Logistic Map f(x) = x(1 x) is Topologically Conjugate to the Map f(x) =(2 ) x(1 x)

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1 Tamkang Journal of Science and Engineering, Vol. 10, No 1, pp (2007) 89 Logistic Map f(x) =x(1 x) is Topologically Conjugate to the Map f(x) =(2) x(1 x) Chyi-Lung Lin* and Mon-Ling Shei Department of Physics, Soochow University, Taipei, Taiwan 111, R.O.C. Abstract We prove that the well-known logistic map, f(x) =x(1 x), is topologically conjugate to the map f(x)=(2) x(1 x). The logistic map thus has the same dynamics at parameter values and 2, and hence has the 2symmetry in dynamics. To examine this symmetry, we study the (, s) n relation of f n, which is obtained by eliminating x from the equations f n (x)=xand s = df n (x)/dx. We then obtain an equation directly relating and s for period-n point of f. We derive the (, s) n relation for period n = 1, 2, 3, and 4, and we show that the (, s) n relations are invariant under the transformation of 2. Key Words: Logistic Map, Topologically Conjugate, Symmetry, Invariant, Periodic Bifurcation 1. Introduction Logistic map is defined by f(x)=x(1 x), where is the parameter. For decades, several iterated functions have been extensively studied, and rich contents have been explored. Logistic map is one of the well-known maps, and has become a standard map for studying iteration. This map contains all the interesting subjects in non-linear dynamics; we list some references in [19]. In general, the values of x and of logistic map are restricted in the ranges, 0 x 1, 0 4so that each x in the interval [0, 1] is mapped onto the same interval [0, 1]. It is known that there is a stable fixed point x*=0in the range 0 1, and another stable fixed point x * 1 1 in the range 1 3. After that, we have pe- riod-doubling bifurcation at = 3, , These numerical results are well known and are easy *Corresponding author. cllin@scu.edu.tw to reproduce on computer. However, it is a puzzle why we have two neighbor regions, 0 1 and 1 3, that each has a stable fixed point of f. According to Sharkovsky ordering [1], the appearance of the order of periods should be , but instead we now have This seems the Sharkovsky ordering is slightly violated. However, it does not. If we extend the range of to include the part of < 0, we will find that period-doubling bifurcation also occurs in the region < 0, see Figure 1, where the abscissa is the -axis. We note that the parameter values at which bifurcations occur are symmetrical to = 1. That is we have two Sharkovsky orderings, one is in the region >1 and the other is in the region < 1. Figure 1, then gives the explanation why we see the ordering in the region > 0. From the period-doubling bifurcation, we see that logistic map has symmetrical dynamics with respect to =1. We also list some of the numerical results on Table 1. The first column indicates the period-doubling bifurca-

2 90 Chyi-Lung Lin and Mon-Ling Shei Figure 1. The period doubling bifurcation diagram for the map f(x)= x(1 x), where is extended to the range <0. Table 1. The parameter values of the bifurcation from period 2 i to period 2 i+1 Bifurcation Parameter values at Bifurcation + ( >0) Parameter values at Bifurcation - ( <0) tion from period-2 i to period-2 i+1. The second column is the corresponding values for the occurrence of bifurcation in the range > 0. We denote such by +. The third column is the values for the occurrence of bifurcation in the range < 0, and we denote such by -. We note that for each row on Table 1 we have = 2. (1) Formula (1) means that if there were a period-doubling bifurcation occurring at a parameter value, then there is also a period-doubling bifurcation occurring at a parameter value 2. This shows that the map f(x)=x(1 x) has a symmetrical dynamics in parameter values and 2. Logistic map thus has the symmetry in replacing by 2. Or we say that logistic map is topologically conjugate to the map f(x)=(2) x(1 x). The logistic map is known to be conjugate to the following maps [1]: f(x)=(2)x x 2, f(x)=cx 2, f(x)= 1 cx 2. To our knowledge, it has not been discussed that the logistic map is conjugate to the map: f(x)=(2) x(1 x). We will prove this conjugacy in section 2. To examine the 2symmetry, we study the (, s) n relation, where is the parameter value that f n (x) has fixed points; that is f n (x)=x; and s is the slope of the function f n (x) calculated at the fixed point of f n (x). We study the (, s) n relation in sec. 3. We will show in sec. 4 and sec. 5. that all these (, s) n relations have the 2 symmetry. The (, s) n relation is important, as from which we can determine the range of having stable period-n points. The beginning of the appearance of period three is known to be at the parameter value 1 8. Strogatz [10] in his book said that this result is often mentioned but always without proof. Saha and he [11] then gave a derivation for that, using an elementary way, and would like to know a more elegant way of derivation. We give here another derivation of this result through the study of the (, s) n relation for n =3.

