Effects of random noise in a dynamical model of love

Chaos, Solitons & Fractals 44 (2011) 490–497

Contents lists available at ScienceDirect

Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Effects of random noise in a dynamical model of love Yong Xu ⇑, Rencai Gu, Huiqing Zhang Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China

a r t i c l e

i n f o

Article history: Received 26 April 2010 Accepted 29 March 2011 Available online 28 May 2011 Keywords: Dynamical model of love Gaussian white noise The top Lyapunov exponent Steady-state probability density P-bifurcation Control

a b s t r a c t This paper aims to investigate the stochastic model of love and the effects of random noise. We first revisit the deterministic model of love and some basic properties are presented such as: symmetry, dissipation, fixed points (equilibrium), chaotic behaviors and chaotic attractors. Then we construct a stochastic love-triangle model with parametric random excitation due to the complexity and unpredictability of the psychological system, where the randomness is modeled as the standard Gaussian noise. Stochastic dynamics under different three cases of ‘‘Romeo’s romantic style’’, are examined and two kinds of bifurcations versus the noise intensity parameter are observed by the criteria of changes of top Lyapunov exponent and shape of stationary probability density function (PDF) respectively. The phase portraits and time history are carried out to verify the proposed results, and the good agreement can be found. And also the dual roles of the random noise, namely suppressing and inducing chaos are revealed. Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction Nonlinear dynamics, especially chaos and its applications have been concerned and applied in many different practical fields such as physics, chemistry, electricity and biology even in psychology of social sciences [1,2]. Nonlinearity is the essential character in the social and life system and psychology is a science of researching psychological phenomenon and behavior which is an important field of structuring epistemology. Love affairs, one kind of typical psychology activities can be described as an evolving process from gestation, found, development, maturation to variation (or wither) [3,4], that can be modeled as a series of ordinary differential equations [5–10]. In [5], Strogatz investigated the love affair between Romeo and Juliet by a series of simple linear ordinary differential equations, which can be written as dR dt dJ dt

¼ aR þ bJ; ¼ cR þ dJ;

⇑ Corresponding author. E-mail address: [email protected] (Y. Xu).

ð1Þ

with R(t) the Romeo’s love (or hate if negative) for Juliet at time t and J(t) the Juliet’s love for Romeo. The parameters a and b specify Romeo’s ‘‘romantic style’’ respectively, and c, d specify Juliet’s ‘‘romantic style’’. Here a denotes the extent to which Romeo is encouraged by his own feelings, and b is the extent to which he is encouraged by Juliet’s feelings. Obviously the love model (1) is definitely unrealistic because it does not take into account the appeal of the two individuals. Thus, Strogatz’s model does not explain why two persons who are initially completely indifferent to each other can develop a love affair. We also note that the system (1) has never been followed on social psychology. Some more realistic cases were considered in [9,10], and a minimal model was given as two ordinary differential equations which had taken into account three mechanisms love growth and decay: the pleasure of being loved (return), the reaction to the partner’s appeal (instinct), and the forgetting processes (oblivion). Their models only considered the linear relationships of two persons but the relationships of two persons involved should be nonlinear. In [11] a series of love models were provided to describe the time-variation of the love (or hate) displayed by individuals involved in a romantic relationship. In [12,13], fractional-order models and time delay models

0960-0779/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2011.03.009

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Y. Xu et al. / Chaos, Solitons & Fractals 44 (2011) 490–497

were considered respectively due to the fact that fractional systems and time delay systems could possess memory. However, the effects of random noise and the modified random models of love-triangle have not been reported in the literature. In this paper we are devoted to constructing the new stochastic model of love-triangle and exploring the random dynamics of the proposed systems including the D-bifurcation and P-bifurcation. This paper is organized as follows: Section 2 is to visit the deterministic systems of love-triangle and basic properties of system will be presented. In Section 3, the random version of love-triangle system will be presented and the corresponding stochastic dynamics are examined numerically, where the D-bifurcation will be observed by the means of top Lyapunov exponent and P-bifurcation via the changes of stationary probability density function. Finally, Section 4 presents the concluding remarks to close this paper.

