A new study of chaotic behavior and the existence of Feigenbaum's constants in fractional-degree Yin-Yang Henon maps

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DOI 10.1007/s11071-013-0981-x O R I G I N A L PA P E R

A new study of chaotic behavior and the existence

of Feigenbaum’s constants in fractional-degree Yin–Yang

Hénon maps

Chun-Yen Ho· Hsien-Keng Chen · Zheng-Ming Ge

Received: 8 September 2012 / Accepted: 11 June 2013 / Published online: 1 August 2013 © Springer Science+Business Media Dordrecht 2013

Abstract In this paper, we firstly develop fractional-degree Hénon maps with increasing and decreasing ar-gument n. Yin and Yang are two fundamental opposites in Chinese philosophy. Yin represents the moon and is the decreasing, negative, historical, or feminine prin-ciple in nature, while Yang represents the sun and is the increasing, positive, contemporary, or masculine principle in nature. Chaos produced by increasing n is called Yang chaos, that by decreasing n Yin chaos, respectively. The simulation results show that chaos appears via positive Lyapunov exponents, bifurcation diagrams, and phase portraits. In order to examine the existence of chaotic behaviors in fractional-degree Yin–Yang Hénon maps, Feigenbaum’s constants are measured in this paper. It is found that the Feigen-baum’s constants in fractional-degree Yin–Yang Hénon maps are of great precision to the first and second Feigenbaum’s constants. A detailed analysis of the chaotic behaviors is also performed for the fractional-degree Hénon maps with increasing (Yang) and de-creasing (Yin) argument n.

C.-Y. Ho· Z.-M. Ge (

_B

)

Department of Mechanical Engineering,

National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan

e-mail:zmg@cc.nctu.edu.tw

H.-K. Chen

Department of Mechanical Engineering, Hsiuping University of Science and Technology, 11 Gongye Rd., Dali Dist., Taichung 412-80, Taiwan

Keywords Chinese philosophy· Yin chaos · Yang chaos· Hénon maps · Feigenbaum’s constants · Fractional-degree Yin–Yang chaos

1 Introduction

Chaos, as an interesting nonlinear phenomenon, has been intensively investigated in the last three decades [1–8]. It is well known that chaotic systems have sen-sitive dependence on initial conditions, and chaotic be-haviors have become an important research subject in nonlinear sciences [9–13]. These nonlinear phenom-ena have been applied in many fields, such as se-cure communication [14–18], chemical reactions [19], and biological systems [20]. Chaos exists in both con-tinuous and discrete nonlinear systems. Hénon map [21–23], is the classical, well-known discrete nonlin-ear systems, and its chaotic behaviors have been stud-ied via a large number of researchers.

I Ching, is known as the Classic of Changes, is the first of the Five Classics in Chinese culture. It is a col-lection of ethical experiences from ancient people and has been used as a guide of being a good person. On the meaning of the name I Ching, it is an ancient docu-ment that describes the philosophy of the universe via the changes of the moon and the sun. The moon and the sun are represented as Yin and Yang, respectively.

In 1975, Feigenbaum discovered two constants, which are called Feigenbaum’s constants δ and α, where δ= 4.66920 . . . , and |α| = 2.5029 . . . when

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Fig. 1 The fractional-degree Yang Hénon map with q1= q2= 1, b1= 0.3: (a) phase portrait with a1= 1.4, bifurcation diagrams with a1is varied: (b) 0≤ a1≤1.5, (d) 0.86≤ a1≤0.9, (c) the largest Lyapunov exponent with a1= 0 ∼ 1.4

