Applied Mathematics and Computation 95 (1998) 181192 Love dynamics: The case of linear couples Sergio Rinaldi 1 Centro Teoria dei Sistemi, CNR, Politecnico di Milano, Via Ponzio 34/5, 20133 Milan, Italy Abstract This paper proposes a minimal model composed of two ordinary dierential equations to describe the dynamics of love between two individuals. The equations take into account three mechanisms of love growth and decay: the pleasure of being loved (return), the reaction to the partner's appeal (instinct), and the forgetting processes (oblivion). Under suitable assumptions on the behavior of the individuals, the model turns out to be a positive linear system enjoying, as such, a number of remarkable properties, which are in agreement with common wisdom on the argument. These properties are used to explore the consequences that individual behavior can have on the community structure. The main result along this line is that individual appeal is the driving force that creates order in the community. Possible extensions of this theory of linear love dynamics are briey discussed. 1998 Elsevier Science Inc. All rights reserved. Keywords: Love dynamics; Linear systems; Positive systems; Population structures; Frobenius theorem 1. Introduction Ever since Newton introduced dierential calculus, dynamic phenomena in physics, chemistry, economics and all other sciences have been extensively studied by means of dierential equations. Surprisingly, one of the most important problems concerning our lives, namely the dynamics of love between two persons, has not yet been tackled in this way. The only exception is a one page paper in which Strogatz [1] describes how the classical topic of harmonic oscillations can be taught to capture the attention of students. He suggests a consideration of ``a topic that is already on the minds of many college students: 1 E-mail: rinaldi@elet.polimi.it. 0096-3003/98/$19.00 1998 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 7 ) 1 0 0 8 1 - 9 182 S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 the time evolution of a love aair between two people''. The model proposed by Strogatz (discussed also in [2,3]) is denitely unrealistic because it does not take into account the appeal of the two individuals. Thus Strogatz's model, for example, does not explain why two persons who are initially completely indierent to each other can develop a love aair. The model proposed in this paper is more realistic, although it is still a minimal model. Three aspects of love dynamics are taken into account: the forgetting process, the pleasure of being loved, and the reaction to the appeal of the partner. These three factors are assumed to be independent and are modeled by linear functions. The resulting model is a linear dynamical system, which turns out to be positive if the appeals of the two individuals are positive. The theory of positive linear systems [48] can therefore be applied to this model and gives quite interesting results. Some of them describe the dynamic process of falling in love, i.e. the transformation of the feelings, starting from complete indierence (when two persons rst meet) and tending toward a plateau. Other results are concerned with the inuence that appeal and individual behavior have on the quality of romantic relationships. Some of these properties are used to identify the consequences that individual appeal and behavior can have on partner choice and on community structure. Although the results are extreme, they explain to some extent facts observed in real life, such as the rarity of couples composed of individuals with very uneven appeal. The conclusion is that the proposed model, besides being a good method of capturing student attention, is also an elegant tool for deriving general properties of love dynamics from purely conceptual arguments. 2. The model The model analyzed in this paper is a dynamic system with only two state variables, one for each member of the couple. Such variables, indicated by x1 and x2 , are a measure of the love of individual 1 and 2 for their respective partners. Positive values of x represent positive feelings, ranging from friendship to passion, while negative values are associated with antagonism and disdain. Complete indierence is identied by x 0. The model is a typical minimal model. Firstly, because love is a complex mixture of dierent feelings (esteem, friendship, sexual satisfaction, F F F) and can be hardly captured by a single variable. Secondly, because the tensions and emotions involved in the social life of a person cannot be encapsulated in such a simple model. In other words, only the interactions between the two individuals are modeled, while the rest of the world is kept frozen and does not participate explicitly in the formation of love dynamics. This means that the present theory cannot be easily related to the well-known attachment theory S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 183 [911], which has been a main investigation tool in adult romantic relationships in the last decade (see, for instance [1217]). Three phenomenon are considered, namely, oblivion, return, and instinct. The rst gives rise to a loss of interest in the partner, and explains, for example, the typical decay of xi , which takes place after the death or estrangement of jY i T j. By contrast, the second and the third are sources of interest. Moreover, the return increases with the love of the partner, while the instinct is sensitive only to appeal (physical, intellectual, nancial, F F F). The following simplifying assumptions are also made. First, the appeals and the personalities of the two individuals do not vary in time: this rules out again, learning and adaptation processes which are often important over a long range of time [18,19]. Thus, the model can only be used for short periods of time (months/years), for example in predicting if a love story will be characterized by regular or stormy feelings. Second, synergism is assumed to be negligible, i.e. oblivion and return depend only upon one state variable. Finally, all mechanisms are supposed to be linear. The result is the following model: x1 t a1 x1 t b1 x2 t c1 e2 Y x2 t a2 x2 t b2 x1 t c2 e1 Y 1 where ai Y bi and ci , as well as the appeals ei , are constant and positive parameters. The negative term ai xi t, due to oblivion, says that the love of i, in the absence of the partner j, decays exponentially xi t xi 0 expai t. The second term bi xj t is the return, and the third ci ej is the reaction to the partner's appeal. Thus, each person is identied by four parameters: the appeal ei , the forgetting coecient ai and the reactiveness bi and ci to the love and appeal of the partner. The determination of the behavioral parameters is undoubtedly a dicult task, although some studies on attachment styles [2022] might suggest ways of identifying categories of individuals with high or low reactiveness or forgetting coecients. This identication problem will not be considered in the present paper, which is only centered on the derivation of the properties of the model. Model (1) is a linear system which can be written in the standard form x ex u with u 1, and \u0012 \u0012 \u0012 \u0012 \u0012 a1 b1 \u0012 \u0012 c 1 e2 \u0012 \u0012Y \u0012 \u0012 e\u0012 \u0012 b \u0012 c e \u0012X a2 \u0012 2 2 1 Such a system is positive because the matrix e is a Metzler matrix (nonnegative o-diagonal elements) and the vector has positive components [5]. Thus, x0 P 0 implies xt P 0 Vt. This means that our assumptions imply that the two persons will never become antagonists, because they are completely indifferent to each other when they meet for the rst time (i.e. x0 0). 