Stochastic nonlinear dynamics of interpersonal and romantic relationships

Applied Mathematics and Computation xxx (2011) xxx–xxx

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Stochastic nonlinear dynamics of interpersonal and romantic relationships

Kamal Barley, Alhaji Cherif ⇑School of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287, United States

a r t i c l e i n f o a b s t r a c t

Keywords:Dyadic relationalStochastic resonanceSustained oscillationMathematical sociologySocial psychology

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2010.12.117

⇑ Corresponding author. Present address: Centre foUnited Kingdom.

E-mail addresses: [email protected] (K. Barl

Please cite this article in press as: K. Barley, AMath. Comput. (2011), doi:10.1016/j.amc.201

Current theories from biosocial (e.g., the role of neurotransmitters in behavioral features),ecological (e.g., cultural, political, and institutional conditions), and interpersonal (e.g.,attachment) perspectives have grounded interpersonal and romantic relationships in nor-mative social experiences. However, these theories have not been developed to the point ofproviding a solid theoretical understanding of the dynamics present in interpersonal andromantic relationships, and integrative theories are still lacking. In this paper, mathemat-ical models are used to investigate the dynamics of interpersonal and romantic relation-ships, via ordinary and stochastic differential equations, in order to provide insight intothe behaviors of love. The analysis starts with a deterministic model and progresses to non-linear stochastic models capturing the stochastic rates and factors (e.g., ecological factors,such as historical, cultural and community conditions) that affect proximal experiencesand shape the patterns of relationship. Numerical examples are given to illustrate variousdynamics of interpersonal and romantic behaviors with particular emphases placed on sus-tained oscillations and transitions between locally stable equilibria that are observable instochastic models (closely related to real interpersonal dynamics), but absent in determin-istic models.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Interpersonal relationships appear in many contexts, such as in family, kinship, acquaintance, work, and clubs, to name afew. The manifestation of interpersonal relationships in society comes in many forms ranging from romantic, parent–child,friendships, comradeship, casual, friend-with-benefits, soul-mates, dating to more recently Internet relationships. The mostintriguing of all these interpersonal relationships, which is also a dominant phenomenon and fundamental in human sociallife and interaction, is romantic relationship [1,2].

Romantic relationships refer to the mutually ongoing interactions among two or more individuals. Recent works showthat romantic relationships are more common among adolescents than has previously been presumed, with more than halfof adolescents in the United States being involved in some form of romantic relationships [3,4]. More than 70% of high schooland college students report having had a special romantic relationship in the previous years, and also report more frequentinteractions with romantic partners than with parents, siblings, and/or friends [4,5]. For adult, the study of romantic behav-iors may provide invaluable insight of why majority of romantic relationships fail or do not make it to engagement and/ormarriage [6]. Surra and Hughes [7] found that more than half (54%) of couples in their studies exhibit unpredictable and non-linear relational trajectories involving large number of turning and tipping points. Partners identified events such as newrivals, unresolved differences, meeting partners family, and job changes as turning and tipping points that greatly changed

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r Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford OX1 3LB,

ey), [email protected], [email protected], [email protected] (A. Cherif).

. Cherif, Stochastic nonlinear dynamics of interpersonal and romantic relationships, Appl.0.12.117

2 K. Barley, A. Cherif / Applied Mathematics and Computation xxx (2011) xxx–xxx

and influenced the nature, quality and success of the relationship. Similar findings by other scholars have appeared in var-ious works [3,4,8,9].

Scholarly works on romantic relationships among adolescents have gained traction with emphases on the quality of rela-tionships and their potential implications for positive and negative developmental and socio-psychological outcomes. The-ories in biosocial (e.g., effect of neurotransmitters in behavioral features, [10,11]), ecological (e.g., cultural, political, andinstitutional conditions, [12]), and interpersonal (e.g., attachment, [13,14]) studies have grounded romantic relationshipsin normative social experiences and paradigms. However, these theories have not been developed to the point of providinga solid theoretical understanding of the various dynamics present in romantic relationships. But integrative theories are stilllacking. The study of relationships has begun to hold both the artistic imaginations and interdisciplinary intellectual inter-ests of various scholars in the fields of sociology, biology, neuroscience, psychology, anthropology, and mathematics [15–29,33–38]. Since experiments in these areas are difficult to design and may be constrained by ethical considerations, math-ematical models can play a vital role in studying the dynamics of relationships and their behavioral features. However, thereare few mathematical models capturing the various dynamics of romantic relationships. In this paper, we study both deter-ministic and stochastic models of relationship from interpersonal perspective.