(c = 7 and d = 2). System (2) has the following basic dynamical properties: (1) Symmetry: Symmetry about the origin (0, 0, 0, 0). If (Rj0, J0, Rg0, G0) is a solution of Equation (2) then (Rj0, J0, Rg0, G0) must be a solution too. (2) Dissipation: The divergence of (2) is:

rF ¼

dG dt

ð3Þ

As long as

2a þ d þ f ¼ 5 < 0;

ð4Þ

then the love-triangle system (2) is dissipative. (3) Equilibrium and their stability: The equilibrium of (2) i can be calculated by solving the equations dR ¼ 0, dt dRG dJ dG ¼ 0, dt ¼ 0 and dt ¼ 0 as dt



dRJ ¼ aRJ þ bðJ  GÞð1  jJ  GjÞ; dt dJ ¼ cRJ ð1  jRJ jÞ þ dJ; dt dRG ¼ aRG þ bðG  JÞð1  jG  JjÞ; dt

    dJ dRG dG RJ þ G RJ þ RG þ dt dt dt

¼ 2a þ d þ f :

E¼

2. Deterministic love-triangle system This section is to investigate the deterministic system and the basic properties of this system. And the chaotic dynamics will be observed later. The model of love-triangle in [11] can be written as

dRJ dt

 c e x;  xð1  jxjÞ; x; xð1  jxjÞ ; d f

where

      de þ cf    jxj  j1  jxjj  adf ¼ 0: x bðde þ cf Þð1  jxjÞ 1    df ð5Þ

ð2Þ

¼ eRG ð1  jRG jÞ þ fG;

where the logistic function y = kx(1  |x|) was introduced and the same notations as Eq. (1). This model describes a ‘love-triangle’, in which Romeo (R) is involved in romantic relationships with Juliet (J) and Guinevere (G), where RJ is Romeo’s feeling for Juliet and l is Romeo’s feeling for Guinevere. The model parameters specify the romantic styles of the parties involved: a and b specify Romeo’s romantic style, while (c, d) and (e, f) specify Juliet’s and Guinevere’s romantic styles respectively. For simplicity, it is assumed that Juliet and Guinevere would not know about one another, and that Romeo would exhibit the same romantic style towards them both (as determined by the parameters a, b). The signs of the parameters determine the type of romantic style, which are quoted here from [11]:  Eager beaver: a > 0, b > 0 (Romeo is encouraged by his own feelings as well as Juliet’s).  Narcissistic nerd: a > 0, b < 0 (Romeo wants more of what he feels, but retreats from Juliet’s feelings).  Cautious (secure) lover: a < 0, b > 0 (Romeo retreats from his own feelings but is encouraged by Juliet’s).  Hermit: a < 0, b < 0 (Romeo retreats from his own feelings as well as Juliet’s). Now we observe the dynamical properties of love-triangle system including symmetry, dissipation, fixed points (equilibrium), chaotic behaviors and chaotic attractors in the case of a cautious lover Romeo (a = 3 and b = 4) and Guinevere (e = 2 and f = 1) and a narcissistic nerd Juliet

So one can define the roots of (5) as x0 = 0, x1,2 = ±0.186, x3,4 = ±.2335, and then five equilibrium of (2) can be given as E1 = E2 = (0.186, 0.53, 0.186, 0.303), E3 = E4 = (1.2335, 1.001, 1.2335,0.567), E0 = (0, 0, 0, 0). The coefficient matrix of the corresponding linearized system is:

2

a bð12jJGjÞ 6 cð12jR jÞ d J 6 A¼ 6 4 0 bð12jJGjÞ 0

0

0 0 a eð12jRG jÞ

bð12jJGjÞ

3

7 7 7 bð12jJGjÞ 5 0 f ð6Þ

The eigenvalues of matrix A at E0 are: k1 ¼ a ¼ 3:0,k2 ¼ 2:386,k3;4 ¼ 0:193  3:985i, as Reðk3;4 Þ > 0, E0 is unstable equilibrium point. Similarly, for equilibrium point E1 and E2, the eigenvalues can be written as: k1 ¼ 3:0, k2 ¼ :5975, k3 ¼ 3:8254, k4 ¼ 1:7721, and as k2 > 0, E1 and E2 are unstable saddle point. For equilibrium point E3 and E4, the eigenvalues are: k1 ¼ 3:0, k2;3 ¼ 0:1312 þ 7:6937i,k4 ¼ 2:2624, AsReðk2;3 Þ > 0,E3 and E4 are unstable saddle point too. (4) Chaotic attractors: This system can exhibit chaos with strange attractors, an example of which appears in Fig. 1. This case represents a cautious lover Romeo (a = 3 and b = 4) and Guinevere (e = 2 and f = 1) and a narcissistic nerd Juliet (c = 7 and d = 2), where the top Lyapunov exponent is 0.38, and the Kaplan–Yorke dimension is 2.026. In Fig. 2 it illustrates the time evolution of Juliet’s love for Romeo and for the same case as Fig. 2 but with two initial conditions that are identical except that Romeo’s love for Juliet differs by 1%.

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Y. Xu et al. / Chaos, Solitons & Fractals 44 (2011) 490–497

0.5

3

2

0

0

λ1

J

1

-1

-0.5 -2

-3 -3

-2

-1

0

1

2

3

-1 -4

RJ

-3.5

-3

-2.5

a Fig. 1. Strange attractor from the nonlinear love triangle in Eq. (2). Fig. 3. The top Lyapunov exponent of system (2) versus parameter a.

2.5 1.2

2 1.5

1

1 0.8

0

0.6

λ1

RJ

0.5

-0.5

0.4

-1 -1.5

0.2

-2 0 -2.5

0

20

40

60

80

100

t Fig. 2. Chaotic evolution of Romeo’s love for Juliet from Eq. (2) showing the effect of changing the initial conditions by 1%.

Fix b = 4.0, c = 7.0, d = 2.0, e = 2.0 and f = 1.0, as 3.272 < a < 2.926, the system exist chaotic attractors. The top Lyapunov exponent of system (2) versus the parameter a is presented in Fig. 3. One can find the dynamic evolution varies as the parameter changes. In fact, the system exhibits stable dynamic behaviors in the certain range of parameter a, and as the increase of the parameter a the system will change from the stable behaviors to chaotic dynamics by the criteria of the sign of top Lyapunov exponent. Namely, the system demonstrates the first bifurcation versus the parameter a. Later, when we increase the values of parameter a the system will evolve from the chaos order to the stable behaviors, and this states the second bifurcation. Similarly, fix other parameters we can get the value range of the sixth parameter where system (2) is a chaotic system. Actually, the regions of parameter space that appear chaos are relatively

-0.2

0

0.02

0.04

0.06

0.08

0.1

m Fig. 4. The top Lyapunov exponent of System (6) versus m.

small, and sandwiched between cases that generate limit cycles and unbounded solutions. 3. Stochastic love-triangle system with parametric noise In this section we wish to investigate the random model of love-triangle and stochastic dynamics are to be considered. Real systems and artificial systems are inevitably subject to effects of uncertain factors [14–18]. For example, the ecosystem and biotic community will be influenced by temperature, climate and other natural conditions [19]. Different neurons have to accept the external common signal’s stimulation in the nervous system. In real life, mental activities will also be subject to some random factors which could intensify the complexity of mental activities. In fact, many factors may affect the evolution of love, including physical fatigue, illness, the weather and some

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Y. Xu et al. / Chaos, Solitons & Fractals 44 (2011) 490–497 -1.0425

b

-1.0435 -1.044

-1.0445

-1.0445

-1.0455

-1.046

-1.046

-1.0465

-1.0465

-1.047

-1.047 1.239

1.24

1.241

1.242

2

1

-1.045

-1.0455

-1.0475 1.238

-2

1.239

RJ

1.24

1.241

1.242

-3 -3

1.243

3

e

2.5

f

1

1.5

1

0

1

0

J

2

-1

0.5

-1

-2

0

-2

-1

0

1

2

-0.5 -2

3

-1.5

RJ

0

1

2

3

1

2

3

3

2

-2

-1

RJ

2

-3 -3

-2

RJ

J

J

d

0

-1

-1.0475 1.238

1.243

3

-1.043

-1.044

-1.045

c

-1.0425

-1.0435

J

J

-1.043

J

a

-1

-0.5

0

-3 -3

-2

-1

0

RJ

RJ

Fig. 5. Phase plot for system when a = 3.4 (a) m = 0; (b) m = 0.00001; (c) m = 0.0008; (d) m = 0.00347;(e) m = 0.02; (f) m = 0.021.