184 S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 Positive linear systems enjoy a number of remarkable properties. Particularly if they are asymptotically stable. In the present case, the necessary and sufcient condition for asymptotic stability is b1 b2 ` a1 a2 Y 2 i.e. the system is asymptotically stable if and only if the (geometric) mean rep activeness to love b1 b2 is smaller than the (geometric) mean forgetting cop ecient a1 a2 . In the following, condition (2) is assumed to hold. When this is not the case, namely when the two individuals are quite reactive to the love of the partner, the instability of the model gives rise to unbounded feelings, a feature which is obviously unrealistic. In that case (i.e. when b1 b2 b a1 a2 ) one must model the couple more carefully by assuming, for example, that the reaction function is increasing but bounded with respect to the partner's love. Such an extension will be considered in another paper. 3. Properties of the model We shall now point out ve simple but interesting properties of model (1), under the assumption that condition (2) holds. Thus the system is asymptotically stable and the love of each individual is bounded. Moreover the positivity of a1 and a2 rules out the possibility of cyclic behavior (Bendixon's criterion [3]), so that one can conclude that xi t tends toward an equilibrium value "i , which must be nonnegative because the system is posx itive. More can be said about this equilibrium, however, as specied by the following remark. Remark 1. The equilibrium " "1 Y "2 of system (1) is strictly positive, i.e. x x x "i b 0, i 1Y 2. x Proof. The proof follows immediately as a result of a general property of positive systems [23] which states that asymptotically stable and excitable systems have strictly positive nontrivial equilibria. (It may be recalled that a positive system x ex u is excitable if and only if each state variable can be made positive by applying a suitable positive input starting from x0 0.) In the present case, the system is excitable because the components of the vector are positive. An alternative proof consists in explicitly computing the equilibrium ", x which turns out to be given by "1 x a2 c1 e2 b1 c2 e1 Y a1 a2 b1 b2 "2 x a 1 c 2 e 1 b 2 c 1 e2 X a1 a2 b1 b2 ( 3 S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 185 Thus, if two individuals meet for the rst time at t 0 x0 0 they will x develop positive feelings xi t tending toward the positive equilibrium value "i . Since positive systems have at least one real eigenvalue (the so-called Frobenius eigenvalue kF , which is the dominant eigenvalue of the system), a second-order system cannot have complex eigenvalues, i.e., the equilibrium of system (1) cannot be a focus. In other words, the transients of xi t cannot be damped oscillations characterized by an innite number of minima and maxima. But even the possibility of a single maximum (minimum) can be excluded, as specied in the following remark. Remark 2. The function xi t, corresponding to the initial condition x0 0, is strictly increasing, i.e. xi t b 0 VtY i 1Y 2. Proof. The isoclines xi 0 are straight lines given by a1 c1 x2 x1 e2 1 0Y x b1 b1 b c x x2 2 x1 2 e1 2 0X a2 a2 These isoclines (see dotted lines in Fig. 1) intersect at point i (representing the strictly positive equilibrium " "1 Y "2 , thus partitioning the state space in x x x four regions. In the region containing the origin, xi t b 0Y i 1Y 2, which proves the stated result. ( Fig. 1. Trajectories (continuous lines) and isoclines (dotted lines) of the system. The straight trajectories are identied by the two eigenvectors. Single and double arrows indicate slow and fast motion. 186 S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 It should be noticed that for nonzero initial conditions, one of the two variables xi t can rst decrease and then increase (see trajectory ei in Fig. 1) or vice versa (see trajectory fi). This can be easily interpreted as follows. Suppose, that a couple is at equilibrium and that individual 2 has, for some reason, a sudden decline of interest in the partner. The consequence (see trajectory ei in Fig. 1) is that individual 1 will suer during the whole transient bringing the couple back to equilibrium. Moreover, for very particular initial conditions (straight trajectories in x Fig. 1) the two functions xi t "i Y i 1Y 2 decay exponentially at the same rate (equal to an eigenvalue of e). The slowest decay occurs along a trajectory which has a positive slope and is identied by the dominant eigenvector. On the other hand, the fastest decay occurs along the other straight trajectory which has a negative slope. The result is a direct consequence of the well-known Frobenius theory [24] which says that in a positive and irreducible system the dominant eigenvector is strictly positive and there are no other positive eigenvectors (it may be recalled that a system is irreducible when it cannot be decomposed into the cascade or parallel connection of two subsystems, a property which is guaranteed in the present case by b1 b2 b 0). Applied to a second-order system, the Frobenius theory states that the dominant eigenvector has components with the same sign, while the other eigenvector has components of opposite sign. We can now focus on the inuence of the various parameters on the equilibrium and dynamics of the system, starting with the reactiveness to love and appeal. Remark 3. An increase in the reactiveness to love [appeal] bi ci of individual i gives rise to an increase in the love of both individuals at equilibrium. Moreover, the relative increase Dxa" is higher for individual i. x Proof. The result can be obtained directly from Eq. (3) by deriving "i with x x respect to bi ci and then dividing by "i . Condition (2), of course, must be taken into account. Nevertheless, this derivation is not needed. Indeed, the rst part of the remark is a direct consequence of the famous law of comparative dynamics [5]. This law states that in a positive system the increase of a positive parameter gives rise to an increase of the components of the state vector at any time, and hence also at equilibrium. The second part of the remark is the consequence of a general theorem concerning positive systems, known as the theorem of maximum relative variation [8]. Such a theorem states that if the ith component of the vector or one element of the ith row of the matrix e of an asymptotically stable and excitable positive system is slightly increased (in such a way that the system remains asymptotically stable and excitable), the