Int. J. Fuzzy Log. Intell. Syst. 2017; 17(1): 17-25
Published online March 31, 2017
https://doi.org/10.5391/IJFIS.2017.17.1.17
© The Korean Institute of Intelligent Systems
Linyun Huang^{1}, and Youngchul Bae^{2}
^{1}Department of Biomedical and Electronic Engineering, Chonnam National University, Yeosu, Korea, ^{2}Electrical·Electronic Communication and Computer Engineering, Chonnam National University, Yeosu, Korea
Correspondence to :
Youngchul Bae (ycbae@chonnam.ac.kr)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper propose a dynamic mathematical model of love with the Fourier series function as an external force. We also investigate the periodic doubling including 1, 2, 4 periodic motion, chaotic attractor and limit cycle in the love models based on Romeo and Juliet with the Fourier series function as an external force, using time series and phase portraits. To show the nonlinear phenomena using time series and phase portraits, we vary the parameter value with the Fourier series function as an external force.
Keywords: Chaotic phenomena, Nonlinear dynamic, Love model, Time series, Phase portrait, External force, Fourier series function
In general, the human society can model as a complex system. Typically, complex systems have the characteristics of a nonlinear dynamical system, which can describe by a differential equation or difference equation. However, when we compare with linear systems, nonlinear systems are more difficult to analyze, synthesize, implement or design. All natural systems in the world or earth such as wind, weather, and human behavior are nonlinear systems, which include fuzzy systems, neural networks, chaotic systems, and complex systems. Among these nonlinear systems, chaotic systems have gained considerable interest over the last three decades among many researches in the field of mathematics, physics, chemistry, engineering, and social science.
In particular, studies on chaotic behaviors in the social sciences, including those related to habits and the human mind, such as addiction [1–4], happiness [5–8], physical exercise [27] and love models [8–13], involve an overlap of the fields of mathematics, biology, psychology, and social science.
Bae and his associates [1–4] and Kim [27] proposed a mathematical model for addictions to digital leisure, internet, tobacco, and physical exercise. Further, they also verified the presence of nonlinear behaviors or chaotic phenomena by using time series and phase portraits.
Sprott [6, 8] proposed a basic dynamic equation for happiness and investigated its behavior. Bae [7] proposed a mathematical happiness model and a synchronization technique for the model.
Love model have proposed several models including the Romeo and Juliet model [9–13], the Laura and Petrarch model [14, 15], the Adam and Eve model [16], and others [17–21].
Among these, the Romeo and Juliet’s love model is most commonly use in the research of nonlinear dynamics and complex systems.
Strogatz [21, 22] was the first to suggest modelling love affairs by using differential equations; for example, a simple model for the ill-fated romance of Romeo and Juliet modelled as a second-linear system. He also defined love affairs according to parameter values.
Sprott [8] who is inspired by Strogatz’s research [21, 22], proposed a love model based on Romeo and Juliet, which was represented by the linear differential equation; Sprott also described the linear as well as nonlinear behaviors of the model.
The existence of periodic motion and chaotic behavior or motion 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, an external force, and different external forces proposed by Bae [16–21, 25, 26].
Bae and his associates recently proposed external force that compose fuzzy membership function such as Gaussian [28, 29], triangular [30], which close to represent human mind. They shows chaotic behavior according to varying parameter value.
However, most previously published papers in the love model of Romeo and Juliet, sufficiently have not provided to represent human mind through an time series or phase portrait to demonstrate chaotic phenomena or behavior, including periodic motion.
In this paper, we propose a dynamic mathematical model of love with the Fourier series function as an external force. We also investigate the chaotic behaviors including periodic doubling in the love models based on Romeo and Juliet with the Fourier series function as an external force based on varying the parameter value using time series and phase portraits.
There are many love models for Romeo and Juliet; we can classify these models as follows:
Strogatz [21, 22] and Sprott [15] proposed the basic love model for Romeo and Juliet, which can formulate as
where parameter a, and b specify Romeo’s romantic style, and c and d specify Juliet’s style.
Sprott [8] also proposed an alternative love model for Romeo and Juliet, which can be written as
Where, parameter a, b, c and d are same to use for the basic love model.
(
or
or
where f (t) and y(t) are external forces applied to the Romeo and Juliet differential equations, respectively.