unexpected events such as wining the lottery or occurring an accident [20–22]. However, the factors which have impacts on evolution of love are stochastic. We assume that ‘‘good’’ and ‘‘bad’’ factors are equally likely. We can model standard Gaussian white noise m with mean zero to express the random factors at the time of m. For simplify, we only consider Romeo’s ‘‘romantic style’’ a which is affected by random factors, and then the corresponding random model is introduced as

3.1. (I) Case of a = 3.4

0.04

ð6Þ

In this case, we describe Romeo’s ‘‘romantic style’’ as a = 3.4, and Fig. 4 is a variation curve of the top Lyapunov exponent k1 versus excitation amplitude m. The phase diagrams of system (6) under different excitation amplitudes

b

0.04 0.03

p(RJ ,J )

0.06

p(RJ , J)

dRJ ¼ að1 þ mnðtÞÞRJ þ bðJ  GÞð1  jJ  GjÞ; dt dJ ¼ cRJ ð1  jRJ jÞ þ dJ; dt dRG ¼ að1 þ mnðtÞÞRG þ bðG  JÞð1  jG  JjÞ; dt dG ¼ eRG ð1  jRG jÞ þ fG: dt

a

Using stochastic order-4 Runge–Kutta algorithm [23] for numerical integration to system (6), we take initial conditions with RJ(0) = RG(0) = 0.001, J(0) = G(0) = 0.001, t0 = 0 and integration step Dt = 0.01. We apply Benettin method [24] to calculate the top Lyapunov exponent k1 of the system. In the next discussions, we take the following parameters b = 4, c = 7, d = e = 2, f = 1 and consider three different cases of a = 3.4, 2.93, 3.0 respectively.

0.02

0.02 0.01

0 0

0

2 0

4 6

-4

5 0

0

-6

8

RJ

5

-2

10

-8 -10

J

-5

J

-5

Fig. 6. The stationary probability density of RJ and J: (a) m = 0.02; (b) m = 0.021.

RJ

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Y. Xu et al. / Chaos, Solitons & Fractals 44 (2011) 490–497

2

1

b

1.8

0.9

1.6

0.8

1.4

0.7

1.2

0.6

p(RJ )

p(RJ )

a

1

0.5

0.8

0.4

0.6

0.3

0.4

0.2

0.2

0.1

0 -2.5

-2

-1.5

-1

-0.5

0

0 -3

-2

-1

0

1

2

3

RJ

RJ

Fig. 7. The stationary marginal probability density of RJ :(a) m = 0.02; (b) m = 0.021.

1

30%

0.8 0.6 20%

λ1

0.4 0.2 10%

0 -0.2 0

0

0.01

0.02

0.03

0.04

0.05

m Fig. 8. The rate of RJ > 0, J > 0 with m.

are displayed in Fig. 5. It can be seen that the system appears rich dynamical behaviors such as stable steady state, periodic state, chaotic and non-chaotic state in the process of m increasing which will be depicted as follows. Firstly without noise effects, i.e., the noise intensity m zero, it can be seen from Fig. 4(a) that the solutions of Eq. (6) move exponentially toward the focus s (1.2407, 1.045, 1.2407, 0.597), and then Fig. 4 (b) shows the solutions fluctuate in the small neighborhood of s as m = 0.00001. When m = 0.0008, the shape of attractor have a large change and the basin of attraction is no longer a small neighborhood of s. Beside, the system is in a large scale periodic state and the attractor is symmetric about its center, which is depicted in Fig. 4(c). With further increase of excitation amplitude m, the structure of attractors will change once again. From Fig. 4(e) and (f) one can find the attractor is in the region of (3  0, 0  3) with m = 0.02, while the basin of attraction extends to

-0.4

0

0.002

0.004

0.006

0.008

0.01

m Fig. 9. The top Lyapunov exponent of system (6) with m as a =  2.93.