ith component "i of the state vector at equilibrium is the most sensitive of all in relative x terms. ( S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 187 The following, somehow intriguing, remark species the inuence of appeal on the equilibrium. Remark 4. An increase in the appeal ei of individual i gives rise to an increase in the love of both individuals at equilibrium. Moreover, the relative increase Dxa" is higher for the partner of individual i. x Proof. The proof results from Remark 3, with notice to Eq. (3) in which ci is multiplied by ej Y i T j. As in the proof of Remark 3, we notice that the result is the direct consequence of the law of comparative dynamics and of the theorem of maximum relative variation, because e1 e2 enters only in the second [rst] state equation. ( The last remark is one concerning the inuence of the reactiveness to love on the dynamics of the system. Remark 5. An increase in the reactiveness to love gives rise to an increase of the dominant time constant of the system, which tends to innity when b1 b2 approaches a1 a2 . By contrast, the other time constant decreases and tends to 1aa1 a2 . Proof. Since the trace of the matrix e is equal to a1 a2 , the sum of the two eigenvalues remains constant and equal to a1 a2 when bi varies. On the other hand, the two eigenvalues remain real (because the system is a secondorder positive system) and one of them (the dominant one kF ) tends to zero because the system loses stability when b1 b2 approaches a1 a2 . This means that the dominant time constant H 1akF increases with bi and tends to innity when b1 b2 tends to a1 a2 . By contrast, the other time constant HH decreases, because 1a H 1a HH a1 a2 . For b1 b2 tending to a1 a2 the time constant HH tends to 1aa1 a2 because H tends to innity. ( The above ve remarks can be easily interpreted. The rst states that individuals with positive appeal are capable of establishing a steady romantic relationship. The emotional pattern of two persons falling in love is very regular beginning with complete indierence, then growing continuously until a plateau is reached (Remark 2). The level of passion characterizing this plateau is higher in couples with higher reactiveness and appeal (Remarks 3 and 4). Moreover, an increase in the reactiveness of one of the two individuals is more rewarding for the same individual, while an increase of the appeal is more rewarding for the partner. Thus, there is a touch of altruism in a woman (man) who tries to improve her (his) appeal. Finally, couples with very high reactiveness respond promptly during the rst phase of their romantic relationship, but 188 S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 are very slow in reaching their plateau (Remark 5). Together with Eq. (3), this means that there is a positive correlation between the time needed to reach the x equilibrium and the nal quality ("1 and "2 ) of the relationship. Thus, passions x that develop too quickly should be expected to be associated with poor romantic relationships. 4. Consequences at community level We can now try to identify the consequences, at community level, of the dynamics of love discussed in Section 3. Let us consider a community composed by x women and x men structured in x couples en Y an Y bn Y cn Y en Y an Y bn Y cn Y 1 1 2 2 1 1 2 2 n 1Y 2Y F F F Y x and suppose that 1 is a woman and 2 is a man. For simplicity, suppose that there are no women (men) with the same appeal, i.e. eh T ek , i i VhY k with h T k. Such a community is considered unstable if a woman and a man of two distinct couples believe they could be personally advantaged by forming a new couple together. In the opposite case the community is stable. Thus, practically speaking, unstable communities are those in which the separation and the formation of couples are quite frequent. Obviously, this denition must be further specied. The most natural way is to assume that individual i would have a real advantage in changing partner, if this change is accompanied by an increase of "i . However, in order to forecast the value x "1 "2 that a woman [man] will reach by forming a couple with a new partner, x x she [he] should know everything about him [her] (in mathematical terms, she [he] should know his [her] appeal e2 e1 and his [her] behavioral parameters a2 Y b2 , and c2 [a1 Y b1 Y and c1 ]). Generally, this is not the case and the forecast is performed with limited information. In this case it is assumed that the only available information is the appeal of the potential future partner and that the forecast is performed by imagining that the behavioral parameters of the future partner are the same as those of the present partner. This choice obviously emphasizes the role of appeal, quite reasonably, because appeal is the only easily identiable parameter in real life. The above discussion is formally summarized by the following denition. Denition 1. A community en Y an Y bn Y cn Y en Y an Y bn Y cn Y n 1Y 2Y F F F Y x is unsta1 1 1 1 2 2 2 2 ble if and only if there exists at least one pair hY k of couples such that "1 eh Y ah Y bh Y ch Y ek Y ah Y bh Y ch b "1 eh Y ah Y bh Y ch Y eh Y ah Y bh Y ch Y x 1 1 1 1 2 2 2 2 x 1 1 1 1 2 2 2 2 "2 eh Y ak Y bk Y ck Y ek Y ak Y bk Y ck b "2 ek Y ak Y bk Y ck Y ek Y ak Y bk Y ck Y x 1 1 1 1 2 2 2 2 x 1 1 1 1 2 2 2 2 4 where the functions "1 X and "2 X are given by Eq. (3). A community which is x x not unstable is called stable. S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 189 We can now prove that stable communities are characterized by a very simple but peculiar property involving only appeal. Remark 6. A community is stable if and only if the partner of the nth most attractive woman n 1Y 2Y F F F Y x is the nth most attractive man. Proof. First notice that Remark 4 implies that condition (4) is equivalent to e k b eh Y 2 2 eh b e k 1 1 5 i.e. a community is unstable if and only if there exists at least one pair hY k of couples satisfying Eq. (5). Condition (5) is illustrated in Fig. 2(a) in the appeal space, where each couple is represented by a point. Consider a community in which the partner of the nth most attractive woman is the nth most attractive man. Such a population is represented in Fig. 2(b), which clearly shows that there is no pair hY k of couples satisfying inequalities (5). Thus, the community is stable. On the other hand, consider a stable community and assume that the couples have been numbered in order of increasing appeal of the women, i.e. e1 ` e2 ` ` ex X 1 1 1 6 e1 Y e1 1 2 e2 Y e2 1 2 to the second point with a segThen, connect the rst point ment of a straight line, and the second to the third, and so on until the last point ex Y ex is reached. Obviously, all connecting segments have a positive 1 2 slope because, otherwise, there would be a pair of couples satisfying condition (5) and the community would be unstable (which would contradict the assumption). Thus, e1 ` e2 ` ` ex . This, together with Eq. (6), states that the 2 2 2 partner of the nth most attractive woman is the nth most attractive man. ( Fig. 2. Population structures in the appeal space: (a) two points corresponding to two couples hY k belonging to an unstable community (see (5)); (b) an example of a stable community (each dot represents a couple). 