Deterministic differential equations have been used extensively to study dynamic phenomena in a wide range of fields,ranging from physical, natural, biological to social sciences. The mathematical models capturing the dynamics of love be-tween two people have recently gained attention among many researchers [20–28] who have provided extensions to Stro-gatz’s seminal model. In a one-page influential work [15] and later in a book [16], Strogatz applied a system of lineardifferential equations to study Shakespearean model of love affair of Romeo and Juliet. Rinaldi [20,21], Sprott [23], Liaoand Ran [24] and Wauer et al. [28] have investigated realistic perturbations and extensions of Strogatzian model by includingfeatures such as attraction factor [20,22,24,28], delay and nonlinear return functions [28], and three-body love affairs or lovetriangles [23,28]. Rinaldi investigated the three mechanisms of love dynamics: instinct, return, and oblivion in [22], makingthe model more realistic because it accounts for the growth of feeling from a state of indifference. In [20], Rinaldi proposed athree dimensional model to describe the cyclical love dynamics of Laura and Patriarch and introduced nonlinear return andoblivion functions, and poems written by Patriarch are used to validate the dynamics. Gottman et al [27] employed discretedynamical models to describe the interaction between married couples; Liao and Ran [24] studied time delays, nonlinearcoupling and Hopf bifurcation conditions. Recently, Wauer et al. [28] examined various models, starting with a time-invari-ant two dimensional linear and nonlinear models and concluding with time-dependent fluctuations in the source-terms andparameters, where the impact of additive random effects, parametric excitation and periodic source terms were investigated.In previous papers [20,21,23–28], dyadic and triadic interactions are also considered, and other effects such as personalitiesand differential appeals of individuals are ignored. As a result, learning and adaptation processes are ruled out. In this paper,we investigate stochastic dynamical models. But first, we summarize previous deterministic models [15,17,20] before devel-oping an equivalent stochastic model.

2. Models and stability analysis

In this section, we study two models with two state variables. The variables X1 and X2 are the measures of love of indi-vidual 1 and 2 for their respective partners, where positive and negative measures represent positive (e.g., friendship, pas-sionate, intimate) and negative (e.g., antagonism and disdain) feelings, respectively. We first propose a deterministic systemof differential equations to model the dynamic of romantic relationship, and later naturally extend the model to a stochasticdynamic model, where the deterministic rates become the stochastic rates. Using the typology of Strogatz [15,16] and Sprott[23], the four romantic styles are summarized below:

(i) Eager Beaver: individual 1 is encouraged by his own feelings as well as that of individual 2 (ai > 0 and bi > 0).(ii) Secure or Cautious lover: individual 1 retreats from his own feelings but is encouraged by that of individual 2 (ai < 0

and bi > 0).(iii) Hermit: individual 1 retreats from his own feelings and that of individual 2 (ai < 0 and bi < 0).(iv) Narcissistic Nerd: individual 1 wants more of what he feels but retreat from the feelings of individual 2 (ai > 0 and

bi < 0).

This classification allows us to characterize the dynamics exhibited by various combinations of different romantic styles.Previous papers [16,17,20,21,23] have considered various dynamics using all possible combinations in the sign of parametersai and bi. In this paper, we focus primarily on dynamics observed when individuals in region (ii) interact with individuals inregion (iii). We later focus on the features of stochastic model, that are not present in the deterministic dynamics [17]. Forinstance, Cherif [17] numerically showed that stochastic models of romantic relationships can exhibit exotic dynamics suchas sustained oscillations (e.g., stochastic resonance) and diffusion of trajectories (e.g., multiple crossing of stability regionscontaining stable equilibria) while deterministic models do not and may only show damped oscillations. In this section,we proceed as follows: deterministic (linear and nonlinear) models are provided, and are then followed by their stochasticequivalents. In the Deterministic Model section, we use standard deterministic linear romantic dynamic model [16] and non-linear model provided in [17]. In Stochastic Dynamic Models, stochastic versions of the models studied in Section 2.1 are

Please cite this article in press as: K. Barley, A. Cherif, Stochastic nonlinear dynamics of interpersonal and romantic relationships, Appl.Math. Comput. (2011), doi:10.1016/j.amc.2010.12.117

K. Barley, A. Cherif / Applied Mathematics and Computation xxx (2011) xxx–xxx 3

investigated with special emphasis on behaviors not observed in the deterministic dynamics. We then provide an extension,where differential parameters are considered.