Table 1 Bifurcation points for each case

Case q1 q2 a(1) a(2) a(3) a(4) a(5) a(6) 1 1 1 0.368042 0.769134 0.849731 0.866331 0.869884 0.870645 2 0.99 1.9 0.579137 1.099208 1.219609 1.246652 1.252507 1.253753 3 0.9 1 0.429799 0.805855 0.869892 0.882955 0.885755 0.886355 4 1 0.9 0.368252 0.720598 0.785837 0.799428 0.802341 0.802965 5 0.98 0.98 0.373616 0.767808 0.84597 0.86194 0.865362 0.866095 6 0.94 0.94 0.38549 0.766374 0.834148 0.84875 0.851875 0.852544 7 1 1 1.267114 1.846424 1.951662 1.972614 1.977059 1.978011 8 0.99 1.9 0.917653 1.387428 1.501890 1.527159 1.532665 1.533842 9 0.9 1 1.233279 1.724023 1.827632 1.849776 1.854514 1.855529 10 1 0.9 1.344044 1.962335 2.061627 2.081358 2.085490 2.086375 11 0.98 0.98 1.281236 1.842318 1.94647 1.967451 1.972007 1.972982 12 0.94 0.94 1.309129 1.833289 1.934771 1.955761 1.960208 1.961156

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Table 2 Measurement results of Feigenbaum’s constants and errors for each case

Case i δi Percentage error of δi |αi| Percentage error of|αi|

1 1 4.976512 6.581 % 3.034366 21.23 % 2 4.855240 3.984 % 2.848865 13.82 % 3 4.672108 0.062 % 2.570944 2.718 % 4 4.668856 0.007 % 2.501618 0.051 % 2 1 4.319490 7.489 % 2.793279 11.6 % 2 4.452205 4.647 % 2.542469 1.58 % 3 4.618787 1.079 % 2.519736 0.67 % 4 4.699036 0.638 % 2.507169 0.17 % 3 1 5.417083 16.01 % 3.174695 26.84 % 2 5.226943 11.94 % 2.906941 16.14 % 3 4.99557 6.989 % 2.532188 1.17 % 4 4.666666 0.054 % 2.503681 0.03 % 4 1 5.043269 8.011 % 3.149333 25.82 % 2 4.894301 4.820 % 2.810731 12.29 % 3 4.666861 0.050 % 2.500140 0.11 % 4 4.668485 0.015 % 2.501854 0.041 % 5 1 5.043269 8.011 % 3.149333 25.82 % 2 4.894301 4.820 % 2.810731 12.29 % 3 4.666861 0.050 % 2.500140 0.11 % 4 4.668485 0.015 % 2.501854 0.041 % 6 1 5.372258 15.057 % 3.202252 27.94 % 2 4.821805 3.268 % 2.716627 8.53 % 3 4.672640 0.073 % 2.511764 0.35 % 4 4.671150 0.041 % 2.503639 0.029 % 7 1 5.504760 17.895 % 2.582036 3.16 % 2 5.022814 7.573 % 2.55625 2.13 % 3 4.713610 0.951 % 2.529411 1.059 % 4 4.669117 0.001 % 2.5 0.11 % 8 1 4.104200 12.100 % 2.738127 9.39 % 2 4.529740 2.986 % 2.454444 1.93 % 3 4.589357 1.710 % 2.492268 0.42 % 4 4.677994 0.188 % 2.505338 0.097 % 9 1 4.736499 1.441 % 3.039444 21.43 % 2 4.678874 0.207 % 2.810799 12.3 % 3 4.673701 0.096 % 2.516620 0.54 % 4 4.667980 0.026 % 2.505288 0.095 % 10 1 6.226997 33.363 % 2.559420 2.25 % 2 5.032284 7.776 % 2.530434 1.1 % 3 4.775169 2.269 % 2.5 0.11 % 4 4.668926 0.005 % 2.504878 0.079 % 11 1 5.387145 15.376 % 2.731021 9.11 % 2 4.964110 6.316 % 2.602681 3.98 % 3 4.605136 1.372 % 2.508035 0.205 % 4 4.672820 0.077 % 2.505307 0.096 %

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Table 2 (Continued)

Case i δi Percentage error of δi |αi| Percentage error of|αi|

12 1 5.165053 10.619 % 2.776734 10.94 %

2 4.834778 3.546 % 2.657534 6.17 %

3 4.720035 1.088 % 2.517241 0.572 %

4 4.690928 0.465 % 2.502720 0.007 %

Fig. 2 The fractional-degree Yang Hénon map with q1= 0.9, q2= 1.99, b1= 0.3: (a), (b) phase portrait with a1= 1.8, bifurcation