190 S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 On the basis of Remark 6, higher tensions and frictions should be expected in communities with couples in conict with the appeal ranking. This result, derived from purely theoretical arguments, is certainly in agreement with empirical evidence. Indeed, partners with very uneven appeals are rarely observed in relatively stable communities. Of course, in making these observations one must keep in mind that appeal is an aggregate measure of many dierent factors (physical, nancial, intellectual, F F F). Thus, for example, the existence of couples composed of a beautiful lady and an unpleasant but rich man does not contradict the theory, but, instead, conrms a classical stereotype. 5. Concluding remarks A minimal model of love dynamics composed of two ordinary dierential equations has been presented and discussed in this paper. The equations take into account three mechanisms of love's growth and decay: the forgetting process, the pleasure of being loved and the reaction to the partner's appeal. for suitable, but reasonable, assumptions on the behavioral parameters of individuals, the model turns out to be an asymptotically stable, positive, linear system, and enjoys a number of remarkable properties. The model predicts, that the feelings of the two partners vary monotonically, growing from zero (complete indierence) to a maximum. The value of this maximum, i.e. the quality of the romantic relationship at equilibrium, is higher if the reactiveness to love and appeal are higher. The same is true, if the time needed to reach the maximum is longer. All these properties are in agreement with traditional wisdom on the dynamics of love between two persons. These properties have been used to derive the characteristics under which the couples of a given community have no tendency to separate (stability). The main result along this line is that the driving force that creates order in the community is the appeal of the individuals. In other words, couples with uneven appeals should be expected to have higher chances of break up. These results are in a sense complementary to those predicted by attachment theory, where appeal has a very limited role. As with any minimal model, the extensions one could propose are innumerable. Aging, learning and adaptation processes could be taken into account allowing for some behavioral parameters to slowly vary in time, in accordance with the most recent developments of attachment theory. Particular nonlinearities, as well as synergism, could be introduced in order to develop theories for classes of individuals with personalities dierent from those considered in this paper. Men and women could be distinguished by using two structurally dierent state equations. The dimension of the model could also be enlarged in order to consider individuals with more complex personalities or the dynamics of love in larger groups of individuals. Moreover, the process followed by each S. Rinaldi / Appl. Math. Comput. 95 (1998) 181192 191 individual in forecasting the quality of the relationship with a potential new partner could be modeled more realistically, in order to attenuate the role of appeal, which has been somewhat overemphasized in this paper. This could be done quite naturally by formulating a suitable dierential game problem. Undoubtedly, all these problems deserve further attention. Acknowledgements This study has been nancially supported by the Italian Ministry of Scientic Research and Technology, under contract MURST 40% Teoria dei sistemi e del controllo. Part of the work has been carried out at the International Institute for Applied Systems Analysis, IIASA, Laxenburg, Austria. The author is grateful to Gustav Feichtinger, Technical University of Wien, Austria, Alessandra Gragnani, Politecnico di Milano, Italy, Lucia Carli, Universit\u0018 a Cattolica, Milano, Italy, and Frederic Jones, University of Wales, Cardi, UK, for their helpful suggestions and encouragement. 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Rinaldi, Excitability, stability and sign of equilibria in positive linear systems, Syst. Control Lett. 16 (1991) 5963. [24] G. Frobenius, On matrices with nonnegative elements, S.-B. Deutsch. Akad. Wiss. Berlin, Math.-Nat. Kl (1912) 456477 (in German). Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 2, No. 4, 1998 Love Dynamics Between Secure Individuals: A Modeling Approach Sergio Rinaldi 1,3 and Alessandra Gragnani2 A mathematical model composed of two differential equations, which qualitatively describe the dynamics of love between secure individuals, is presented in this paper with two goals. The general goal is to show how dynamic phenomena in the field of social psychology can be analyzed following the modelling approach traditionally used in all other fields of sciences. The specific goal is to derive, from very general assumptions on the behavior of secure individuals, a series of rather detailed properties of the dynamics of their feelings. The analysis shows, in particular, why couples can be partitioned into fragile and robust couples, how romantic relationships are influenced by behavioral parameters and in which sense individual appeal creates order in a community. KEY WORDS: dyadic relationships; love dynamics: partner selection: community structure: stability; multiple equilibria. INTRODUCTION This paper deals with love dynamics, a subject that falls in the field of social psychology, where interpersonal relationships are the topic of major concern. Romantic relationships are somehow the most simple case since they involve only two individuals. The analysis is performed following the modelling approach, which has recently received some attention in the sci1 Centro Teoria dei Sistemi, CNR, Politecnico di Milano, Milano, Italy. -International Institute for Applied Systems Analysis. Laxenburg, Austria. 3 Correspondence should be directed to Professor Sergio R i n a l d i , Centro Teoria dei Sistemi, CNR, Politecnico di M i l a n o , Via Ponzio 34/5. 