2.1. Deterministic models

In [17], a more general model was proposed. In this paper, we focus on dyadic relational dynamics characterized by (ii)and (iii) interactions (e.g., secure or cautious love dynamics). To model the behavioral features of romantic dynamics, thefollowing deterministic model is proposed:

PleaseMath.

dX1

dt¼ �a1X1 þ b1X2 1� eX2

2

� �þ A1 ð1Þ

dX2

dt¼ �a2X2 þ b2X1 1� eX2

1

� �þ A2 ð2Þ

for (X1,X2) 2 R � R, where ai > 0 is non-negative, and bi and Ai i = 1, 2 are real constant, respectively. These parameters areoblivion, reaction and attraction constants, respectively. For bi and Ai, we relax positivity condition. In the equations above,Eqs. (1) and (2), we assume that feelings decay exponentially fast in the absence of partners. The parameters specify theromantic style of individuals 1 and 2. For instance, ai describes the extent to which individual i is encouraged by his/herown feeling. In other words, ai indicates the degree to which an individual has internalized a sense of his/her self-worth.In addition, it can be used as the level of anxiety and dependency on other’s approval in romantic relationships. The param-eters bi represent the extent to which individual i is encouraged by his/her partner, and/or expects his/her partner to be sup-portive. It measures the tendency to seek or avoid closeness in a romantic relationship. Therefore, the term �aiXi say that thelove measure of i, in the absence of the partner decay exponentially; and 1/ai is the time required for love to decay. We pro-pose this structure as opposed to model described in previous literature [20–28], because romantic relationships (as in anyinterpersonal relationship) are not linear, especially return factors. Although the return factors chosen herein are unboundedand unrealistic in its global structure, the functional structure of return factors are motivated by dynamics often portrayed inromance novels and the tragic outcomes illustrated in them (Shakespeare’s Romeo and Juliet, Patriarch’s Canzoniere and Pos-teritati, Tolstoy’s Anna Karenina, Flaubert’s Madame Bovary, more recently the tragic myspace suicide of Megan Meier). Theconstant ein the return function can be interpreted as the compensatory constant. For example, the romantic dynamics be-tween Laura Winslow and Steve Urkel in the sitcom Family Matter can serve as a popular media example. When Steve Urkeldespairs, Laura Winslow feels sorry for him and her antagonism is overcome by feeling of pity. As a result, she reverses herreaction to passion. This behavioral characteristic is captured by the function of reaction or return function (e.g.,b1X2ð1� eX2

2Þ). This expression captures the compensation for antagonism with flattery, or pity, for positive and negativevalues of X2 in b1X2ð1� eX2

2Þ, respectively. For e = 0, the model reduces to the models proposed by Strogatz [15,16], and oth-ers [20–24,26–28], and see the analysis therein. For the analysis of the dynamics of the model, we obtain the followingstatement:

Theorem 1. Let ðX1;X2Þ denotes the equilibrium point, then it is said to be asymptotically stable if and only if: the diadicrelationship threshold Rd

Rd ¼b1b2

a1a2d1d2 < 1 ð3Þ

where the correction term dj ¼dgðXjÞ

dXjwith j = 1, 2 and g(u) is the linearized return function. The equilibrium is otherwise unstable.

The dj = 1, j = 1, 2 for Strogatzian model where e = 0. We can interpret Theorem 1 as follows: for asymptotic stability, thesquared geometric mean of the ratio of reactiveness to love and oblivion coefficients must be less than 1. The proof of theabove theorem relies on linearization around the steady states and positivity of the determinant of the Jacobian matrix (e.g.,see [24] for similar results). Whenever this statement does not hold, Strogatzian model causes the solution to be unbounded,which is obviously unrealistic. Therefore, we restrict our study to the stable condition and state the following corollary forthe linear model:

Corollary 2.1. The following statements hold:

(i) IfRd < 1 or more specifically 0 < Rd < 1, then the equilibrium point of the system does not admit stable focus point or center.The transients of Xi(t) cannot have transient (damped) oscillations or other cyclic dynamics.

(ii) If Rd < 0 and for some parameter values, then the equilibrium point admits stable focus.

