Generalized Fibonacci Sequences and the Triangular Map Chyi-Lung

CHINESE JOURNAL OF PHYSICS

VOL. 32, NO. 5-I

OCTOBER 1994

Generalized Fibonacci Sequences and the Triangular Map Chyi-Lung Lin Department of Physics, Soochow University, Taipei, Taiwan 111, R.O.C. (Received April 18, 1994; revised manuscript received June 8, 1994)

By defining F,,, maps, in which m = 2,3,4, ..., we show that these maps generate the generalized F,,, Fibonacci sequences that include the customary Fibonacci sequence and its generalized sequences. All these F,,, maps and the F, Fibonacci sequences are found in the triangular map with its parameter p chosen with the specified values C,. For each m, C, has the meaning that C, is the largest parameter value among those m-cycle parameters pm for which I = 0 is a period-m point, and Cm+1 = X,, w h e r e X, is the limit ratio of the F, Fibonacci sequence. PACS. 05.45.+b - Theory and models of chaotic systems. PACS. 03.2O.+i - General mathematical aspects.

I. INTRODUCTION The interesting Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, .... Nature shows that these numbers occur quite frequently in various areas [l]. These numbers form the Fibonacci sequence {F(n)}~=, that satisfies the difference equation: F(n) = F( n - 1) + J’(n - 2). That is, each element in this sequence is the sum of its preceding two elements. In general, the first two numbers may be chosen arbitrarily. The Fibonacci numbers correspond to the choices F( 1) = 1 and F( 2) = 1. The 1imit ratio, defined by ,IIW F(n)/F(n - 1) E X, satisfies the equation X = 1 + X- r. The two roots of this equation are: (1 + a)/2 and (l-&)/2. If the positive one is chosen, then X = (1+&)/2 which is a number well known in the golden section as the golden mean or golden ratio. The appearance of the golden mean is one reason that Fibonacci numbers are interesting. The limit ratio X, although an irrational number, is the characteristic number of the Fibonacci sequence, because all the numbers in this sequence can be expressed in terms of this characteristic number. For example, the Fibonacci numbers can be expressed as F(n) = (X” - (-X)-*)/(X -t- X-l). Therefore the limit ratio has a special meaning to the Fibonacci sequence. 467

@ 1994 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

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We can obtain these Fibonacci numbers easily by iteration. By starting with a line segment in the range 0 2 z 2 1, and disturbing it iteratively with the triangular map, which is also called the tent map [2-41, the geometry of the shape becomes a zigzag line, if the parameter value is chosen arbitrarily. However, if we choose the parameter value to be the golden mean, then the shape becomes simple: It contains line segments of only two types-one is long and the other is short, just like the large rabbits and the small rabbits in the original Fibonacci problem. The numbers of the line segments are also those of the Fibonacci sequence under iteration. This behavior may be generalized to other suitable parameter values, and we present here this interesing property of the triangular map. We now generalize the customary Fibonacci sequence to form the F,-type Fibonacci sequence. That is, the sequence in which each element is the sum of its preceding m elements, in which m takes on the values 2, 3, 4, 5, - - -. The original Fibonacci sequence is then the FZ Fibonacci sequence. Another number sequence, for instance 1, 1, 2, 4, 7, 13, 24, 44, . . ., satisfying the difference equation F(n) = F(n - 1) + F( R. - 2) + F(n - 3), is the Fs Fibonacci sequence. It seems that these generalized Fibonacci sequences are not yet appreciated in nature. The limit ratio of the F, Fibonacci sequence, ,Jirnm F,(n)/F,(n-1), satisfies the equation X = 1 + X-’ + Xm2 + Xm3 + -. e + X-(m-*). There are m roots to this equation; the positive one is denoted by X, and 1 < X, < 2. We take this X, as the limit ratio of the F,,, Fibonacci sequence. We will show the importance of this X, to the triangular map; that is, if we choose the value of the parameter in this map to be this limit ratio, then the F, Fibonacci sequence appears in the triangular map. Our studies are of the fixed points of the iterated map on the finite interval. The iterated map [5] is an interesting area of research. The triangular map may be taken to be the simplest non-trivial map but it is still complicated enough. In general, the range of parameter p in this map is restricted to (0,2]. The map is simple for 0 I p 5 1, but complicated for 1 < p 2 2. The map is interesting in this complicated region which, naively, is a chaotic region. The function fn(,) in the range 0 < z 5 1 contains many line segments that connect to form a zigzag shape which varies irregularly under iteration. The reason is that not only the number of line segments increases but also the type of line segments increases. It is then hard to predict the final shape. However, we may find interesting results if we focus on the parameter p for certain values such that z = 0 becomes a periodic point. The values of p at which z = 0 is a period-n point are denoted by pn and are called the n-cycle parameters. The largest one is denoted by C,. We show that when /I = Es, after some iterations the type of the line segments does not increase under iteration; however, the number of the line segments does increase. These numbers are those of the Fibonacci numbers. We note that C3 = (1 + &)/2 = ~2, the limit ratio of the Fibonacci sequence. We may focus on the next cycle parameter, i.e. p = Cd; we find Cd = X3, the limit ratio