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Fig. 3 The fractional-degree Yang Hénon map with q1= 0.9, q2= 1, b1= 0.3: (a) phase portrait with a1= 1.4, (b) phase

portrait with a1= 1.2. Bifurcation diagrams with a1is varied:

(c) 0≤ a1≤1.6 (e) 0.91≤ a1≤0.95 (d) the largest Lyapunov ex-ponent with a1= 0 ∼ 1.6

the dynamical system approaches chaotic behavior via period-doubling bifurcation [24]. Feigenbaum’s con-stants were found in many chaotic systems [25–29]. The measurement of Feigenbaum’s constants in a con-tinuous time fractional-order system was firstly per-formed in detail by Chen et al. [29]. Ge and Li [30] investigated the dynamics of continuous time chaotic

system with negative time and the chaotic behavior of continuous time nonlinear system with negative time is called “Yin chaos.” Contrarily, the classical positive time counterpart is called “Yang chaos.” Ho, Chen, and Ge [31] investigated Yin and Yang chaos of discrete time maps. In order to make the research of Yin and Yang chaos more complete, we firstly study the Yin and

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Fig. 4 The fractional-degree Yang Hénon map with q1= 1, q2= 0.9, b1= 0.3: (a) phase portrait with a1= 1.3. Bifurcation diagrams

with a1is varied: (b) 0≤ a1≤1.4 (d) 0.79≤ a1≤0.85 (c) the largest Lyapunov exponent with a1= 0 ∼ 1.4

Yang chaos that appeared in fractional-degree maps. Meanwhile, the Feigenbaum’s constants in this frac-tional degree map are also measured.

This paper is organized as follows: The fractional-degree Yang and Yin Hénon maps are introduced in Sect.2, together with a review of the definition of the first and second Feigenbaum’s constants. The chaos characteristics of the two maps are analyzed using Lyapunov spectra, phase portraits, and bifurcation di-agrams in Sect.3. The measurement of Feigenbaum’s constants of the two maps is carried out in the same section. Finally, conclusions are drawn in Sect.4.

2 Chaos of fractional-degree Yin–Yang Hénon maps

The research of Hénon map extends to a new era by replacing the integer degrees to fractional ones. Two

novel fractional-degree Hénon maps are introduced in the following. Moreover, the definition of Feigen-baum’s constants is reviewed in this section.

2.1 Fractional-degree Yang Hénon map

The equations of the fractional-degree Yang Hénon map with increasing n1are described as follows:

x1[n1+ 1] = −a1x₁2q1[n1] +x₂q2[n1] +1

x2[n1+ 1] = b1x1[n1]

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where · is the norm of a complex number, a1, b1∈

R are the system parameters, q1 and q2 are positive

real numbers, n1= 0, 1, 2, 3, . . . , n, n is a nonnegative

integer sequence, x1and x2are the states of the map.

The initial conditions are chosen to be{x1[0], x2[0]} =

{0.63, 0.19}, and b1is fixed as b1= 0.3 throughout the

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Fig. 5 The fractional-degree Yang Hénon map with q1= q2= 0.98, b1= 0.3: (a) phase portrait with a1= 1.4. Bifurcation diagrams

2.2 Fractional-degree Yin Hénon map

The fractional-degree Yin Hénon map with decreasing n2is defined as

y1[n2− 1] = b2y2[n2]

y2[n2− 1] = a2y22q1[n2] +y1q2[n2] −1

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where a2, b2∈ R, and b2= 0.3 are the system

pa-rameters, q1 and q2 are positive real numbers, n2=

−1, −2, −3, . . . , −n, −n is a nonpositive integer se-quence, y1and y2are the states of the map. The initial

conditions are set as{y1[0], y2[0]} = {0.19, 0.63}.