(3  3, 3  3) as m = 0.021. Using Monte Carlo method, we can get the joint stationary probability density function (PDF) of state variables RJ and J, as shown in Fig. 6. The corresponding marginal probability density is given in Fig. 7. From Fig. 6 we can observe the changes of shape of PDF, namely, as m = 0.02 only one peak in the curve of the probability density, and then when m = 0.021, two peaks happen in Fig. 7, that implies probability bifurcation versus the bifurcation parameter, i.e., the excitation amplitudes m vary from 0.02 to 0.05. Corresponding to psychology, we find an interesting love affair. Romeo loves Juliet but hates Guinevere while Juliet doesn’t love him while Guinevere does, and then the relationships are stable. With the impacts of random factors, which are modeled as the Gaussian white noise, the relationships between Romeo and Juliet (RJ > 0, J > 0) appear fluctuation gradually when the strength of random

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Y. Xu et al. / Chaos, Solitons & Fractals 44 (2011) 490–497 3

b

2

0

J

J

1

-1 -2 -3 -3

-2

-1

0

1

2

c

25

20

15

15

10

10

5

5

0

-5

-10

-10

-15

-15

-20

-20 -10

-5

RJ 40

e 50

30

40

10

15

-25 -20 -15

20

-10

-5

0

5

10

15

20

RJ

f

60 40

20

10

20

10

0

J

J

5

30

20

0 -10

-10

-30

-30

0 -20

-20

-20

-40 -30

0

RJ

J

d

0

-5

-25 -20 -15

3

25

20

J

a

-40

-40 -20

-10

0

10

20

-50 -40 -30

30

-20

-10

RJ

0

10

20

30

40

-60 -50

0

50

RJ

RJ

Fig. 10. Phase plot for system when a = 2.93 (a) m = 0.0; (b) m = 0.0002; (c) m = 0.0005; (d) m = 0.05; (e) m = 0.12; (f) m = 0.20.

2.5

0.06

m=0.05 m=0.12 m=0.20

0.05

2

0.04

λ1

J

p(R )

1.5 0.03

1 0.02

0.5

0.01

0

-40

-20

0

20

40

60

RJ Fig. 11. The stationary marginal probability density of RJ with different m.

force meet a certain level. However, in most of the time, either Romeo loves Juliet but Juliet hates him or Romeo hates Juliet but Juliet loves him, there is no love between them. But the opportunity of Romeo and Juliet will increase with the noise intensity of random noise increases (see Fig. 8). It seems to imply that a certain strength of random force could guide people who can’t love originally love each other. In this sense, random factors are positive to our lives. From 6 or 7, we can also find that a slight change of random factors may lead to a great transition of someone’s feeling, which conforms to our real life.

0

0

0.05

0.1

0.15

0.2

m Fig. 12. The top Lyapunov exponent of system (6) with m as a =  3.0.

3.2. (II) Case of a = 2.93 We figure the top Lyapunov exponent k1 of system (6) with excitation amplitudes m in Fig. 9 when the parameter is taken as a = 2.93. From Fig. 9 one can find that the system converts from chaotic order to periodic states via the means of the signs of the top Lyapunov exponent with the changes of the noise intensity m and then it changes from periodic states to aperiodic states as excitation amplitude m increases. Phase portraits of Fig. 10 agree with the conclusion quite well. If m = 0, the system is chaotic and its

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Y. Xu et al. / Chaos, Solitons & Fractals 44 (2011) 490–497

a

c

b 25

3

20 2

30

15 20

10

1

10

0

J

0

J

J

5

0

-5 -1

-10

-10

-15

-2

-20

-20 -3 -3

40

-2

-1

0

1

2

3

-25 -20

-15

-10

-5

RJ

0

5

10

15

20

-30 -30

-20

-10

RJ

0

10

20

30

RJ

Fig. 13. Phase plot for system when a = 3.0 (a) m = 0.006; (b) m = 0.01; (c) m = 0.06.

changes in a gradual manner in case (II) while changes suddenly in case (I).