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of the F3 Fibonacci sequence, and the number of line segments under iteration are those of the F3 Fibonacci sequence. bloving on to p = Cn+i, we find Err+1 = X,, and the number of line segments under iterations are those of the F,, Fibonacci sequence. The purpose of this paper is to introduce these interesting phenomena of the triangular map in this complicated region. We show that these F, maps and F,,, Fibonacci sequences can all be found in this single map with a suitable choice of value for its parameter /J. These suitable values for generation of these types of Fibonacci sequences can be considered either in terms of C,: the m-cycle parameter, or X,.* the limit ratio of the F, Fibonacci sequence. In Sec. II, we show that the usual Fibonacci sequence can be generated from the F2 map. By generalizing the F2 map to the F,,, map, we show that the F, map generates the F,,, Fibonacci sequence. In Sec. III, we discuss the triangular map in three ranges of p : p < 1, p = 1, p > 1; the functional shape is of particular interest in the range p > 1. In Sec. IV, we show that if the parameter p in the triangular map is chosen with the value Es, then we have the F2 map. In Sec. V, we show that for p = Cd, we have the F3 map. In Sec. VI, we discuss the general case for 11 = &+I. In Sec. VII, we discuss the limit case: /l = c, = 2. II. F,,, M A P S A N D G E N E R A L I Z E D F I B O N A C C I S E Q U E N C E S We use an improved langua.ge to express an old problem, and to solve or to see the solution more clearly. The famous Fibonacci problem can be expressed in terms of an iterated map, which we call the F2 map, given by

F2: T -+ R

R - R+r,

(1)

in which T represents the small rabbit and R represents the large rabbit. This map simply represents the original ideal of Fibonacci: That after a definite period, the small rabbit becomes a large rabbit and the large rabbit bears a new small rabbit. So if we start from a small rabbit T, the rabbit families under iteration are accordingly T + R + R -I- T + 2R + T + 3R + 2r + 5R + 3r - 8R + 5r + 13R + 8r,. --. The numbers of rabbits in each generation are therefore 1, 1, 2, 3, 5, 8, 13, 21, .. a. These are the so-called Fibonacci numbers. Let R(n) and r(n) b e, respectively, the number of large and small rabbits in generation 7~. We see from formula (1) that the small rabbits come from the preceding large rabbits, and the large rabbits come from the preceding large and small rabbits; so we have r(n)

= R(n- 1 )

R(n) = R(n - 1) + r(n - 1).

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We can then easily find that R(n) = R(n - 1) + R(n - 2) and r(n) = r(n - 1) + r( n - 2). This then defines the Fibonacci sequence that satisfies the difference equation F(n) = F(n - 1) + F(n - 2).