2.3 The first and second Feigenbaum’s constants The first Feigenbaum’s constant δ is defined in [24] as δ= lim

i→∞

ai+1− ai

ai+2− ai+1

(3)

where ai is the value of the parameter at the ith

bifur-cation point, and the value of δ is 4.66920 . . . . The second Feigenbaum’s constant|α| is defined as |α| = Mxi

Mxi+1

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where Mxiis the width of the widest bifurcation fork

of ith bifurcation, and|α| = 2.5029 . . . .

3 Chaotic behaviors and measurement of Feigenbaum’s constants

In this section, the measurement of the first and second Feigenbaum’s constants and various chaotic behaviors are studied on different types of q1, q2. The

fractional-degree Yang Hénon map is analyzed in the first 6 cases, and the Yin Hénon map in the remaining 6 cases.

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Fig. 6 The fractional-degree Yang Hénon map with q1= q2= 0.94, b1= 0.3: (a) phase portrait with a1= 1.4. Bifurcation diagrams

with a1is varied: (b) 0≤ a1≤1.5 (c) the largest Lyapunov exponent with a1= 0 ∼ 1.5

Case 1, 7: q1, q2are integer degree.

Case 2, 8: q1, q2are different fractional degree.

Case 3, 9: q1 is fractional degree and q2 is integer

degree.

Case 4, 10: q1 is integer degree and q2 is fractional

degree.

Case 5, 6, 11, and 12: q1, q2are same fractional

de-gree.

The values of a at the ith bifurcation point are shown in Table 1, where i = 1, 2, 3 . . . , 6. The first and second Feigenbaum’s constants can be calculated by Table1. The measurement results of the first and second Feigenbaum’s constants for these 12 cases are listed in Table2.

3.1 Case 1 Yang Hénon map with q1= q2= 1

In this case, the q1, q2are integer degree and the

nu-merical results are shown in Figs.1. A phase portrait

is plotted in Fig.1(a), where a1= 1.4. Figure1(b) and

(d) are the bifurcation diagrams of x1, which can be

shown the period-doubling bifurcations clearly. The largest Lyapunov exponent for a1= 0 to 1.5 is shown

in Fig.1(c). The dynamics approaches to infinity when a1>1.426.

The measurement results of the first and second Feigenbaum’s constants for i = 1, 2, 3, and 4 are shown in Table2. It is shown that measurement val-ues of the first and second Feigenbaum’s constants are great precision to 4.66920 . . . with error percentage 0.007 % and 2.5029 . . . with error percentage 0.051 %, respectively when i= 4. More detailed measurement results of δ and |α| for i = 1, 2, 3 are shown in Ta-ble2.

3.2 Case 2 Yang Hénon map with q1= 0.99, q2= 1.9

By setting q1 = 0.99, q2= 1.9 in Eq. (1), the

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Fig. 7 The fractional-degree Yin Hénon map with q1= q2= 1, b2= 0.3: (a) phase portrait with a2= 2.5. Bifurcation diagrams with a2is varied: (b) 0≤ a2≤2.7 (d) 1.97≤ a2≤2.1 (c) the largest Lyapunov exponent with a2= 0 ∼ 2.7

map are shown in Figs. 2. A phase portrait is plot-ted in Fig.2(a), where a1= 1.8. Comparing with this

case and case 1, it is found that the chaotic behavior of this case is squeezed so that the motion is simi-lar to a parabola pattern (Fig.2(a)), where Fig. 2(b) shows an enlarged view of Fig. 2(a). A more com-plicated chaotic behavior can be seen in the bifurca-tion diagrams of x1 in Figs.2(c) and (e). The largest

Lyapunov exponent for a1 = 0 to 1.9 is shown in

Fig.2(d). Various dynamic behaviors for varied a1are

shown in Figs.2(c) and (d), such as the period-3 when 1.602≤ a1≤ 1.616, and the dynamics approaches to

infinity when a1>1.855.

The first Feigenbaum’s constant measures δ = 4.699036, with an error percentage of 0.63 % when i= 4. The second Feigenbaum’s constant measures |α| = 2.5071692, with an error percentage of 0.17 % when i= 4.