0.12

m=0.02 m=0.06 m=0.10 m=0.16 m=0.25

p(R ) J

0.1

0.08

4. Concluding remarks

0.06

0.04

0.02

0 -50

0

50

R J Fig. 14. The stationary marginal probability density of RJwith different m.

top Lyapunov exponent is 0.39 (see Fig. 10.(a)). As m = 0.0002,k1 ¼ 0, the system is in periodic states. Fig. 11 show marginal probability density of m = 0.05, 0.12, 0.20, from which we can see that, the shape of probability density changed greatly with m varying, from crater to singlepeak type, besides, the minimum point evolved into a maxima point at RJ = 0. 3.3. (III) Case of a = 3.0 The state of m = 0.0 have been discussed adequately in Section 2 in the case of a = 3.0. Fig. 12 shows the top Lyapunov exponent with m and Fig. 13 shows RJ  J phase portraits of different excitation amplitude. It is seen that there is no case of k1 < 0 appears in the process of m varying. The system will stay in chaotic state and the basin of attraction is almost invariant until m increased to 0.006. Then, the basin of attraction will expand and the system will become instability at last if m reached to a certain value. The stationary marginal probability density of RJ is given in Fig. 14 which is similar with Fig. 11 in case (II). However, comparing with case (I), there is some obvious difference exists. The stationary probability density

In this paper, the aim of this work is to create interest and spark research effort in the field of ‘psychology and life sciences’, where stochastic modeling might offer more insights towards understanding the dynamical behaviors of these systems. We constructed and considered a love-triangle model with parametric random excitation. The deterministic model love-triangle was revisited by discussing basic properties including fixed points and chaotic dynamical behaviors. The phase portrait and time history are carried out to agree with the results of top Lyapunov exponents. Furthermore, the range of the interested parameters is presented numerically in which the system demonstrates chaotic state. To analyze the more realistic love and to reflect the impacts of random noise, a new love model of stochastic version is established by using the standard Gaussian white noise to model the stochastic effects. Under different three cases of ‘‘Romeo’s romantic style’’, stochastic dynamics are investigated and the dynamic and probability bifurcation are observed versus the bifurcation parameter namely the noise intensity m. Also the phase portraits, the top Lyapunov exponent and the steady-state probability density are explored and one can find rich random dynamics of the stochastic model of love-triangle. For example, in the case 1, as m = 0.02, there is only one peak in the curve of the probability density, but if m = 0.021, there are two peaks, so it implies that random noise induces a Pbifurcation. From another viewpoint, we can say appropriate noise intensity can change the dynamic evolution of a system, where it can induce chaotic behaviors, or convert the periodic attractor into chaotic attractor, or generate chaotic order. While random noise can also tame the chaotic behaviors and convert a chaotic attractor to a periodic attractor. Thus, the dual roles of noise which are taming or inducing chaos in nonlinear dynamical systems are revealed. Meanwhile, the Lyapunov exponent curve could reflect the system states accurately when the noise excitation amplitude is small and phase portraits and steady stationary probability density function are applied to

Y. Xu et al. / Chaos, Solitons & Fractals 44 (2011) 490–497

verify the proposed results, and excellent agreements can be found right now. In addition, one can find some people are more sensitive to random factors so that whose psychology should deserve more attentions. In this report we consider three different kinds of lovers (with different ‘‘romantic style’’ a) excited by the random noise, and one may see that different lovers have different reaction to random factors, which may play a key role in some people’s love affairs (or some other relationships between two or more people). Acknowledgements This work was supported by the NSF of China (10972181), NPU Foundation for Fundamental Research (2007A10), and Aoxiang Star Plan of NPU. References [1] Susan A. The Application of Chaos Theory to Psychology , Theory Psychology, 1997;7(3):373–398 DOI:10.1177/0959354397073005. [2] Robin Robertson, Chaos Theory and the relationship between psychology and science, 3–15. In: Robin Robertson, Allan Combs, editors. Chaos Theory in psychology and the life sciences, Lawrence Erlbaum Associates, Publishers, 1995 Mahwah, New Jersey Hove, UK. [3] Terry M. The self as a dynamical system. Nonlinear Dynam Psychol Life Sci 1999;3:311–45. [4] Li XP. Nonlinear science and its application in psychology. J Nanjing Normal University (Social Science) 2005;2:84–8. [5] Strogatz SH. Love affairs and differential equations. Math Mag 1988;61:35. [6] Gottman JM. The mathematics of marriage dynamic nonlinear models. The MIT Press Massachusetts; 2002. [7] Sprott JC. Can a monkey with a computer create art? Nonlinear Dynam Psychol Life Sci 2004;1:103–14.

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