In this paper, we only focus on the stable dynamics of where Rd < 0 because their stochastic counterparts exhibit differ-ent dynamics for some parameter value regimes. Fig. 1 shows the phase portrait and the time series of romantic dynamicsrepresenting relationship between secure and hermit individuals.

For stochastic dynamics, our emphasis is placed on the dynamics that exhibit damped oscillations in the deterministicmodels (e.g., Fig. 1), for the stochastic models fairly follow the deterministic behaviors of other equilibrium-type (e.g., stable

cite this article in press as: K. Barley, A. Cherif, Stochastic nonlinear dynamics of interpersonal and romantic relationships, Appl.Comput. (2011), doi:10.1016/j.amc.2010.12.117

−3 −2 −1 0 1 2 3 4 5−2

−1.5

−1

−0.5

0

0.5

1Determinstic Model with β1 =5.7, α1=+0.1, β2 =−1, α2=+0.01

x1

x 2

0 20 40 60 80 100−4

−2

0

2

4

6Determinstic Model with ε = 0

time t

x 1

0 20 40 60 80 100−2

−1.5

−1

−0.5

0

0.5

1Determinstic Model with ε = 0

time t

x 2

Fig. 1. A deterministic evolution of love measure for a romantic relationship between secured or cautious and hermit lovers. Top plot shows the phaseportrait of Eqs. (1), (2) with e = 0, a1 = 0.1, a2 = 0.01, b1 = 5.7, and b2 = �1, linearized near a stable equilibrium. The bottom panel shows the time seriessolutions of romantic feelings, which exhibit damped oscillations.

4 K. Barley, A. Cherif / Applied Mathematics and Computation xxx (2011) xxx–xxx

nodes and limit cycle and/or centers, except in the case where the stability domain is small). In the next section, we provide astochastic dynamical system approach to illustrate some interesting dynamics observed in interpersonal and romanticrelationships.

2.2. Stochastic dynamical models

In the previous mathematical papers on the subject, all factors of relationship are independent of one another and theyconsider time-invariant personalities and the appeal of individuals, ignoring long-term aging, learning, adaptation processes,fast fluctuation of feelings, and external forces and influences such as familial approval and disapproval of loved ones. Therole of oxytocin or vasopressin in the behavioral features, cultural and institutional conditions, and attachment dynamics arealso ignored. Accumulating all these forces as external factors that play major role on the quality of relationships, we con-sider stochastic variation of previous models [17] to investigate the stochastic nature of romantic dynamics by consideringthe rates as stochastic rates, rather than including additive random drift and/or periodic parametric excitation terms as donein [28]. In the following section, we provide a method of using a deterministic formulation to derive a stochastic model.

2.2.1. Derivation of stochastic love dynamicsIn [17], similar method of deriving a stochastic equivalent of deterministic model was outlined. The method used to de-

rive the stochastic differential equations for dynamical process naturally lead to Ito stochastic differential equations, as op-pose to other stochastic calculi (e.g., Stratonovich). This paper only summarizes the approximation procedures, andinterested readers should consult the work of Kurtz [30], Cherif [17] and Kuske and collaborators [32] for more detailedtreatment of the diffusion equation approximation, which corresponds to a continuous time Markov process. We obtainthe stochastic dynamical model for the processes by:

(i) listing all the possible changes DX = [DX1,DX2] along with the probabilities for each change in a short time step Dt (seeTable 1);

(ii) taking the expected changes E[DX] and covariance matrix E[DX (DX)T] are calculated for the Markov process.

Note that E[DX](E[DX])T = o(D t2) and can be ignored. The rates in Table 1 become the conditional transition rates of thestochastic process, that is, P(X1, (t+Dt) = x1 � 1jX1 = x1) = �a1X1D t + o(Dt) and so on. To each of the increments, we add and

Please cite this article in press as: K. Barley, A. Cherif, Stochastic nonlinear dynamics of interpersonal and romantic relationships, Appl.Math. Comput. (2011), doi:10.1016/j.amc.2010.12.117

Table 1Transition rate.