3 Logistic Map f(x) = x(1 x) is Topologically Conjugate to the Map f(x) = (2 ) x(1 x) The Map g(x) =(2)x(1 x) is Topologically Conjugate to Logistic Map We are to prove that the map g(x)=(2)x(1 x)is topologically conjugate to logistic map f(x)=x(1 x). Mathematically, if two maps, f: A A and g: B B are topologically conjugate, then there is a homeomorphism h: A B such that hof= goh, where h(x) is a continuous, one-to-one function, and has an inverse. We may rewrite such topologically conjugate map g as: g = hofoh -1. (2) We thus need to find an h relating these two maps. We consider a linear conjugacy, h(x)=ax+ b, where a and b are to be properly chosen. With such an h(x), we have g = hofoh -1 = A + Bx+ Cx 2 b, with Ab( 1 ), a b B ( 1 2 ), andc 1. We Choose ba( 1), then a a we get A =0,andB =2. Choosing a, then we 2 have C = (2 ). And therefore we have the final result: g(x)=(2) x(1 x). (3) 1 Thus through the function hx ( ) x, the 2 2 map g(x) =(2)x(1 x) is shown to be topologically conjugate to logistic map f(x)=x(1 x). And hence logistic map has the symmetry in replacing by 2. Due to this symmetry, the whole contents of logistic map may only be studied in the range 1. We will examine this symmetry below in the determination of the range of that has stable periodic points. 3. The (, s) n Relation For notations, we write f 2 (x)=f(f(x)), the second iterate of x for f. And f n (x)=f(f n 1 (x)), the nth iterate of x for f. We say an x is a period-n point of f if f n (x) x = 0, and if in addition, x f i (x), where, i =1,2,,n-1. For period-n points, in general, we need to study the two equations: f n (x)=x, (4) s = df n (x)/dx. (5) Eq. (4) contains the relation of and the period-n point x. Eq. (5) defines the slope function of f n (x), and it relates the slope s to and x. If we eliminate the period-n point x from these two equations, we then obtain an equation directly relating and s, where is the parameter value at which the map f has period-n points, and s is slope of f n (x) calculated at the period-n point. We call this equation the (, s) n relation of f n (x). From the (, s) n relation, we can determine the value of s for a given, and we can therefore check whether there exists period-n points or not, depending on the value of s is real or not. The value of s might be complex due to that the period-n points x solved from (4) might be complex. Of course, if such an x is complex, it means this x is not a period-n point. If s is real, we can check the stability of the period-n points depending on whether the value of s is in the range -1 s 1. For stable period-n points, the slope s should be in the range -1 s 1. Taking this requirement to the (, s) n relation, we can determine the range of for the existence of stable period-n points. We show this in sections 4 and 5. If the number n in (4) is not a prime, then the equation f n (x) x = 0 contains also those points with smaller period m for which n is an integer. To determine truly period-n points, we need to exclude those shorter period m points from (4). For instance, if n = 6, then Eq. (4), f 6 (x) x = 0, contains points with periods: 1, 2, 3, and 6. To have six as the prime period of f, Eq. (4) should be replaced by the equation:

4 92 Chyi-Lung Lin and Mon-Ling Shei (f 6 (x) x)/(f 1 (x)f 2 (x)f 3 (x)) = 0, (6) of f 3 : where f 1 (x)=f(x) x, f 2 (x)=(f 2 (x) x)/(f(x) x), f 3 (x)= (f 3 (x) x)/(f(x) x). The division of f 1 (x) in (6) is to exclude the fixed points of f from the equation f 6 (x) x = 0. And the division of f 2 (x) is to exclude the prime period-2 points of f. We note that f 2 (x) contains only the prime period-2 points of f, that is, it has excluded the fixed points of f. Otherwise, we will do twice the elimination of fixed points of f. And the same, the division of f 3 (x) is to exclude the prime period-3 points of f from the equation f 6 (x) x = 0. Then the x in (6) is guaranteed to be a prime period-6 point, and won t be a fixed point, or a period-2 point, or a period-3 point of f. In the following sections, we show that the (, s) n relation has the symmetry of 2 and its applications. We consider n = 3 first, since period-3 is more interesting. 4. The (, s) n Relation of f 3 The parameter value for the birth of period 3 of logistic map is known to be at =1 8. Below we show how to derive this value through studying the (, s) n relation. The equation for period-3 of f is (f 3 (x) x)/(f(x) x) = 0. Expanding this equation, we obtain x ( 3 ) x ( 4 3 ) x ( 3 5 ) x ( ) x ( 2 2 ) x (1 ) 0 (7) For the slope equation of f 3 (x), we have s df 3 ( x)/ dx x 28 x ( ) x (30 20 ) x ( ) x (6 6 6 ) x ( ) x (8) We use J. J. Sylvester method [12] to eliminate the x from Eqs. (7) and (8). We then obtain the (, s) n relation ( 16 2 s) ( 32 4 s) 2 ( 6416 ss ) 0 (9) We can use (9) to determine the range of for the existence of stable period-3 points. Solving the s from (9), we obtain s (8 2 ) (10) From (10), we see that for s to be real, it needs , that is 1+ 8,or1 8. Therefore, in the range > 0, period-3 starts at = 1 8. We can also check that the values of x of period-3 points at this parameter value are all real. For stable period-3 points, it needs -1 s 1. Setting s = 1 in (9), we get only one value of that is real and positive; the value is = Setting s = -1, we get Therefore stable period-3 points exist and only exist in the range We can directly check that Eq. (9) is invariant under the transformation of 2. Or, it may be easier to see this symmetry by changing the valuable from to y = 1. Then (9) becomes y y s y s s 11 (35 2 ) ( 18 89) 0 (11) Eq. (11) contains only even powers of y, and is thus obviously symmetrical to y and -y. Thus Eq. (9), the (, s) n relation for n = 3, reflects the 2symmetry of the logistic map. 5. The (, s) n Relation for Other Periodic Points 5.1 The (, s) n Relation of f The equation for fixed point is x(1 x) =x. The slope equation is s = (1 2x). From these two equations, and considering x 0, we easily obtain the (, s) relation

