Abstract— Over the last three decades, the chaotic dynamics in the field of mathematics, physics, chemistry, engineering, and social science have been studied in many researches and by many scholars. In this paper, we review period and chaotic behaviors in the fractional-order love model of Romeo and Juliet with Gaussian fuzzy membership function as an external force. To do this, we set a = −1, b =−5, c = d = 1 and vary theαandβ in the fractional-order differential equation of Romeo and Juliet. Each α and β means the magnitude of love status of Romeo and Juliet. If α and β are the same, the magnitude of love status of Romeo and Juliet are the same. If αand βare different, the magnitude of love status of Romeo and Juliet are different. To verify the periodic motion and chaotic motion for proposed love equation., we use time-series and phase portrait.

I. INTRODUCTION

Over the last three decades, the chaotic dynamics in the field of mathematics, physics, chemistry, engineering, and social science have been studied in many researches and by many scholars. In particular, the study of chaotic behaviors in the social and social sciences including habit and mind of human are addiction [1-4], happiness [5-8], love model [8-15] and family model [16-17].

The mathematical love model can be defined in several forms such as the Romeo and Juliet model [8-15], the Laura and Petrarch model [18], and the Adam and Eve model [19]. However, the most famous model is the Romeo and Juliet model in a study for nonlinear dynamics.

Love model based on Romeo and Juliet with a linear differential equation was proposed by Sprott and Strogatz [6, 8-10]. The studies of other love models have been conducted by many researchers [20-28]

Recently, researchers concern about fractional-order love model[29-30]. Particularly, Bae et.al [31-32] proposed love model of fractional-order with fuzzy membership function as an external force. They also verify the chaotic behavior using time series and phase portrait. However, this paper did not provide exact definition about magnitude of love status for

Linyun Huang are with the Chonnam National University, Yeosu, Jeonnam,

59626, South Korea. (e-mail: hly911121@gmail.com).

Romeo and Juliet. Therefore, we need the exact definition of magnitude of love status for Romeo and Juliet.

In this paper, we review period and chaotic behaviors in the fractional-order love model of Romeo and Juliet with Gaussian fuzzy membership function as an external force. To do this, we set a = −1, b = −5, c = d = 1 and vary the αand β in the fractional-order differential equation of Romeo and Juliet. The αand β means the magnitude of love status of Romeo and Juliet. If αand βare the same, the magnitude of love status of Romeo and Juliet are the same. If αand β are different, the magnitude of love status of Romeo and Juliet are different. To verify the periodic motion and chaotic motion for proposed love equation., we use time-series and phase portrait.

II. ROMEO AND JULIET’S LOVE MODEL

Typically, Romeo and Juliet’s love model can be divided into three different models: basic model, basic model with nonlinear term and basic model with nonlinear terms and with external force.

A. Basic Romeo and Juliet’s love model

Linear differential equation (1) represents the basic love model for Romeo and Juliet that was firstly proposed by Sprott [15]. = aR + bJ

= cR + dJ (1)

Where a and b specifies Romeo’s romantic style, and c and d specifies Juliet’s style.

B. Basic Romeo and Juliet’s love model with nonlinear term

We can rewritten equation (2) from the linear differential equation (1) by using an advanced form that may have nonlinear characteristics. = aR + bJ(|1 − J|) = cR(|1 − R|) + dJ (2)

Youngchul Bae was with Gwangwoon University, Seoul, 01897, South Korea. He is now with the Division of Electrical•Electronic Communication and Computer Engineering, Chonnam National University, Yeosu, Jeonnam, 59626, South Korea. (phone: 82-61-659-7315; fax: 82-61-659-7319; e-mail: ycbae@chonnam.ac.kr).

Nonlinear Behavior in Fractional-order Romeo and Juliet’s Love Model Influenced by External Force with Fuzzy Function

Linyun, Huang, and Youngchul Bae

B. Romeo and Juliet’s love model with external force

In order to make third-order system in equation (2), we add external force in Romeo or Juliet item of equation (2). Figure 1depicts the black diagram of love model of Romeo and Juliet with external force that can generate from their family, friends and relatives. Equation (2) also represents the third-order system of Romeo and Juliet’s love model with external force from figure 1.

Figure 1. Block diagram of love model of Romeo and Juliet with

external force

= aR + bJ(|1 − J|) + ( ) = cR(|1 − |) + dJ (3)

Where f(t) is an external force. The external force represents the influence from other people including their friends and relatives except Romeo and Juliet themselves.

III. FRACTIONAL-ORDER ROMEO AND JULIET’S LOVE MODEL

Fractional order of Romeo and Juliet’s love dynamical system can be described as equation (4), = + (1 − | |)

= (1 − | |) + (4)

Where, α and β are real number.