Transition Rate

X1 ? X1 � 1 a1X1

X1 ? X1 + 1 b1X2ð1� eX22Þ þ A1

X2 ? X2 � 1 a2X2

X2 ? X2 + 1 b2X1ð1� eX21Þ þ A2

K. Barley, A. Cherif / Applied Mathematics and Computation xxx (2011) xxx–xxx 5

subtract its conditional expectation, conditioned on the value of the process at the beginning of the time increment of lengthDt. This allows us to then decompose each increment into the sum of the expected value of the increment and sum of cen-tered increment. That is, DX1 ¼ ½�a1X1 þ b2X2ð1� eX2

2Þ þ A1�Dt � DZ1 þ DZ2 with the expected value of EðDX1Þ ¼ ½�a1X1þb2X2ð1� eX2

2Þ þ A1�Dt, where the centered increment DX1 � E(DX1) is given as the difference of two increments, DZ2 � DZ1.The terms D Zi are the difference of two centered Poisson increments. These terms are then replaced by increment of Brown-ian motion dWi with corrected standard deviations or conditional variance. The stochastic equations of the process can thenbe expressed in a form easily comparable to their deterministic equation counterpart.

Alternatively, we can also arrive at the stochastic model by dividing the expected changes and the square root of thecovariance matrix by Dt. In the limit as Dt ? 0, the former becomes the drift term l(t,X1,X2), and the latter becomes thediffusion coefficient D(t,X1,X2), respectively. Both procedures yield similar stochastic differential equations of the form:

PleaseMath.

dX ¼ lðt;X1;X2Þdt þ Dðt;X1;X2ÞdW; ð4Þ

where W = [W1, . . . ,W4]T is an independent Wiener process. Notice that from the above formalism, the following statementsare also true and can be verified:

E EDXDt

� �� lðt;X1;X2Þ

��������2

" #! 0 as Dt ! 0 ð5Þ

and

E EDXðDXÞT

Dt

!� Dðt;X1;X2ÞDðt;X1;X2ÞT

����������2

24

35! 0 as Dt ! 0: ð6Þ

The alternative procedure relies on using the discrete deterministic model and using similar argument as in the first method.Using this approach, we arrive at a discrete stochastic model. Conditions (Eqs. (5) and (6)) provide a justification for a weakapproximation of moving from a discrete stochastic model to a continuous stochastic model. This weak approximation isequivalent to the convergence of a family of discrete state-space Markov chains to a continuous stochastic process. Thatis, for some class of smooth functions G : R2 ! R and let the solution to the stochastic differential equations be X(T) at timeT and the solution to the discrete stochastic equation be denoted by XD(T), then E[G(X(T))] � E[G(XD(T))] ? 0 as D t ? 0. Thisprovides a definition for weak convergence of discrete to continuous stochastic differential equations. Kurtz [30], and Klodenand Platen [31] have given detailed expositions on the methodology outlined in this section.

Using one of the methodologies sketched above, the stochastic differential equations describing the dynamics of interper-sonal and romantic relationships are given as follows:

dX1 ¼ �a1X1 þ b1X2 1� eX22

� �þ A1

h idt �

ffiffiffiffiffiffiffiffiffiffiffia1X1

pdW1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib1X2 1� eX2

2

� �þ A1

rdW2 ð7Þ

dX2 ¼ �a2X2 þ b2X1 1� eX21

� �þ A2

h idt þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2X1 1� eX2

1

� �þ A2

rdW3 �

ffiffiffiffiffiffiffiffiffiffiffia2X2

pdW4 ð8Þ

for (X1,X2) 2 R � R, where Wi, i = 1, . . . ,4 are independent standard Wiener processes. This approach allows us to extenddeterministic models to stochastic models. One can analyze the dynamics of the stochastic models with the help of the sta-bility analysis of the deterministic equations. In fact, the solution to the deterministic model corresponds to the mean of thestochastic model. This framework provides a step towards understanding of the dynamics that are exhibited by the stochas-tic model. It should be noted that the dynamics of the stochastic differential Eqs. (7) and (8) are closely related to the dynam-ics of the deterministic model (e.g., Eqs. (1) and (2)), but can exhibit important differences. Additional dynamics can emergein the stochastic model for some parameter values (e.g., Figs. 2–4) for some b1b2 < 0 (e.g., for Strogatzian model, e = 0). In [17],similar and more exotic behaviors were observed for nonlinear case and some of the results are included herein, for bothlinear and nonlinear systems.