2013? Milano. I t a l y ; e-mail: rinaldi(a elet.polimi.it. 283 1090-0578,98/1000-0283$15.00/0 C 1998 Human Sciences Press. Inc. 284 Rinaldi and Gragnani entific community (Butz, Chamberlain & McCown, 1996; Vallacher & Nowak, 1994). Real-life observations tell us that very frequently love-stories develop very regularly and then level off for a very long time, as described by Levinger (1980) in his famous graph on the longitudinal course of partners involvement. But there are also love-stories characterized by remarkable temporal fluctuations. This spontaneously suggests the use of differential equations for modelling close relationships, because differential equations can explain stationary regimes as well as cyclic and chaotic regimes (Vallacher & Nowak, 1994). Although Strogatz (1988) has indicated this possibility in a one page pioneering paper, such an approach has never been followed in social psychology. The only exception is a very recent study (Rinaldi, 1998) on the dynamics of love between Petrarch, an Italian poet of the 14-th century, and Laura, a beautiful and married lady. Their love story developed over 21 years and has been described in the Canzoniere, a collection of 366 poems addressed by the poet to his platonic mistress. In such a study, the main traits of Petrarch's and Laura's personalities are identified by analyzing the Canzoniere; he is very sensitive and transforms emotions into poetic inspiration; she protects her marriage by reacting negatively when he becomes more demanding and puts pressure on her, but at the same time, following her genuine Catholic ethic, she arrives at the point of overcoming her antagonism by strong feelings of pity. Then, these traits are encapsulated in a model composed of differential equations where the variables are the emotions of the two individuals. Finally, the model is analyzed and the result is that its solution tends toward a cyclic behavior. In other words, the peculiarities of the personalities of the two lovers inevitably generate a love story characterized by recurrent periods of ecstasy and despair. This is indeed what happened, as empirically ascertained by Jones (1995) through a detailed stylistic and linguistic analysis of the dated poems. It is interesting to note, however, that with the modelling approach the existence of the emotional cycle is fully understood and proved to be inevitable, while with the empirical approach it is only discovered. We follow in this paper the same modelling approach to discuss the dynamics of the feelings between two persons. But instead of concentrating our attention on a specific and well documented case, we deal with the most generic situation, namely that of couples composed of secure individuals (Bartholomew & Horowitz, 1991; Griffin & Bartholomew, 1994). For this we will first assume that secure individuals are characterized by specific properties concerning their reactions to partner's love and appeal. The analysis of the corresponding model will show how the feelings between two individuals evolve in time. Intentionally, we do not support these Love Dynamics Between Secure Individuals: A Modeling Approach 285 properties with data, because we like only to highlight the power of the method, namely the possibility of deriving the main characteristics of love dynamics from individual behavior. It is worth anticipating, however, that all the properties discussed in the paper are in agreement with common wisdom on the argument. The paper is organized as follows. In the next section we assume that secure individuals follow precise rules of behavior and we write two differential equations encapsulating these rules. Then, we analyze these equations and derive properties which formally describe the process of falling in love, i.e., the transformation of the feelings, starting from complete indifference and tending toward a plateau. Other results are concerned with the influence that appeal and individual behavior have on romantic relationships. Some of these properties are used to derive the consequences on partner selection. Although the results are extreme, they explain to some extent facts observed in real life, such as the rarity of couples composed of individuals with very uneven appeal. Merits and weaknesses of the present analysis, as well as directions for further research, are briefly discussed at the end of the paper. Although this is the first serious attempt of studying close relationships by means of differential equations, in a number of previous studies other kinds of dynamical models have been used for predicting the evolution of romantic relationships. In the most relevant of these studies (Huesmann & Levinger, 1976) a model, called RELATE, was proposed and extensively used for simulations. Such a model, discussed with other models in a recent survey (Baron, Amazeen & Beek, 1994), is a sort of computer model (based on exchange theory) involving recursive manipulations of suitable "dyadic outcome matrices" representing different interpersonal states. It is certainly a dynamic model, but it is much more complex and less elegant than a pair of differential equations. THE MODEL The model proposed in this paper is a minimal model, in the sense that it has the lowest possible number of state variables, namely one for each member of the couple. Such variables, indicated by x1 and x2, are a measure of the love of individual 1 and 2 for the partner. Positive values of x represent positive feelings, ranging from friendship to passion, while negative values are associated with antagonism and disdain. Complete indifference is identified by x = 0. Details on how love can be measured with a single variable, how graphs of such a variable versus time can be 286 Rinaldi and Gragnani obtained through interviews, and what are the limitations of such procedures can be found in Levinger (1980). The model is a crude simplification of reality. Firstly, because love is a complex mixture of different feelings (esteem, friendship, sexual satisfaction, . ..) and can be hardly captured by a single variable. Secondly, because the tensions and emotions involved in the social life of a person cannot be included in a minimal model. In other words, only the interactions between the two individuals are taken into account, while the rest of the world is kept frozen and does not participate explicitly in the formation of love dynamics. This means that rather than attempting to be complete, the aim is to check which part of the behaviors observed in real life can in theory be explained by the few ingredients included in the model. It is important to state that the time scale to which we like to refer is an intermediate time scale (e.g., phases A, B, C in Levinger (1980)). More precisely, we are not interested in fast fluctuations of the feelings, like those controlled by the daily or weekly activities or by the sexual rhythms of the couple, nor in the long term dynamics induced by life experiences. Thus, the model can only be used for relatively short periods of time (months/years), for example in predicting if a love story will be smooth or turbulent. This implies that the present study can only weakly relate to attachment theory (Bowlby, 1969, 1973, 1980), which has been a main investigation tool in adult romantic relationships in the last decade (see, for instance, Collins & Read, 1990; Feeney & Noller, 1990; Hazan & Shaver, 1987; Kirkpatrick & Davis, 1994; Shaver & Brennan, 1992; Simpson, 1990). Three basic processes, namely return, instinct, and oblivion are assumed to be responsible of love dynamics. More precisely, the instantaneous rate of change dX i (t)ldt of individual's i love is assumed to be composed of three terms, i.e., where Ri, Ii, and Oi must be further specified. The return Ri describes the reaction of individual i to the partner's love and can therefore be assumed to depend upon xj , j = i. Loosely speaking, this term could be explained by saying that one "loves to be loved" and "hates to be hated." In order to deal with the most frequent situation, we