3.3 Case 3 Yang Hénon map with q1= 0.9, q2= 1

By setting q1= 0.9, q2= 1 to the Yang Hénon map,

the numerical results of the fractional-degree Yang Hénon map are shown in Figs.3. Figure3(a) and (b) show the chaotic motion for a1= 1.4 and a1= 1.2,

re-spectively. It is clearly shown that the chaotic motion of Fig.3(a) expands wider than Fig.3(b). Figure3(c) and (e) are the bifurcation diagrams of x1. The largest

Lyapunov exponent for a1 from 0 to 1.6 is shown in

Fig.3(d). The dynamics approaches to infinity when a1>1.523.

The first Feigenbaum’s constant measures δ = 4.666666, with an error percentage of 0.054 % when i= 4. The second Feigenbaum’s constant measures |α| = 2.503681, with an error percentage of 0.03 % when i= 4.

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Fig. 8 The fractional-degree Yin Hénon map with q1= 0.99, q2= 1.9, b2= 0.3: (a) phase portrait with a2= 2.1. Bifurcation diagrams

3.4 Case 4 Yang Hénon map with q1= 1, q2= 0.9

By setting q1= 1, q2= 0.9 in Eq. (1), the

numeri-cal results of the fractional-degree Yang Hénon map are shown in Figs.4. A phase portrait is plotted in Fig.4(a), where a1= 0.3. The bifurcation diagrams

of x1are shown in Figs.4(b) and (d). The largest

Lya-punov exponent is shown in Fig.4(c). The dynamics approaches to infinity when a1>1.351.

Feigenbaum’s constant measure 4.668269 and 2.504215, having error percentage of 0.02 % and 0.05 % when i= 4.

3.5 Case 5 Yang Hénon map with q1= q2= 0.98

By setting q1= q2= 0.98 in Eq. (1), the

numeri-cal results of the fractional-degree Yang Hénon map are shown in Figs.5. A phase portrait is plotted in Fig.5(a), where a1= 1.4. Figure5(b) and (d) are the

bifurcation diagrams of x1. The largest Lyapunov

ex-ponent for varied a1= 0 to 1.5 is shown in Fig.5(c).

The dynamics approaches to infinity when a1>1.434.

The first Feigenbaum’s constant measures δ = 4.668485, with an error percentage of 0.015 % and the second Feigenbaum’s constant measures |α| = 2.5018542, with an error percentage of 0.04 % when i= 4.

3.6 Case 6 Yang Hénon map with q1= q2= 0.94

By setting q1= q2 = 0.94 in Eq. (1), the

numeri-cal results of the fractional-degree Yang Hénon map are shown in Figs. 6. A phase portrait is plotted in Fig.6(a), where a1= 1.4. Figure6(b) and (d) are the

bifurcation diagrams ofx1. The largest Lyapunov

ex-ponent for varied a1= 0 to 1.5 is shown in Fig.6(c).

The dynamics approaches to infinity when a1>1.447.

The first Feigenbaum’s constant measures δ = 4.671150, with an error percentage of 0.04 % and

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Fig. 9 The fractional-degree Yin Hénon map with q1= 0.9, q2= 1, b2= 0.3: (a) phase portrait with a2= 2.4. Bifurcation diagrams

the second Feigenbaum’s constant measures |α| = 2.5036392, with an error percentage of 0.03 % when i= 4.

3.7 Case 7 Yin Hénon map with q1= q2= 1

The integer degrees are q1= q2= 1 for the Yin Hénon

map with decreasing n2. The phase portrait (Fig.7(a)),

bifurcation diagrams (Figs.7(b), (d)) and the largest Lyapunov exponent (Fig.7(c)) show the chaotic be-haviors and period-doubling bifurcations with q1=

q2= 1. It is found that the dynamics approaches to

infinity when a2>2.523.

The first Feigenbaum’s constant measures δ = 4.669117, with an error percentage of 0.0017 % and the second Feigenbaum’s constant measures|α| = 2.5, with an error percentage of 0.12 % when i= 4.