For a stochastic Strogatzian romantic model, the stochastic system exhibits sustained oscillations (e.g., Fig. 2) whereas adeterministic model shows damped oscillation (e.g., Fig. 1). For a nonlinear model (e > 0) with appropriate conditions beingsatisfied, we observe various dynamics including those observed in the linear case (e.g., sustained oscillation via stochasticresonance). Fig. 2 shows oscillations can persist or are sustained for some parameter values, whereas the deterministic equa-tions exhibit damped oscillations (not shown). In addition to sustained oscillations, we also see that the systems can have

cite this article in press as: K. Barley, A. Cherif, Stochastic nonlinear dynamics of interpersonal and romantic relationships, Appl.Comput. (2011), doi:10.1016/j.amc.2010.12.117

−2 −1 0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1Stochastic Model with β1 =5.7, α1=+0.1, β2 =−1, α2=+0.01

x1

x 2

0 20 40 60 80 100−2

−1

0

1

2

3

4

5Stochastic Model with ε = 0

time t

x 1

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1Stochastic Model with ε = 0

time t

x 2

−2 −1 0 1 2 3 4 50

5000

10000

15000Histogram

−1.5 −1 −0.5 0 0.5 10

0.5

1

1.5

2x 104 Histogram

Fig. 2. A the existence of sustained oscillation in the Stochastic model, while deterministic model does not exhibit such behaviors with the same parametervalues e = 0, a1 = 0.1, a2 = 0.01, b1 = 5.7, and b2 = �1. The deterministic system exhibits damped oscillations. It provides the contrast between thedeterministic and stochastic models for the dynamics of romantic relationship. Left figure shows the distributions associated with the dynamics shown onthe left.

−8 −6 −4 −2 0 2 4 60

1

2

3

4

5Stochastic Model with β1 =5.7, α1=+0.1, β2 =−1, α2=+0.1

x1

x 2

0 20 40 60 80 100−8−6−4−20246

Stochastic Model with ε = 0.1

time t

x 1

0 20 40 60 80 1000

1

2

3

4

5Stochastic Model with ε = 0.1

time t

x 2

−8 −6 −4 −2 0 2 4 60

1000200030004000500060007000

Histogram

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2000

4000

6000

8000

10000

12000Histogram

Fig. 3. A diffusion between two locally stable equilibria in stochastic dynamics of love affair. It corresponds to the nonlinear stochastic dynamics withsimilar bi parameter values as in Fig. 3, where we use the following parameters: e – 0, a1 = a2 = 0.1, b1 = 5.7, and b2 = �1. The distribution for X1 is bi-modalwhile that of X2 is unimodal.

6 K. Barley, A. Cherif / Applied Mathematics and Computation xxx (2011) xxx–xxx

more exotic dynamics. We observed, in Figs. 3and 4, both sustained oscillations at the equilibria and ‘‘jumping’’ or ‘‘switch-ing’’ phenomena. For ‘‘switching,’’ there are trajectories which diffuse from equilibrium to equilibrium. A more detailedinvestigation of such dynamics (e.g., sustained and ‘‘jumping’’ oscillations or transition between locally stable (spiral) equi-libria) can be done through multiple scales with Ito-Doeblin Formula (or Ito formula) correction and/or stochastic bifurcation(e.g., Phenomenological (P) and Dynamical (D) Bifurcations). The system exhibiting sustained oscillations with transitionsbetween local stable equilibria is only observed in fragile interpersonal and romantic relationships (e.g., Figs. 3 and 4), whilerobust interpersonal and romantic relationships show only sustained oscillations (e.g., Fig. 2). These behavioral dynamics aredependent on the variance associated with the diffusion terms, which can easily be verified with multiple time-scale methodoutlined by Kuske et al. [32]. We also observed that for some parameter values, the transitions (jump) between equilibria orsustained oscillations are transient and are not sustained for these values. In other cases, for different conditions, the trajec-tories visit most of the equilibria of the system in a sustained way. For example, when there are more than two locally stableequilibria satisfying Corollary 2.1(ii) condition, the trajectories can visit most (if not all) of the steady states in fragile rela-tionships, hence exhibiting multi-modal distributions. Therefore, this paper illustrates the need for more mathematical anal-ysis of interpersonal and romantic relationships from the perspective of nonlinear stochastic differential equations.