restrict our attention to the class of secure individuals, who are known to increase their reaction when the love of the partner increases. More precisely, we assume that the return function Ri, is positive, increasing, concave and bounded for positive values of xj and negative, increasing, convex and bounded for negative values ofxj. Figure 1 shows the graph of a typical Love Dynamics Between Secure Individuals: A Modeling Approach 287 Fig. 1. The return function Ri of a secure individual i vs. the partner's love xj. return function. The boundedness of the return function is a property that holds also for non secure individuals: it interprets the psycho-physical mechanisms that prevent people from reaching dangerously high stresses. By contrast, increasing return functions are typical of secure individuals, since non-secure individuals reduce their reaction when pressure and involvement are too high (Griffin & Bartholomew, 1994). The second term Ii, describes the reaction of individual i to the partner's appeal Aj, Of course, it must be understood that appeal is not mere physical attractiveness, but, more properly and in accordance with evolutionary theory, a suitable combination of different attributes among which age, education, earning potential and social position. Moreover, there might be gender differences in the relative weights of the combination (Feingold, 1990; Sprecher, Quintin & Hatfield, 1994). For reasons similar to those mentioned above, we assume that instinct functions I i (A j ) enjoy the same properties that hold for return functions. Nevertheless, for simplicity, we will restrict our attention to the case of individuals with positive appeals. Notice that the reaction to the partner's appeal Ii(Aj) is greater than the reaction to the partner's love Ri(xj) when Xj is small (or negative). In particular, at the beginning of the relationship (i.e., in the phase called "attraction" by Levinger (1980)),x1 and x2 are small, so that the instinct term is the dominant term. In other words, the rate of change of the feeling is initially determined by the appeal of the partner. Finally, the third term Oi can be easily understood by looking at the particular case in which Ri and Ii, are identically equal to zero, as in the 288 Rinaldi and Gragnani case in which an individual has lost the partner. If we assume that in such extreme conditions Xj(t) vanishes exponentially at a rate ai, we must write so that we can obtain In the following, the parameter ai. is called forgetting coefficient and its inverse 1/ai is a measure of the time needed to forget the partner after separation. Of course, a, is a behavioral parameter of the individual, but it is certainly influenced by culture and religion. In conclusion, our model for secure individuals, is where the functions Ri, and Ii annihilate for xj = 0 and Aj = 0 and satisfy the following properties (see Fig. 1) Each individual i is identified in the model by two parameters (the appeal A and the forgetting coefficient ai) and two functions (the return function Ri and the instinct function Ii). Such parameters and functions are assumed to be constant in time: this rules out aging, learning and adaptation processes which are often important over a long range of time (Kobak & Hazan, 1991; Scharfe & Bartholomew, 1994) and sometimes even over relatively short periods of time (Fuller & Fincham, 1995). The a-priori estimate of these parameters and functions is undoubtedly a difficult task, although some studies on attachment styles (Bartholomew & Horowitz, 1991; Love Dynamics Between Secure Individuals: A Modeling Approach 289 Carnelly & Janoff-Bulman, 1992; Griffin & Bartholomew, 1994) might suggest ways for identifying categories of individuals with high or low reactiveness to love and appeal. By contrast, the accurate estimate of the parameters appears forbiddingly difficult: it would require sitting down with a couple for weeks to obtain data from both partners on a long list of questions (Levinger, 1980). This identification problem will not be considered in this paper, which is centered only on the derivation of the properties of the model. CONSEQUENCES AT INDIVIDUAL LEVEL The analysis of model (1), taking into account properties (2), points out a series of properties concerning the dynamics of the feelings and the impact that individual behavior has on the quality of romantic relationships. In the following, we simply state and interpret these properties, while their formal derivations are reported in Appendix. As already stated, we consider only the case of positive appeals. Property 1 // the feelings are non-negative at a given time, then they are positive at any future time. This property implies that an undisturbed love story between secure individuals can never enter a phase of antagonism. In fact when two individuals first meet, say at time 0, they are completely indifferent one to each other. Thus, x1 (0) = x 2 (0) = 0 and Property 1 implies that the feelings immediately become positive and remain positive forever. Actually it can be noted that model (1) implies that at time 0 i.e., at the very beginning of the love story the instantaneous rates of change of the feelings are determined only by the appeals and are therefore positive if the appeals are such. The fact that antagonism can never be present in a couple of secure individuals is obviously against observations. Indeed, in real life the feelings between two persons are also influenced by facts that are not taken into account in the model. We can therefore imagine that most of the times model (1) describes correctly the dynamics of the 290 Rinaldi and Gragnani feelings and this is what we have called "undisturbed" behavior. But from time to time unpredictable, and hence unmodelled, facts can act as disturbances on the system. A typical example is a temporary infatuation for another person giving rise to a sudden drop in interest for the partner. Of course, heavy disturbances can imply negative values of the feelings. It is therefore of interest to know what happens in such cases after the disturbance has ceased. The answer is given by the following property. Property 2 Couples can be partitioned into robust and fragile couples. As time goes on, the feelings x 1 (t) and x 2 (t) of the individuals forming a robust couple tend toward two constant positive values no matter what the initial conditions are. By contrast, in fragile couples, the feelings evolve toward two positive values only if the initial conditions are not too negative and toward two negative values, otherwise. Figure 2 illustrates Property 2 by showing in state space the trajectories of the system starting from various initial conditions. Each trajectory represents the contemporary evolution of X 1 ( t ) and x2(t) and the arrows indicate the direction of evolution. Note that, in accordance with Property 1, no trajectory leaves the first quadrant. In Fig. 2a, corresponding to robust couples, all trajectories tend toward point E + = (x 1 + , x 2+ ), which is therefore a globally attracting equilibrium point. By contrast, fragile couples (Fig. 2b) have two