(3)

The Fz map then generates the Fibonacci sequence. By this mechanism (1) nature produces Fibonacci numbers in various areas. We define the limit ratio X b y X - J$m F(n)/F(n - 1).

(4)

From formula (3), we see that X satisfies the equation x=1+x-’

or with X > 1.

x2-x-1=0,

(5)

The solution is the number (1 + &)/2, the famous golden mean. We denote this number by

x2.

The easiest generalization of the Fibonacci sequence is to consider a sequence in which each element is the sum of its preceding 3 elements. We achieve this sequence by generalizing the types of large rabbits. We define the Fs map by

F3:

T

+

RI

RI

--f RI +

R2

---)

R2

(6)

RI + T .

We see that T arises from the preceding R2, R2 from the preceding RI, and RI from the preceding RI, R2 and T . We then have

RI(~)

=

Rl(n - 1) + Rz(n - 1) t

T(72 -

1)

R2(n) = Rl(n - 1 ) T(n)

=

(7)

R2(n - 1 ) .

Then we easily find that all RI(n), R2(n) and r(n) satisfy the difference equation

F(n) = F(n - 1) t F(n - 2) t F(n - 3).

(8)

This relation defines the F3 Fibonacci sequence. Then the F3 map generates the F3 Fibonacci sequence. The limit ratio lim Fs(n)/Fx(n - 1) satisfies the equation n-+oo x = 1 -I- x-’ + x-2

-_a,

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x3-x2-x-1=0,

471

with X > 1.

(9)

The solution is denoted by X3, and X2 < X3 < 2. The generalization to the F, map for an arbitrary integer m is then clear. We define the F,,, map by

Ri

+

(10)

R I + Ri+ly

in which 1 5 i 5 m - 1 and R, G T. We have Rj(n) = Rj_l(n - l), in which 2 5 j 2 m. Hence Rj(n) = Rl(n - j + 1 ) . Al so as RI(n) arises from all preceding Ri(n - l), then RI(~) = 2 R;(n - 1) = e Rl(n - i). Th US all R;(n), 1 5 i 5 m, satisfy the difference equation

i=l

i=l

F(a)=FF(n-i).

(11)

i=l

This condition defines the F, Fibonacci sequence. Then the F, map generates the F, Fibonacci sequence. The limit ratio ,brrnm F,(n)/F,(n - 1) satisfies the equation

i=O

xm =

m-l

c Xi)

with X > 1 .

(12)

i=O

The solution is denoted by X,, and X,-r < X, < 2. The limit case of the F, map is found by taking m + co in formula (10); we then have the F, map. According to this map, the orbit that starts from T is: -+ RI -+ RI + R2 + 2R1+ R2 + R3 + 4R1+ 2R2 + R3 + Rq + a... The population doubles under interation (except for the first iteration), so the total populations of each generation are accordingly as: 1, 1,2,4,8,16,. . . , 2n,. - . . This is th F, Fibonacci sequence in which each element is the sum of all its preceding elements. The F, Fibonacci sequence is the same type as the (2”) sequence. The limit ratio X, z J@m Fw(n)/F,(n - 1) = 2. The F, map, due to its property of being doubled under iteration, is described more easily as T

R +

~_.

._

2R.

,-.

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So, starting from r, we have under iteration T + R + 2R + 4R + 8R * .. 0. The populations in each generation are accordingly 1 + 1 + 2 + 4 -+ 8 + 16 + .- +, the J’, Fibonacci sequence. In what follows, we show that all these generalized Fibonacci sequences can be found in the triangular map by choosing proper values for the parameter. III. THE TRIANGULAR MAP The triangular map is defined by

or x,+1 =

1 - PM,

(14)