Authors [24, 25] demonstrated the existence of chaotic phenomena when f (t) = sin
In this paper, we consider the chaotic behaviors in (
In the following section, the time series and phase portrait for (
First, we investigate the time series and phase portrait to show the behavior of love for Romeo and Juliet, when we set the parameters “a” from −7 to −5.6 in (
We investigate the time series and phase portrait to show the behavior of love for Romeo and Juliet when we set the parameters as a = −5.5, b = −2, c = 1, and d = 1 in (
We now investigate the time series and phase portrait to show the behavior of love for Romeo and Juliet when we set the parameters as a = −5, b = −2, c = 1, and d = 1 in (
We now investigate the chaotic attractor to discover the nature of the chaotic phenomena, when we set the parameters as a = −4.5, b = −2, c = 1, and d = 1, with the Fourier series function in (
Generally chaotic phenomena happens following as order: periodic motion, chaotic motion, periodic motion which is called periodic windows and limit cycle, or periodic motion, chaotic motion and limit cycle. Thus, we can also expect that two different way, periodic motion and chaotic attractor, and limit cycle occur when we apply another parameter value.
We investigate that periodic motion or limit cycle take place when we set the parameters as a = −4, b = −2, c = 1, and d = 1, with the Fourier series function in (
From result of Figures 5 and 6, we can easily expect that next parameter value shows the chaotic motion because of pattern according to order of periodic doubling, chaotic attractor, periodic window and limit cycle.
We investigate that periodic motion or limit cycle take place when we set the parameters as a = −3.0, b = −2, c = 1, and d = 1, with the Fourier series function in (
From result of Figures 5 and 7, we recognize that the 2 chaotic attractors are little different.
We investigate the limit cycle when we set the parameters as a = −2.5, b = −2, c = 1, and d = 1, with the Fourier series function in (
In this paper, we proposed a dynamic mathematical model of love with a Fourier series function as an external force. We also investigated the chaotic behaviors that have periodic doubling and chaotic attractor in love models of Romeo and Juliet with the Fourier series function as an external force, using time series and phase portraits. This paper shows that proposed dynamic mathematical model of love with a Fourier series function as an external force have chaotic behaviors including periodic doubling, chaotic attractor and limit cycle.
No potential conflict of interest relevant to this article was reported.
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
E-mail: hly911121@gmail.com
Tel: +82-61-659-7315, Fax: +82-61-659-7310
E-mail: ycbae@chonnam.ac.kr
Int. J. Fuzzy Log. Intell. Syst. 2017; 17(1): 17-25
Published online March 31, 2017 https://doi.org/10.5391/IJFIS.2017.17.1.17
Copyright © The Korean Institute of Intelligent Systems.
Linyun Huang^{1}, and Youngchul Bae^{2}
^{1}Department of Biomedical and Electronic Engineering, Chonnam National University, Yeosu, Korea, ^{2}Electrical·Electronic Communication and Computer Engineering, Chonnam National University, Yeosu, Korea
Correspondence to: Youngchul Bae (ycbae@chonnam.ac.kr)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper propose a dynamic mathematical model of love with the Fourier series function as an external force. We also investigate the periodic doubling including 1, 2, 4 periodic motion, chaotic attractor and limit cycle in the love models based on Romeo and Juliet with the Fourier series function as an external force, using time series and phase portraits. To show the nonlinear phenomena using time series and phase portraits, we vary the parameter value with the Fourier series function as an external force.
Keywords: Chaotic phenomena, Nonlinear dynamic, Love model, Time series, Phase portrait, External force, Fourier series function
In general, the human society can model as a complex system. Typically, complex systems have the characteristics of a nonlinear dynamical system, which can describe by a differential equation or difference equation. However, when we compare with linear systems, nonlinear systems are more difficult to analyze, synthesize, implement or design. All natural systems in the world or earth such as wind, weather, and human behavior are nonlinear systems, which include fuzzy systems, neural networks, chaotic systems, and complex systems. Among these nonlinear systems, chaotic systems have gained considerable interest over the last three decades among many researches in the field of mathematics, physics, chemistry, engineering, and social science.
In particular, studies on chaotic behaviors in the social sciences, including those related to habits and the human mind, such as addiction [1–4], happiness [5–8], physical exercise [27] and love models [8–13], involve an overlap of the fields of mathematics, biology, psychology, and social science.
Bae and his associates [1–4] and Kim [27] proposed a mathematical model for addictions to digital leisure, internet, tobacco, and physical exercise. Further, they also verified the presence of nonlinear behaviors or chaotic phenomena by using time series and phase portraits.
Sprott [6, 8] proposed a basic dynamic equation for happiness and investigated its behavior. Bae [7] proposed a mathematical happiness model and a synchronization technique for the model.
Love model have proposed several models including the Romeo and Juliet model [9–13], the Laura and Petrarch model [14, 15], the Adam and Eve model [16], and others [17–21].
Among these, the Romeo and Juliet’s love model is most commonly use in the research of nonlinear dynamics and complex systems.
