In this paper, we propose a dynamic mathematical model of love involving various external forces, in order to analyze the chaotic phenomena in a love model based on Romeo and Juliet. In addition, we investigate the nonlinear phenomena in a love model with external forces using time series and phase portraits. In order to describe nonlinear phenomena precisely using time series and phase portraits, we vary the type of external force, using models such as a sine wave, chopping wave, and square wave. We also apply various different parameters in the Romeo and Juliet model to acquire chaotic dynamics.
As the world inhabited by humans is organized into complex systems, human society can also be described as a complex system. A complex system is an example of a nonlinear dynamical system. It is well known that nonlinear systems are difficult to analyze, synthesize, or implement in comparison with linear systems.
All systems that naturally occur are nonlinear in nature. Furthermore, all kinds of phenomena that are modeled by fuzzy neural networks, chaotic systems, and complex systems correspond to systems that are nonlinear. During recent decades, the analysis and applications of chaotic phenomena in nonlinear systems has been widely researched in connection with fields including natural sciences, engineering, psychology, and social sciences [1–7].
Recently, some researchers have been interested in models for addiction [8–11], happiness [12–15], and love [15–20], which involve an overlap of mathematics, biology, psychology, and social science.
Bae [8–10] and Bae and Kim [11] proposed a mathematical model for addictions to digital leisure, internet use, tobacco use, and physical exercise. He also confirmed the presence of nonlinear or chaotic phenomena, using time series and phase portraits. Sprott [13, 15] proposed a basic dynamic equation for happiness, and investigated its behavior. Kim et al. [12] and Bae [14] proposed a mathematical happiness model and a synchronization technique for the model.
Several models have been proposed for love, such as the Romeo and Juliet model [13, 15], the Laura and Petrarch model [21], and the Adam and Eve model [22]. Among these, the love model based on Romeo and Juliet is most commonly employed in the study of nonlinear dynamics.
Sprott [15] first proposed a love model based on Romeo and Juliet using a linear differential equation, and he also described both the linear and nonlinear behavior of the model. The existence of periodic and chaotic behavior in the love model based on Romeo and Juliet is represented through time series and phase portraits, with either the same or different time delays, and an external force proposed by Bae and his colleagues [16–20]. However, the majority of previously published papers have not provided an exact time series or phase portrait for demonstrating chaotic phenomena in the Romeo and Juliet model. In order to analyze such chaotic behavior, we should formulate a dynamic mathematical model using ordinary differential equations or difference equations, following the models of addiction, happiness, and love.
In this paper, we propose a dynamic mathematical model of love with various external forces. We also investigate the nonlinear phenomena present in love models with external forces using time series and phase portraits. In order to describe the nonlinear phenomena precisely using time series and phase portraits, we vary the type of external force, using models such as a sine wave, chopping wave, and square wave. We also apply various parameters in the Romeo and Juliet model to acquire chaotic dynamics.
The basic love model proposed by Sprott [15] can be formulated by the equations
where a and b specify Romeo’s romantic style and c and d specify Juliet’s style.
An alternative love model, also proposed by Sprott [15], can be written as
where a and b specify Romeo’s romantic style and c and d specify Juliet’s style.
where f(t) and y(t) are external forces.
Kim et al. [16] and Bae [17, 18] demonstrated the existence of chaotic phenomena when f(t) = sin ωt and y(t) = 0 in
In this paper, we investigate the chaotic phenomena occurring in
We investigate the time series and chaotic attractor to discover the nature of chaotic phenomena when we fix the parameters as a = −2, b = −2, c = 1, and d = 1. We also apply different external forces, with y(t) = sin ωt and f(t) = triplus(t, w) with w = 100 for Romeo and Juliet, respectively. In this case, we can consider that the external forces applied for Romeo and Juliet are completely different. This means that different people are modeled by Romeo and Juliet, who have different ways of thinking and opinions.
Figure 1 illustrates the results of the time series and phase portrait when we apply these different external forces for Romeo and Juliet.
We now investigate the chaotic attractor to discover the nature of the chaotic phenomena when we fix the parameters as a = −2, b = −2, c = 1, and d = 1. We also apply different external forces, with y(t) = sinωt and f(t) = rectplus(t,w) with w = 100 for Romeo and Juliet, respectively. Figure 2 illustrates the results of the time series and phase portrait with such parameters.
We now investigate the chaotic attractor to discover the nature of the chaotic phenomena when we fix the parameters as a = −3, b = −2, c = 1, and d = 1. We also apply different external forces, with y(t) = sinωt and f(t) = square(t,w) with w = 50 for Romeo and Juliet, respectively. Figure 3 illustrates the results of the time series and phase portrait with these parameters.
By comparing Figures 1
Now, we investigate the chaotic attractor to discover the nature of the chaotic phenomena when we fix the parameters as a = −3, b = −2, c = 1, and d = 1. We also apply different external forces, with y(t) = sinωt and f(t) = rectplus(t,w) with w = 100 for Romeo and Juliet, respectively. Figure 4 illustrates the results of the time series and phase portrait with these parameters.
By comparing Figures 1 and 4, we can see that there is a clear difference when we vary the parameter a. This means that intensity of love between Romeo and Juliet can vary according to the personality of Romeo. From this result, we infer that the status of love between Romeo and Juliet can be affected by the parameter settings of a, b, c, and d, which represent the states of mind of both participants.
We now investigate the chaotic attractor to discover the nature of the chaotic phenomena when we fix the parameters as a = −3, b = −11, c = 1, and d = 1. We also apply different external forces, with y(t) = triplus(t,w) with w = 50 and f(t) = square(t,w) with w = 100 for Romeo and Juliet, respectively. Figure 5 illustrates the results of the time series and phase portrait with these parameters.
By comparing Figures 1 and 5, we recognize that there is a clear difference when we vary both the parameters and the external forces. Therefore, we conjecture that the love between Romeo and Juliet is affected by many factors, including their own states of mind and current circumstances. Hence, in order to maintain a relationship between a man and woman, both should give their best efforts to maintain their love status.
In this paper, we have proposed a dynamic mathematical model of love, involving different external forces. Furthermore, we investigated the chaotic phenomena occurring in a love model with external forces using time series and phase portraits.
From the time series and phase portraits that we acquired using computer simulation, we were able to recognize two important facts. First, we recognize that there is a clear difference in the results as the external force f(t) is varied. This implies that the status of love between Romeo and Juliet can be affected by external forces such as the opinions of parents, friends, and other family members. Second, there is a clear difference in the results in cases where both the parameters and external forces are changed. Therefore, we conjecture that the love between Romeo and Juliet is affected by many factors, such as their own states of mind and present circumstances. Hence, in order to maintain a relationship between a man and woman, both should give their best efforts to maintain their love status.
The author wishes to acknowledge that the research is done as a part of BK21 Plus Center for Training Top-Level Human Resources in Future Fusion Bio Electronic Medical Technology in Chonnam National University made in the program year 2014.
No potential conflict of interest relevant to this article was reported.
Results of the time series (a) and phase portrait (b) when we apply the external forces as y(t) = sin
Results of the time series (a) and phase portrait (b) when we apply the external forces as y(t) = sin
Results of the time series (a) and phase portrait (b) when we apply the external forces as y(t) = sin
Results of the time series (a) and phase portrait (b) when we apply the external forces as y(t) = sin
Results of the time series (a) and phase portrait (b) when we apply the external forces as y(t) = tripius(t,w),w = 50, and f(t) = square(t,w), w = 100, witha = −3,b = −2, c = 1, and d = 1.
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E-mail: ycbae@chonnam.ac.kr