Please cite this article in press as: K. Barley, A. Cherif, Stochastic nonlinear dynamics of interpersonal and romantic relationships, Appl.Math. Comput. (2011), doi:10.1016/j.amc.2010.12.117

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2Stochastic Model with β1 =1, α1=0.001, β2 =0.00036, α2=1

x 1

x 2

0 20 40 60 80 100−3

−2

−1

0

1

2

3Stochastic Model with ε = 0.3

time t

x 1

0 20 40 60 80 100−3

−2

−1

0

1

2Stochastic Model with ε = 0.3

time t

x 2

−3 −2 −1 0 1 2 30

500

1000

1500

2000

2500Histogram

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 20

500

1000

1500

2000

2500Histogram

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3Stochastic Model with β1 =1, α1=0.001, β2 =0.00036, α2=1

x1

x 2

0 20 40 60 80 100−3

−2

−1

0

1

2

3Stochastic Model with ε = 0.3

time t

x 1

0 20 40 60 80 100−3

−2

−1

0

1

2

3Stochastic Model with ε = 0.3

time t

x 2

−3 −2 −1 0 1 2 30

500

1000

1500

2000

2500Histogram

−3 −2 −1 0 1 2 30

500

1000

1500

2000

2500Histogram

Fig. 4. A diffusion between multiple locally stable equilibria in stochastic dynamics of love affair with the following parameters: e – 0, a1 = 0.001, a2 = 1.0,b1 = 1.0, and b2 = 0.00036. The distribution for Xi is multi-modal.

K. Barley, A. Cherif / Applied Mathematics and Computation xxx (2011) xxx–xxx 7

2.2.2. Extension: differential romantic stylesUsing similar methods as outlined above, a more complex stochastic dynamic of love affairs can be proposed in various

ways: (i) deterministic parameters and states which result in Eqs. (1) and (2), and other deterministic variant of romanticmodels (e.g., Refs. [15,16,20,21,23–28]); (ii) time-varying stochastic parametric dynamics ai, bi and Ai which result in Eqs.(1), (2) and (11), (12), (13), (14) (e.g., in similar spirit of [28]); (iii) stochastic states with deterministic parameters whichis represented by Eqs. (7) and (8), and other parametric forcing (e.g., combination of Eqs. (7) and (8) with time-varying para-metric forcing); and (iv) both time-varying stochastic parameters and states which will yield Eqs. (9)–(14). In this section, weonly provide the latter assumptions where the parameters and the states are both time varying stochastic dynamical sys-tems. The ansatz describing (iv) is given as follows:

PleaseMath.

dX1 ¼ �a1X1 þ b1X2 1� eX22

� �þ A1

h idt �

ffiffiffiffiffiffiffiffiffiffiffia1X1

pdW1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib1X2 1� eX2

2

� �þ A1

rdW2 ð9Þ

dX2 ¼ �a2X2 þ b2X1 1� eX21

� �þ A2

h idt þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2X1 1� eX2

1

� �þ A2

rdW3 �

ffiffiffiffiffiffiffiffiffiffiffia2X2

pdW4 ð10Þ

da1 ¼ a10½a11 � a1�dt þffiffiffiffiffiffiffia12p

dW5 ð11Þ

da2 ¼ a20½a21 � a2�dt þffiffiffiffiffiffiffia22p

dW6 ð12Þ

db1 ¼ b10½b11 � b1�dt þffiffiffiffiffiffiffib12

pdW7 ð13Þ

cite this article in press as: K. Barley, A. Cherif, Stochastic nonlinear dynamics of interpersonal and romantic relationships, Appl.Comput. (2011), doi:10.1016/j.amc.2010.12.117

8 K. Barley, A. Cherif / Applied Mathematics and Computation xxx (2011) xxx–xxx

PleaseMath.