3.8 Case 8 Yin Hénon map with q1= 0.99, q2= 1.9

By setting q1= 0.99, q2= 1.9 in Eq. (2), the

numeri-cal results of the fractional-degree Yin Hénon map are shown in Figs.8. The largest Lyapunov exponent for varied a2= 0 to 2.2 is shown in Fig.8(d). Figure8(b)

and (d) are the bifurcation diagrams of y1. Various

dynamic behaviors for varied a2 can be shown by

Figs.8(b) and (d), such as the period-3 is shown when 1.869≤ a1≤1.884, and the dynamics approaches to

in-finity when a2>2.109.

3.9 Case 9 Yin Hénon map with q1= 0.9, q2= 1

By setting q1= 0.9, q2= 1 in Eq. (2), the

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Fig. 10 The fractional-degree Yin Hénon map with q1= 1, q2= 0.9, b2= 0.3: (a) phase portrait with a2= 2.6. Bifurcation diagrams

are shown in Figs.9. A phase portrait is plotted in Fig.9(a), where a2= 2.4. Figure9(b) and (d) are the

bifurcation diagram of y1, which can be shown the

period-doubling bifurcations clearly. The largest Lya-punov exponent is shown in Fig.9(c). The dynamics approaches to infinity when a2>2.413.

3.10 Case 10 Yin Hénon map with q1= 1, q2= 0.9

By setting q1= 1, q2= 0.9 in Eq. (2), the

numer-ical results of the fractional-degree Yin Hénon map are shown in Figs.10. A phase portrait is plotted in Fig.10(a), where a2= 2.6. Figure10(b) and (d) are

the bifurcation diagram of y1, which can be shown the

3.11 Case 11 Yin Hénon map with q1= q2= 0.98

By setting q1= q2 = 0.98 in Eq. (2), the

numeri-cal results of the fractional-degree Yin Hénon map are shown in Figs. 11. A phase portrait is plotted in Fig.11(a), where a2= 2.5. Figure11(b) and (d) are

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Fig. 11 The fractional-degree Yin Hénon map with q1= q2= 98, b2= 0.3: (a) phase portrait with a2= 2.5. Bifurcation diagrams

3.12 Case 12 Yin Hénon map with q1= q2= 0.94

By setting q1= q2= 0.94 in Eq. (2), the

numeri-cal results of the fractional-degree Yin Hénon map are shown in Figs.12. A phase portrait is plotted in Fig.12(a), where a2= 2.5. Figure12(b) and (d) are

period-doubling bifurcations clearly. The largest Lya-punov exponent for varied a2= 0 to 2.6 is shown in

Fig.12(c). The dynamics approaches to infinity when a2>2.515.

The first Feigenbaum’s constant measures δ = 4.690928, with an error percentage of 0.465 % and the second Feigenbaum’s constant measures |α| =

2.502722, with an error percentage of 0.007 % when i= 4.

4 Conclusions

In this paper, we firstly develop the Yin–Yang fraction-al-degree Hénon maps and measure the Feigenbaum’s constants to testify the existence of the chaotic behav-iors on different types of q1, q2, which can be

summa-rized as follows:

1. The variation of fractional degrees affects the val-ues of bifurcation points.

2. The variation of fractional degrees affects the chaotic behaviors; Period-3 motion is found among them.

3. The measurement precision of Feigenbaum’s con-stants is independent of q1 and q2, it depends on

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Fig. 12 The fractional-degree Yin Hénon map with q1= q2= 0.94, b2= 0.3: (a) phase portrait with a2= 2.5. Bifurcation diagrams

It is worthwhile to notice that further development of Yin and Yang systems for real engineering prob-lems. The extension of the Yin and Yang concept to other kinds of nonlinear systems like fractional-order systems may produce even more interesting dynamical characteristics.

Acknowledgements The research was partially supported by a grant (NCS101-2221-E-164-008) from the National Science Council, R.O.C.

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