alternative attractors (points E+ and E-) with basins of attraction delimited by the dotted line. Figure 2b points out an interesting fact. Suppose that the couple is at the positive equilibrium E+, and that for some reason individual 2 has a drop in interest for the partner. If the drop is not too large the couple recovers to the positive equilibrium after the disturbance has ceased (see trajectory starting from point 1). But if the disturbance has brought the system into the other basin of attraction (see point 2), the couple will tend inevitably toward point E-, characterized by pronounced and reciprocal antagonism. This is why such kind of couples have been called fragile. In conclusion, robust couples are capable of absorbing disturbances of any amplitude and sign, in the sense that they recover to a high quality romantic relationship after the disturbance has ceased. On the contrary, individuals forming a fragile couple can become permanently antagonist after a heavy disturbance and never recover to their original high quality mode of behavior. Of course, it would be interesting to have a magic formula that could predict if a couple is robust or fragile. Unfortunately, this is not possible without specifying the functional form of the return functions, a choice that Love Dynamics Between Secure Individuals: A Modeling Approach 291 Fig. 2. Evolution of the feelings in a robust couple (a) and in a fragile couple (b) for different initial conditions. we prefer to avoid, at least for the moment. Nevertheless, we can in part satisfy our curiosity with the following property. Property 3 If the reactions I1 and I2 to the partner's appeal are sufficiently couple is robust. high, the 292 Rinaldi and Gragnani This property is rather intuitive: it simply says that very attractive individuals find their way to reconciliate. Figure 2 shows that the trajectory starting from the origin tends toward the positive equilibrium point E+ without spiraling around it. This is true in general, as pointed out by the following property. Property 4 Two individuals, completely indifferent one to each other when they first meet, develop a love story characterized by smoothly increasing feelings tending toward two positive values. This property implies that turbulent relationships with more or less regular and pronounced ups and downs are typical of couples composed of non secure individuals. Thus, we can, for example, immediately conclude that Laura and/or Petrarch were not secure individuals. We can now focus on the influence of individual parameters and behavior on the quality of romantic relationships at equilibrium. The first property we point out specifies the role of the appeals. Property 5 An increase of the appeal Ai of individual i gives rise to an increase of the feelings of both individuals at equilibrium. Moreover, the relative increase is higher for the partner of individual i, The first part of this property is rather obvious, while the second part is more subtle. Indeed it states that there is a touch of altruism in a woman [man] who tries to improve her [his] appeal. We conclude our analysis by considering perturbations of the behavioral characteristics of the individuals. A positive perturbation of the instinct function of individual i gives rise to a higher value of Ii(Aj) and is therefore equivalent to a suitable increase of the appeal Aj. Thus, one can rephrase Property 5 and state that an increase of the instinct function Ii, of individual i gives rise to an increase of the feelings of both individuals at equilibrium and that the relative improvement is higher for individual i. A similar property holds for perturbations of the return functions as indicated below. Love Dynamics Between Secure Individuals: A Modeling Approach 293 Property 6 An increase of the return function Ri of individual i gives rise to an increase of both feelings at equilibrium, but in relative terms such an improvement is more rewarding for individual i. This is the last relevant property we have been able to extract from model (1). Although some of them are rather intuitive, others are more intriguing. But what is certainly surprising is that they are all mere logical consequences of our assumptions (2). CONSEQUENCES AT COMMUNITY LEVEL Now that we have identified the individual consequences of properties (2), we can use them to extend the study to a more aggregated level. For this we will develop a purely theoretical exercise, dealing with a hypothetical community composed only of couples of secure individuals. Nevertheless, since secure individuals are a relevant fraction of population, we can be relatively confident and hope that our results retain, at least qualitatively, some of the most significant features of real societies. Consistently with model (1), an individual i is identified by appeal, forgetting coefficient, return function and instinct function, i.e., by the quadruplet (A i , ai, Ri ,Ii,). Thus, a community composed of N women and N men structured in couples is identified by [A1 n, a1n, R 1 n ,I 1 n , A2n, a2n, R 2n , I2n], where the integer n = 1, 2, ..., N is the ordering number of the couple. For simplicity, let us assume that individual 1 is a woman and 2 is a man and suppose that there are no women or men with the same appeal, i.e., Aih =Aik for all h = k. This means that the couples can be numbered, for example, in order of decreasing appeal of the women. Since we have assumed that all individuals are secure, properties (2) hold for all individuals of the community. Moreover, in order to maintain the mathematical difficulties within reasonable limits, we perform only an equilibrium analysis and assume that all fragile couples are at their positive equilibrium E+. In conclusion, our idealized community is composed of couples of secure individuals in a steady and high quality romantic relationship. Such a community is called unstable if a woman and a man of two distinct couples believe they could be personally advantaged by forming a new couple together. In the opposite case the community is called stable, Thus, practically speaking, unstable communities are those in which the separation and the formation of couples are quite frequent. Obviously, this definition must be further specified. The most natural way is to assume that individual i would have a real advantage in changing the partner, if this change is accompanied by an increase of the quality of the romantic 294 Rinaldi and Gragnani relationship, i.e., by an increase of xi+. However, in order to forecast the value x1+[x2+] that a woman [man] will reach by forming a couple with a new partner, she [he] should know everything about him [her]. Generally, this is not the case and the forecast is performed with limited information. Here we assume that the only available information is the appeal of the potential future partner and that the forecast is performed by imagining that the forgetting coefficient and the return and instinct functions of the future partner are the same as those of the present partner. Thus, the actual quality x1+ of the romantic relationship for the woman of the