db2 ¼ b20½b21 � b2�dt þffiffiffiffiffiffiffib22

pdW8 ð14Þ

for (X1,X2,a1,a2,b1,b2) 2 R2 � R4, where Wi, i = 1, . . . ,8 are independent standard Wiener processes. Eqs. (9)–(14) represent astochastic model of romantic and interpersonal relationships incorporating variabilities associated with romantic styles dueto factors such as dislike of family members and/or friends against one’s partner, ecological and institutional conditions,bio-sociological factors, etc. Variations due to these factors can also affect the measure of relationship in an unpredictable man-ner. One way to capture these variations is to have different variables for each one of the factors of interest. However, this willcomplicate the analysis, and render the analysis of the system intractable. The equations above simplify and assume that, forexample, b1(t,x1 , . . . ,xn), where xi, i = 1, . . . ,n, there are different romantic factors that affect the romantic style of individuals.Note that each of these parametric dynamics a1, a2, b1, and b2 in Eqs. (11)–(14) is Ornstein–Uhlenbeck process, and can be inte-grated and solved exactly. We can then take their limiting cases (e.g., steady state conditions, limt?1 E(ai) and limt?1E(bi) fori = 1, 2 since they are decoupled from Xi) and substitute them into Eqs. (9) and (10) to reduce our system to Eqs. (7) and (8).Assuming that romantic styles vary stochastically, one can also investigate the effects of their inclusions on the dynamics.One such effect is the spread in distribution of evolution of feelings, which might exhibit earlier observation of more exoticdynamics of sustained oscillations or stochastic resonance. In such cases, the dynamics of Eqs. (7) and (8) are sufficient to under-stand the dynamics of Eqs. (9)–(14). However, this analysis is left to the future paper and was not investigate herein.

3. Conclusion

In summary, we provide a new direction for analyzing relationships using stochastic differential equation extension. Inparticular, we have considered both deterministic and stochastic models with nonlinear return functions. The stochasticmodel has a structure related to a deterministic model which allows us to study most of its dynamics through the lens ofdeterministic analysis. We have focused on particular subsets of interesting dynamics that are not observed in deterministicmodels. While a deterministic model exhibited damped oscillations with certain parameter values, the stochastic modelsshowed sustained oscillations and/or diffusivity of equilibria and multiple stability boundary crossings with the sameparameter values. The results show that deterministic linear and nonlinear models tend to approach locally stable emotionalbehaviors. However, in the presence of stochasticity in the models, these complex and exotic patterns of emotional behaviorsare observed. The stochastic differential equation extension provides insight into the dynamics of romantic relationshipsthat are not captured by deterministic models, which assumes that love is scalar and individuals respond predictably to theirfeelings and that of others without external influences, such as ecological factors. Stochastic models capture the fluctuationsdue to the dynamics of love and that of external influences. In the context of sociology, the patterns of behavior observed instochastic model (e.g., Eqs. (7) and (8) and Figs. 2–4), represent stable periods in relationships are subject to the effects ofemotional, psychological, social structural, and cultural force [35]. The transition between states could be interpreted asinstabilities caused by these forces. In the case where both partners’ love and affection are drawn from mutual attraction,the instabilities can be attributed to external triggers such as cultural and familial ties. Figs. 3 and 4 depict examples wherethese external perturbations result in emotional conflict of in-and-out-of-love behavior [33–38]. When the love is not inde-pendently drawn from mutual attraction, internal (e.g., individual personalities such abusive personalities) triggers causethese instabilities. The observed transition states are consistent with traumatic bonding theory [34] and theories of psycho-logical and economic entrapment [33], where relationships are characterized by attachments formed from repeated abuseand entrapment relationships include relationships where the abused partner is coerced to remain in the relationship di-rectly and indirectly with threats of abandonment and loss, respectively.

The paper provides a new direction to the study of interpersonal relationship. Further directions toward more realisticmathematical and theoretical modeling of romantic and interpersonal relationships are possible through the lens ofagent-based modeling, where community interaction and age structures are included, and/or integrative models (interper-sonal, bio-sociological and ecological models are integrated) can be investigated. Calculation of the distribution of time indif-ference or apathy of individuals is a new possibility arising out of the stochastic dynamics of romantic relationships. Themost fruitful direction from mathematical purview is developing methods to analysis systems that exhibit multiple ‘‘stabilityboundary crossing’’ or ‘‘jump between locally stable equilibria’’ dynamics. The analysis of these dynamics is worth studyingin another paper.

Acknowledgements

The authors were supported in part by National Science Foundation through Graduate Research Fellowship (to A.C.) andLSAMP Bridge to the Doctorate Fellowship (to A.C., K.B.), the Sloan Fellowship (to A.C., K.B.), the ASU Graduate College Doc-toral Enrichment Fellowship (to A.C.). The authors thank Carl Ballard, Marcel Hurtado, Dr. Priscilla Greenwood and the twoanonymous reviewers for fruitful discussions and suggestions.

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Please cite this article in press as: K. Barley, A. Cherif, Stochastic nonlinear dynamics of interpersonal and romantic relationships, Appl.Math. Comput. (2011), doi:10.1016/j.amc.2010.12.117