in which p is the parameter. We restrict to the range -1 5 z 5 1 and 0 5 ~1 < 2, so that it is a map for z from the interval [-1, l] to [-I, l]. We let f”(~;p) E f o f+r(z;p). The shape of f”(z;p) is symmetrical for z > 0 and z < 0; hence we consider the shape in only the z 2 0 range. The starting shape, f(z; p) = 1 - ~LZ, is a line segment from point (0,l) to point (1,l -p) with slope -p. We see how this starting line segment changes its shape under iteration. Consider a line segment described by g(z) = a + bz; then the upper piece of this line segment, for which g(z) 2 0, iterates to 1 - al_~ - bpx. Hence the slope and the length of the upper piece have been resealed and the sign of the slope alters as well; the slope resealing factor is seen to be -p. The lower piece, for which g(z) < 0, iterates to 1+ a~ + bpx. Hence the slope and the length of the lower piece have been resealed, but the sign of the slope does not alter; the slope resealing factor is /I. As the two pieces of this line segment behave differently under iteration, we conclude that a line segment breaks into two connected pieces if it intersects the z-axis, and remains a line segment if it does not intersect the x-axis. By this basic mechanism we understand how the functional shape varies under iteration. We now proceed to discuss the functional shape under iteration for the three ranges of p. III-l. /L < 1 A s f(z;p) = 1 - +I, so for /L < 1, this represents a starting line segment lying above the x-axis in the range 0 5 z 5 1. We see easily that under iteration it remains a line segment lying above the x-axis. The iterations then only rescale the slope of this line segment but do not brea.k it. The slope resealing factor in each iteration is (-p), and as -1 < (-p> < 1, the Islope] decreases in each iteration. So, this line segment becomes flatter and flatter; finally for n - co, it becomes a flat line segment with height x* = l/(1 + p). We know that Z* is just the fixed point of this f(~;p) map. Iteration cannot change the

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fixed point, so the final flat line segment can only be at this height. Therefore, it is a simple map in the range p < 1. The fate for each point z under iteration is determined. 111-2. j4 = 1 This case is the same as above but now p = 1. The slope resealing factor is then -1. As the starting line segment has a slope -1, so we have slopes of only two kinds, 1 and -1. The shapes under iteration are then a line segment oscillating with these two slopes. The line segment with slope 1 is from point (0,O) to point (l,l), and the line segment with slope -1 is from point (0,l) to point (1,O). These are the two shapes for this case. 111-3. ~1 > 1 In this case, the starting line segment intersects the z-a_xis; then it breaks in the next iteration. We may also see this breaking in the following way: denoting the intersection point of the line segment with the x-axis by #, called the “pre-breaking point”, then as the height of the line segment at this point is 0, so for the next iteration the height increases to 1, which is the maximum height; therefore, the line segment breaks into two connected pieces at the pre-breaking point. Then, new pieces are generated under iteration. Some of the new pieces will intersect the z-axis again and will then break in the next iteration, and so on. Therefore, the starting line segment continues to break under iteration; we have line segments that breed in the range p > 1. In the following, by choosing proper values for ~1, the numbers of line segments under iteration are shown to be those of the Fibonacci sequence or the generalized Fibonacci sequences. IV. THE CASE FOR p = Cs In general, in the case p > 1, the map is complicated. However there are cases that are easier to analyze. We may require that there be periodic motion at the point z = 0. The simplest case is for 1: = 0 being a period-2 point; this corresponds to f2(O;p) = 1 - p = 0, i.e. p = 1, and the line segment under iteration has two shapes alternatively. The next is to require z = 0 to be a period-3 point; this condition corresponds to f3(O;~) = 0. There is no solution for /J < 1. Requiring p > 1, then f2(O;p) = 1 - ~1 < 0; therefore, f3(O;p) = 1 - p[-f2(O;p)]. That is p2 - p - 1 = 0; this is simply formula (5) for the limit ratio X2. So /L = x2 = (1 + J5)/2. We denote this p by Cs; it is the 3-cycle parameter and C3 M 1.618. The three iteration heights at z = 0 are O,l, 1 - Cs = -l/C3 < 0. We use symbols A, B, C to represent respectively the 3 heights: A is for height 1, B is for height 0 and C is for height - l / C 3. Under iteration, A goes to C, C goes to B, and B goes to A ; also the pre-breaking point #, which has height 0, goes to A. These height-varying rules are all we need to analyze the iteration behavior.