Strogatz [21, 22] was the first to suggest modelling love affairs by using differential equations; for example, a simple model for the ill-fated romance of Romeo and Juliet modelled as a second-linear system. He also defined love affairs according to parameter values.
Sprott [8] who is inspired by Strogatz’s research [21, 22], proposed a love model based on Romeo and Juliet, which was represented by the linear differential equation; Sprott also described the linear as well as nonlinear behaviors of the model.
The existence of periodic motion and chaotic behavior or motion 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, an external force, and different external forces proposed by Bae [16–21, 25, 26].
Bae and his associates recently proposed external force that compose fuzzy membership function such as Gaussian [28, 29], triangular [30], which close to represent human mind. They shows chaotic behavior according to varying parameter value.
However, most previously published papers in the love model of Romeo and Juliet, sufficiently have not provided to represent human mind through an time series or phase portrait to demonstrate chaotic phenomena or behavior, including periodic motion.
In this paper, we propose a dynamic mathematical model of love with the Fourier series function as an external force. We also investigate the chaotic behaviors including periodic doubling in the love models based on Romeo and Juliet with the Fourier series function as an external force based on varying the parameter value using time series and phase portraits.
There are many love models for Romeo and Juliet; we can classify these models as follows:
Strogatz [21, 22] and Sprott [15] proposed the basic love model for Romeo and Juliet, which can formulate as
where parameter a, and b specify Romeo’s romantic style, and c and d specify Juliet’s style.
Sprott [8] also proposed an alternative love model for Romeo and Juliet, which can be written as
Where, parameter a, b, c and d are same to use for the basic love model.
(
or
or
where f (t) and y(t) are external forces applied to the Romeo and Juliet differential equations, respectively.
Authors [24, 25] demonstrated the existence of chaotic phenomena when f (t) = sin
In this paper, we consider the chaotic behaviors in (
In the following section, the time series and phase portrait for (
First, we investigate the time series and phase portrait to show the behavior of love for Romeo and Juliet, when we set the parameters “a” from −7 to −5.6 in (
We investigate the time series and phase portrait to show the behavior of love for Romeo and Juliet when we set the parameters as a = −5.5, b = −2, c = 1, and d = 1 in (
We now investigate the time series and phase portrait to show the behavior of love for Romeo and Juliet when we set the parameters as a = −5, b = −2, c = 1, and d = 1 in (
We now investigate the chaotic attractor to discover the nature of the chaotic phenomena, when we set the parameters as a = −4.5, b = −2, c = 1, and d = 1, with the Fourier series function in (
Generally chaotic phenomena happens following as order: periodic motion, chaotic motion, periodic motion which is called periodic windows and limit cycle, or periodic motion, chaotic motion and limit cycle. Thus, we can also expect that two different way, periodic motion and chaotic attractor, and limit cycle occur when we apply another parameter value.
We investigate that periodic motion or limit cycle take place when we set the parameters as a = −4, b = −2, c = 1, and d = 1, with the Fourier series function in (
From result of Figures 5 and 6, we can easily expect that next parameter value shows the chaotic motion because of pattern according to order of periodic doubling, chaotic attractor, periodic window and limit cycle.
We investigate that periodic motion or limit cycle take place when we set the parameters as a = −3.0, b = −2, c = 1, and d = 1, with the Fourier series function in (
From result of Figures 5 and 7, we recognize that the 2 chaotic attractors are little different.
We investigate the limit cycle when we set the parameters as a = −2.5, b = −2, c = 1, and d = 1, with the Fourier series function in (
In this paper, we proposed a dynamic mathematical model of love with a Fourier series function as an external force. We also investigated the chaotic behaviors that have periodic doubling and chaotic attractor in love models of Romeo and Juliet with the Fourier series function as an external force, using time series and phase portraits. This paper shows that proposed dynamic mathematical model of love with a Fourier series function as an external force have chaotic behaviors including periodic doubling, chaotic attractor and limit cycle.
Fourier series function.
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Linyun Huang, Suk-Seung Hwang, and Youngchul Bae
Int. J. Fuzzy Log. Intell. Syst. 2016; 16(4): 262-269 https://doi.org/10.5391/IJFIS.2016.16.4.262Youngchul Bae
Int. J. Fuzzy Log. Intell. Syst. 2015; 15(4): 283-288 https://doi.org/10.5391/IJFIS.2015.15.4.283Youngchul Bae
Int. J. Fuzzy Log. Intell. Syst. 2016; 16(1): 64-71 https://doi.org/10.5391/IJFIS.2016.16.1.64Fourier series function.
|@|~(^,^)~|@|Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (
Results of time series (a) and phase portrait (b) when (