### 1. Introduction

Over the last three decades, nonlinear dynamics including chaotic dynamics has been studied in many science fields, such as mathematics, physics, chemistry, engineering, and social science. In particular, studies of chaotic behavior in love models [1â€“5] have increased.

The dynamical love model has been defined in several forms, such as the Romeo and Juliet model [1, 3â€“7], the Laura and Petrarch model [8], and the Adam and Eve model [9]. Among these love models, the Romeo and Juliet model is mostly used in the study of nonlinear dynamics.

The love model based on Romeo and Juliet with a linear differential equation was first proposed by Sprott [1, 2], who described its linear and nonlinear behaviors. While the other love models have been the subject of some research [3â€“7], the Romeo and Juliet-based love model has been the most studied. Bae and his associates [3â€“7] proposed that the existence of periodic and chaotic behaviors based on the Romeo and Juliet love model can be represented through time series and phase portraits with identical and different time delays and an external force.

In this paper, we propose a love model for Romeo and Juliet with a discontinuous external force. The discontinuous external force closely reflects the real love between a man and woman. The discontinuous external force implies that the relation of love between Romeo and Juliet is not always the same. Sometimes they do not have thoughts about each other or sometimes they do not have concerns for each other. In addition, we investigated the periodic motion and chaotic behavior in the proposed Romeo and Juliet love model, with a discontinuous external force, by using time series and phase portraits.

### 2. Love Model

### 2.1 Romeo and Juliet Love Model

The basic love model for Romeo and Juliet was proposed by Sprott [1] and can be described as the following Eq. (1):

##### (1)

$$\begin{array}{l}\frac{\text{dR}}{\text{dt}}=a\text{R}+b\text{J}\\ \frac{\text{dJ}}{\text{dt}}=c\text{R}+d\text{J},\end{array}$$where

*a*and*b*specify Romeoâ€™s romantic style, and*c*and*d*specify Julietâ€™s style.According to parameters

*a*,*b*,*c*, and*d*, we divided behavior into four types, as proposed by Strogatz [10]. When*a >*0 and*b >*0, Romeo is an eager beaver. That is, Romeo is encouraged by his and Julietâ€™s feelings. When*a >*0 and*b <*0, Romeo is a narcissistic nerd. This implies that Romeo wants more of what he feels but retreats from Julietâ€™s feelings. When*a <*0 and*b >*0, Romeo is a cautious or secure lover. That is, Romeo retreats from his own feelings but is encouraged by Julietâ€™s. Finally, when*a <*0 and*b <*0, Romeo is a hermit. That is, Romeo retreats from both his and Julietâ€™s feelings.##### (2)

$$\begin{array}{l}\frac{\text{dR}}{\text{dt}}=a\text{R}+b\text{J}(\xe2\u02c6\pounds (1-\text{J})\xe2\u02c6\pounds \\ \frac{\text{dJ}}{\text{dt}}=c\text{R}(\xe2\u02c6\pounds (1-\text{R})\xe2\u02c6\pounds +d\text{J.}\end{array}$$From Eq. (2), we can describe a novel love model with a discontinuous external force, as in the following Eq. (3):

##### (3)

$$\begin{array}{l}\frac{\text{dR}}{\text{dt}}=a\text{R}+b\text{J}(\xe2\u02c6\pounds (1-\text{J})\xe2\u02c6\pounds +\mathrm{\xe2\u20ac\u2030}f(t)\\ \frac{\text{dJ}}{\text{dt}}=c\text{R}(\xe2\u02c6\pounds (1-\text{R})\xe2\u02c6\pounds +d\text{J},\end{array}$$where f (

*t*)= 5 sin*Ï€t*_{1}is a discontinuous external force. In this paper, we gives 2 seconds per every 10 seconds as t_{1}to have intermittence.### 3. Nonlinear Behavior Analysis

For Eq. (3), we investigated the chaotic behavior by using time series and phase portraits according to parameter variation.

### 3.1 *a* = âˆ’7, *b* = âˆ’2, *c* = 1, and *d* = 1

### 3.6 *a* = âˆ’1.5, *b* = âˆ’2, *c* = 1, and *d* = 1

The time series and phase portrait of the model obtained using this parameter set are shown in Figures 11 and 12, respectively. The figures show a quasi-periodic motion or quasi-attractor.

### 3.7 *a* = âˆ’1.4, *b* = âˆ’2, *c* = 1, and *d* = 1

The time series and phase portrait of the model obtained using this parameter set are shown in Figures 13 and 14, respectively, which show that the model has a chaotic attractor.

### 3.8 *a* = âˆ’1.3, *b* = âˆ’2, *c* = 1, and *d* = 1

The time series and phase portrait of the model obtained using this parameter set are shown in Figures 15 and 16, respectively. The figures confirm that the model has a chaotic motion.

### 3.9 *a* = âˆ’1.1, *b* = âˆ’2, *c* = 1, and *d* = 1

The time series and phase portrait of the model obtained using this parameter set are shown in Figures 17 and 18, respectively. These figures confirm a chaotic attractor with different type compare to other chaotic attractor such as Lorenz attractor, Chuas attractor and Chen attractor.

### 3.10 *a* = âˆ’1.0, *b* = âˆ’2, *c* = 1, and *d* = 1

The time series and phase portrait obtained using this parameter set are shown in Figures 19 and 20, respectively. These figures confirm a quasi-periodic motion.

Figures 1â€“20 confirm that the behavior of the love model with a discontinuous external force changes from periodic motion to quasi-attractor, chaotic attractor, and quasi-periodic motion as confirmed by the various time series and phase portraits. Thus, these results confirm that the system has a typical chaotic dynamics.