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The starting line segment, using the notation above, is AC. As AC crosses the z-axis, then in the next iteration, it has to break at the pre-breaking point. We therefore write A C as A#C, then iterating and according to the height-varying rules, we have AC = A#C --$ CAB = CA + AB. This means that AC breaks into two pieces: AC and AB (we do not distinguish between AC and CA, or AB and B A ) . For 1 ine segment AB, as it does not cross the z-axis, it goes to AC under iteration without breaking. So AB does not break but only gets longer. Under iteration, we then have line segments of only two types-AB and AC, and their iteration behavior corresponds to the map

F:

AB

+

A C -+

AC AC+AB.

(15)

This is the same as the Fz map in formula (1). Th e 1 ine segment AB is like the small rabbit that needs first to be mature, and the line segment AC is like the large rabbit that has the ability to breed. We conclude that the starting single line segment breaks into more and more line segments under iteration and that their numbers are those of the Fibonacci numbers. Hence the Fz map, and therefore the Fibonacci numbers, are found in the triangular map with its parameter p chosen as the value Es which just equals X2. These are interesting phenomena of the triangular map, since for parameter /L with this value X2, which is the limit ratio of the Fibonacci sequence, we do find the Fibonacci numbers in the triangular map, and the cycle parameter and the limit ratio are related: C3 = X2. In the following, we show that these interesting phenomena are maintained in generalized cases.

V. THE CASE FOR /L = C4

Requiring x = 0 to be a period-4 point, the needed 4-cycle parameter ~4 is determined from the equation: f4(O;p) = 0. The largest 4-cycle parameter Cd can be determined by requiringp > CJ. Then f2(O;p) < 0 and f3(O;~) < 0; therefore, f4(O;p) = l-p[-f”(O;p)], that is p3 - p2 - p - 1 = 0. This is the same as formula (9) for X3. So C4 = X3 M 1.839. The 4 iteration heights at the point 2 = 0 are A, B, C, D, in which A is for height 1, B is for height 0, C is for height 1+ C4 - Ci = -CT’, and D is for height 1 - C4 = -CT’ - CT2. The rules of height-varying under iteration are A ---t D, D + C, C -+ B, and B + A; also # + A. As the starting line segment is AD, then under iteration, we have AD = A#D i DAC = DA + AC; AC = A#C + DAB = DA + AB, and finally AB + AD. We see that AB is the only line segment that does not break. We have line segments of only three types under iteration-AB, AC, AD. The behavior of these line segments in an iteration corresponds to the map

i-

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F:

AB

-+

AD

(16)

AD+AC

AD + AC

475

+ AD+AB.

This is the same as the F3 map in formula (6). S o we conclude that with p = C4 = X3, we have the F3 map. Hence the original line segment AD breaks into increasingly numerous line segments, and their numbers are those of the F3 Fibonacci sequence. VI. THE CASE FOR p = C,+1 We extend the above results to an arbitrarily large n. For 5 = 0 to be a period(n + 1) point, the needed (n + 1)-cycle parameter p,+r is determined from the equation f”+‘(o;p) = 0. I n general, there are several p,+r that satisfy this equation. We choose the largest among them, so as to ensure that we have a line segment of only one type that remains unbroken under iteration. We denote the largest (n + l)-cycle parameter by &+I, which is determined from the equation f”+‘(O;p) = 0 and ~1 > C,. Then fi(O; p) < 0 for 2 5 i 5 12, and therefore f’+‘(O;i~) = 1- p[--f’(O;p)]. From this, we see that Cn+i satisfies the equation n-l pn - Cp’ =

0,

P>l

i=O

or n-1 p = c g-n+1 = 1+ p-l + p-z + p3 +

. . . + /+-l).

