rev. Biophys. 18 (1985) 165–225. A. Babloyantz, J.M. Salazar, C. Nicolis, Evidence of chaotic dynamics of brain activity during the sleep cycle, Phys. Lett. A. 111 (1985) 152–156. B. West, Fractal Physiology and Chaos in Medicine, World Scientific Publishing Company, Singapore, 1990. A.R. Smith, Plants, fractals and formal languages, Comput. Graph. 18 (1984) 1–10. X.Y. Wang, The Fractal Mechanism of the Generalized M-J Set, Dalian University of Technology Press, Dalian, 2002. G. Pastor, M. Romera, An approach to the ordering of one-dimensional quadratic map, Chaos Solition Fract. 7 (1996) 565–584. M. Romera, G. Pastor, F. Montoya, Graphic tools to analyse one-dimensional quadratic maps, Comput. Graph. 20 (1996) 333–339. G. Pastor, M. Romera, G. Alvarez, et al, How to work with one-dimensional quadratic maps, Chaos Solition Fract. 18 (2003) 899–915. B.B. Mandelbrot, On the quadratic mapping z ! z2 l for complex l and z: the fractal structure of its M set and scaling, Phys. D 7 (1983) 224–239. G. Pastor, M. Romera, G. Alvarez, et al, Chaotic bands in the Mandelbrot set, Comput. Graph. 28 (2004) 779–784. H.O. Peitgen, D. Saupe, The Science of Fractal Images, Springer Verlag, Berlin, 1988. G. Alvarez, M. Romera, G. Pastor, et al, Determination of Mandelbrot set’s hyperbolic component centres, Chaos Solition Fract. 9 (1998) 1997–2005. K.W. Philip, Field lines in the Mandelbrot set, Comput. Graph. 16 (1992) 443–447. H.O. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals, Springer Verlag, New York, 1992. M. Frame, A.G.D. Philip, A. Robucci, A new scaling along the spike of the Mandelbrot set, Comput. Graph. 16 (1992) 223–234. B.W. Sun, Fractal Algorithm and Program Design, Science Press, Beijing, 2004.