5 Logistic Map f(x) = x(1 x) is Topologically Conjugate to the Map f(x) = (2 ) x(1 x) 93 of f: or, (12) (13) From (13), Eq. (12) is obviously invariant under 2. From (12), we see that the stable fixed points, x* 0, exist in the range The (, s) n Relation of f 2 The equation for period-2 point is (f 2 (x) x)/(f(x) x) = 0. Together with the slope equation s = df 2 (x)/dx,we obtain the (, s) n relation of f 2 : or, 2 ( s 4) 0, (14) (15) From (15), Eq. (14) is thus invariant under 2. From (14), we see that stable period-2 points exist in the range The (, s) n Relation of f 4 The equation for period-4 point is (f 4 (x) x)/(f 2 (x) x) = 0. Together with the slope equation s = df 4 (x)/dx,we obtain the (, s) n relation of f 4 : or, 2 (2 ss ) y ( s1) y ( s5) ( 192 s) (384 8 s) (64 20 s) 8s ( ss ) ( s4 s ) ( s4 s ) ( s48 s s ) 0, (16) y 18 y (123 sy ) ( sy ) (15112 ss ) y ( s2 s ) y s s s ( ) 0 (17) From (17), Eq. (16) is invariant under 2. From (16), we see that stable period-4 points exist in the range , and may also be in the range Above examples all show that the (, s) n relation reflects the 2symmetry. The (, s) n relation of f n becomes complicated when n getting larger and larger, and is difficult to obtain. We hope that Eqs. (1217) may give some hints for obtaining the (, s) n relation for higher period of 2 n. This will be helpful for obtaining the Feigenbaum constant in an analytical form. Acknowledgments The first author would like to thank the National Science Council of the Republic of China for support. (NSC Grants No M ), and to Professor Pierre Collet for helpful discussions and suggesting the similarity transformation, and to Laboratoire de Physique Théorique et Haute Energies, Université Paris-Sud, France, for the hospitality received there, while he was a visitor there. References [1] Denny Gulick, Encounters with Chaos, Mc-Graw- Hill, Inc., p. 110 (1992). [2] Steven, H., Strogaze, Nonlinear Dynamics and Chaos, Addition-Wesley Publishing Company (1994). [3] Robert, L., Devaney, An Introduction to Chaotic Dynamical Systems, 2nd-Edition, Addison-Wesley Publishing Company (1989). [4] Robert, L., Devaney, A First Course in Chaotic Dynamical Systems Theory and Experiment, Addison- Wesley Publishing Company, The Advanced Book Program (1992). [5] Mitchell Feigenbaum, Quantitative Universality for a Class of Nonlinear Transformations, J. of Stat. Physics, Vol. 19, p. 25 (1978). [6] Mitchell Feigenbaum, The Universal Metric Prop-

6 94 Chyi-Lung Lin and Mon-Ling Shei erties of Nonlinear Transformations, J. of Stat. Physics., Vol. 1, p. 69 (1979). [7] Pierre Collet and Jean-Pierre Eckmann, Iterated Maps on The Interval As Dynamical Systems, Birkhauser Boston, (1980). [8] Edward Ott, Chaos in Dynamical Systems, Cambridge University Press (1993). [9] Alligood, K. T., Sauer, T. D. and Yorke, J. A., Chaos, Spring-Verlag, New York, Inc. (1997). [10] Ibid. 2, p [11] Saha, P. and Strogatz, S. H., The Birth of Period Three, Mathematics Magazine, Vol. 68, pp (1995). [12] Hillman, A. P. and Alexanderson, G. L., Abstract Algebra, PWS Publishing Company, Boston, MA. U.S.A. (2000). Manuscript Received: May. 12, 2005 Accepted: Mar. 21, 2006