(17)

i=o

This is the same as formula (la), with m = n. Therefore, C,,+r = X,. The 7~ + 1 iteration heights at z = 0 are A, B,C; (3 5 i 2 n + l), in which A is for height 1, B is for height 0, Cn+r is for height -p- r, C, is for height -CL-’ - p-2, C,_r is for height -p-r - pm2 - p-3 and so forth. Finally C3 is for the lowest height -p-’ -/_L-~ -pe3 -. . .-p-(n-*) = 1 -p. All Ci < 0 and are ordered, i.e. C3 < Cd < *. . < Cn+r < 0, according to Ci+r -C; = /L-(~-~+‘). Hence in an iteration, only the line segment AB, which lies above the z-axis, does not break, but the line segments ACi, which intersect the z-axis, break. We have the iteration heightvarying rules A --+ C3, Ci + Ci+r, Cn+r + Cn+2 _ B, a n d B +- A; a l s o # + A. T h e starting line segment is AC3, and we have AC; = A#Ci -+ C3ACi+l = C3A + AC;+17 and AB + C3A. So under iteration, we have line segments of only n types-AB and AC’;. Their iteration behavior corresponds to the map F:

AB AC;

.-.

--f -+

AC3 AC3 t AC;+1 3

(18)

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in which 3 5 i 2 n + 1. This map is the same as the F, map in formula (lo), with m = n. Thus the F, map, and therefore the F,, Fibonacci sequence, is found in the triangular map with the parameter /L chosen with the value &+I, and &+I = X,. VII. THE CASE FOR p = C, = 2 The limit case is for z = 0 to be a period-n point with n -+ 03. Taking n + 00 in formula (17), we have the required parameter value C, = 2. Then C, = X,. The triangular map with /.L = 2 is simple. In this case, f(z;p) = 1 - 2121. So the orbit at point 2 = 0 is 0 -+ 1 + -1 + -1 + ..a. According to this limit case, z = 0 is in fact an eventually fixed point. The 3 heights that we need to consider are 1, 0, -1, which we denote respectively by A, B, C. Then under iteration, we have A --+ C, B -+ A, # + A, and C + C. The starting line segment is AC. In order to have a complete similarity to the F, map, we let the starting line segment be BA which goes to AC under iteration. We have AC = A#C --f CAC = CA $ AC = 2AC. Then we find the following map: F:

AB

--f

AC

AC

+ 2AC.

(19)

This is the same as the F, map in formula (13). So by taking p = X, = 2 in the triangular map, we have the F, map. The numbers of line segments starting from BA under iteration are those of the F, Fibonacci sequence: 1, 1,2,4,8,16,. . . . In conclusion, in the triangular map, the largest m-cycle parameter C,, the limit ratio X,, and the corresponding F, Fibonacci sequences are all related. This reveals interesting contents of the triangular map. ACKNOWLEDGMENT We thank the National Science Council of the Republic of China for support (NSC Grant No. 82-0208-m-031-014). REFERENCES

111

The New Encyclopaedia Britannica (Encyclopaedia Britannica, Inc, 1991) Vol. 7, page 279; H.S.M Coxeter, Introdzlction to Geometry (New York, Wiley, 1969), chapter 11.

PI

Heinz Georg Schuster, Deterministic Chaos (Physik-Verlag, Weinheim, 1984), chapter

PI

Denny Gulick, Encounters with Chaos (McGraw-Hill, Inc, 1992), chapter 1, 2.

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[ 4 ] Edward Ott, Chaos in Dynamical Systems (Cambridge Gniversity Press, 1993), chapter 2. [ 5 ] P. Collet and J. P. Eckmann, Iterated Maps on the Interval us Dynumicul (Birkhauser, Boston, 1980), part I.

Systems