We consider fractional-order love model with membership function as an external force. The equation can be write as following equation (5). = + (1 − | |) + ( )

= (1 − | |) + (5)

Figure 2. Block diagram of fractional-order love model of Romeo and

Juliet with external force

IV. NONLINEAR BEHAVIOR OF FRACTIONAL-ORDER ROMEO

AND JULIET’S LOVE MODEL WITH FIZZY MEMBERSHIP

FUNCTION AS EXTERNAL FORCE

In order to make external force in Romeo and Juliet’s love model, we consider fuzzy Gaussian membership function. In this section, we consider nonlinear behavior of fractional –order of Romeo and Juliet’s love model with fuzzy membership function as Gaussian function [37].

Gaussian(x; σ, c) = ( ) (6)

Where, (σ, c) = ([2,5] [2,10] [2,15] [2,20] [2,25] [2,30] [2,35] [2,40] [2,45] [2,50] [2,55] [2,60] [2,65] [2,70] [2,75] [2,80] [2,85] [2,90] [2,95] [2,100]).

We can integrate external force with Gaussian membership function to use the external force as Eq. (7) shows and can illustrate for Fig. 3.

f(t) = ∑ i ∗ gaussian(x: σ, c) (7)

where, i = 1, −1,

Figure 3. Gaussian membership function using external function

In order to perform computer simulation by using equation (3), (4) and (4) for fractional-order of Romeo and Juliet with an external force of Gaussian fuzzy membership, we fixed

parameters,asa = −1, b = −5, c = d = 1, and we vary α and

β parameter.

A. = 1, = 1

When we set α = 1andβ = 1 , which means the magnitude of love status for Romeo and Juliet are the same. It also means it is integer-order differential equation. We can obtain the time-series and phase portrait as shown in figure 4 and figure 5, respectively.

Figure 4. Time series of Romeo and Juliet when α = 1, β = 1, a =−1, b = −5, andc = d = 1

Figure 5. Phase portrait for Romeo and Juliet when α = 1, β = 1, a =−1, b = −5, andc = d = 1

From figure 3 and 4, we recognize that the time series and phase portrait are providing near chaotic behaviors.

B. = 0.9, = 0.9

When we set α = 0.9andβ = 0.9 , we can obtain the time-series and phase portrait as shown in figure 6 and figure 7, respectively. It means the magnitude of love status for Romeo and Juliet are the same. However it has fractional-order differential equation.

Figure 6. Time series of Romeo and Juliet when α = 0.9, β = 0.9, a =−1, b = −5, andc = d = 1

Figure 7. Phase portrait Time series of Romeo and Juliet when α =0.9, β = 0.9, a = −1, b = −5, andc = d = 1

From figure 6 and 7, we recognize that the time series and

phase portrait shows quasi-periodic behaviors.

C. = 0.9, = 0.99

When we set α = 0.9andβ = 0.99 , which means the magnitude of love status between Romeo and Juliet are different each other, we can obtain the time-series and phase portrait. Figure 8 represents the time series of Romeo and Juliet. Figure 9 shows phase portrait for Romeo and Juliet.

Figure 8. Time series of Romeo and Juliet when α = 0.9, β = 0.99, a =−1, b = −5, andc = d = 1

Figure 9. Phase portrait Time series of Romeo and Juliet when α =0.9, β = 0.99, a = −1, b = −5, andc = d = 1

From figure 8 and 9, we know that the time series and

phase portrait shows quasi-periodic behaviors like figure 6 and figure 7.

D. = 0.94, = 0.99

When we set α = 0.94andβ = 0.99, which means the magnitude of love status between Romeo and Juliet are different each other, we can obtain the time-series and phase portrait as shown in figure 10 and figure 11.

Figure 10. Time series of Romeo and Juliet when α = 0.94, β = 0.99,a = −1, b = −5, andc = d = 1

Figure 11. Phase portrait Time series of Romeo and Juliet when α =0.94, β = 0.99, a = −1, b = −5, andc = d = 1

From figure 10 and 11, we recognize that the time series

and phase portrait are providing near chaotic behaviors. When comparing figure 8 and figure 10 for time series, figure 9 and figure 11 for phase portrait, we recognize that the behaviors in fractional-order love model of Romeo and Juliet are different. As the gap for magintute of love status between Romeo and Juliet become smaller, its behaviors show chaotic characteristic.

V. CONCLUSION

In this paper, we proposed the fractional-order love model of Romeo and Juliet with external force of Gaussian fuzzy membership function. We review the magnitude of love status between Romeo and Juliet for the case of the same and the different. As a result of computer simulation, we recognize that Romeo and Juliet’s love status keep chaotic behaviors, even though the magnitude of love status between Romeo and Juliet are different case. However, when the difference is small,

their love status shows chaotic dynamics, if the difference is large, love status shows periodic motion.

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