h-th couple is x1+( A1 a1 h , R f , /A A2h, o.2h, R2h, I2h) while the quality she forecasts by imagining to form a new couple with the man of the k-th couple is x 1 +(A 1h , a1h, R1h, I1h, A2k, a2h, R2h, I2h). This choice of forecasting the quality of new couples obviously emphasizes the role of appeal. Quite reasonably, however, because appeal is the only easily identifiable parameter in real life. The above discussion is formally summarized by the following definition. Definition 1 A community [A 1 n , a1n", R1n, I1n, A2n, a2n, R2n, I 2n ), n = 1, 2 , N is unstable if and only if there exists at least one pair (h, k) of couples such that A community which is not unstable is called stable. We can now show that stable communities are characterized by the following very simple but peculiar property involving only appeal. Property 7 A community is stable if and only if the partner of the n-th most attractive woman (n = 1, 2, ..., N) is the n-th most attractive man. In order to prove this property, note first that Property 5 implies that a community is unstable if and only if there exists at least one pair (h, k) of couples such that Love Dynamics Between Secure Individuals: A Modeling Approach 295 Condition (3) is illustrated in Fig. 3a in the appeal space, where each couple is represented by a point. Consider now a community in which the partner of the n-th most attractive woman is the n-th most attractive man (n = 1, 2, ..., N). Such a community is represented in Fig. 3b, which clearly shows that there is no pair (h, k) of couples satisfying inequalities (3). Thus, the community is stable. On the other hand, consider a stable community and assume that the couples have been numbered in order of decreasing appeal of the women, i.e. Then, connect the first point (A 11 , A 21 ) to the second point (A 12 , A22) with a segment of a straight-line, and the second to the third, and so on until the last point (A1N, A2N) is reached. Obviously, all connecting segments have positive slopes because, otherwise, there would be a pair of couples satisfying condition (3) and the community would be unstable (which would contradict the assumption). Thus, A21 > A22 > ... > A2N. This, together with (4), states that the partner of the n-th most attractive woman is the n-th most attractive man. On the basis of Property 7, higher tensions should be expected in communities with couples in relevant conflict with the appeal ranking. This result, derived from purely theoretical arguments, is certainly in agreement with empirical evidence. Indeed, partners with very uneven appeals are rarely observed. Of course, in making these observations one must keep in mind that appeal is an aggregated measure of many different factors and Fig. 3. Community structures in the appeal space: (a) two points corresponding to two couples (h, k) belonging to an unstable community (see (3)); (b) an example of a stable community (each dot represents a couple). 296 Rinaldi and Gragnani that gender differences might be relevant. Thus, for example, the existence of couples composed of a beautiful young lady and an old but rich man does not contradict the theory, but, instead, confirms a classical stereotype. CONCLUDING REMARKS A minimal model of love dynamics between secure individuals has been presented and discussed in this paper with two distinct goals, one generic and one specific. The generic goal was to show how one could possibly deal with dynamic phenomena in social psychology by means of the modelling approach based on differential equations and traditionally used in other fields of sciences. This approach is quite powerful for establishing a hierarchy between different properties and distinguishing between causes and effects. The specific goal was to derive a series of properties concerning the quality and the dynamics of romantic relationships in couples composed of secure individuals. The model equations take into account three mechanisms of love growth and decay: the pleasure of being loved, the reaction to the partner's appeal, and the forgetting process. For reasonable assumptions on the behavioral parameters of the individuals, the model turns out to enjoy a number of remarkable properties. It predicts, for example, that the feelings of two partners vary monotonically after they first meet, growing from zero (complete indifference) to a maximum. The value of this maximum, i.e., the quality of the romantic relationship at equilibrium, is higher if appeal and reactiveness to love are higher. The model explains also why there are two kinds of couples, called robust and fragile. Robust couples are those that have only one stationary mode of behavior characterized by attachment. By contrast, fragile couples can also be trapped in an unpleasant mode of behavior characterized by antagonism. All these properties are in agreement with common wisdom on the dynamics of love between two persons. These properties have been used to derive the characteristics under which the couples of a given community have no tendency to separate (stability). The main result along this line is that the driving force that creates order in the community is the appeal of the individuals. In other words, couples with uneven appeals should be expected to have higher chances to brake off. As for any minimal model, the extensions one could propose are innumerable. Aging, learning and adaptation processes could be taken into account allowing for some behavioral parameters to slowly vary in time, in accordance with the most recent developments of attachment theory. Particular nonlinearities could be introduced in order to develop theories for Love Dynamics Between Secure Individuals: A Modeling Approach 297 non secure individuals (Gragnani, Rinaldi & Feichtinger, 1997). Men and women could be distinguished by using two structurally different state equations. The dimension of the model could also be enlarged in order to consider individuals with more complex personalities or the dynamics of love in larger groups of individuals. Moreover, the process followed by each individual in forecasting the quality of the relationship with a potential new partner could be modelled more realistically, in order to attenuate the role of appeal, which has been somehow overemphasized in this paper. This could be done quite naturally by formulating a suitable differential game problem. Undoubtedly, all these problems deserve further attention. ACKNOWLEDGMENTS Preparation of this article was supported by the Italian Ministry of Scientific Research and Technology, contract MURST 40% Teoria dei sistemi e del controllo. We are grateful to Gustav Feichtinger, Lucia Carli, and Frederic Jones for their suggestions and encouragement. REFERENCES Baron, R. M., Amazeen, P. G., & Beek, P. J. (1994). Local and global dynamics of social relations. In R. Vallacher & A. Nowak, (Eds.), Dynamical systems in social psychology (p. 111-138). New York: Academic Press. Bartholomew, K., & Horowitz, L. M. (1991). 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Symmetrically, forx 1 > 0 and x2 = 0 eq. (1b) gives dx2/dt > 0. Moreover, for x\\ = x2 = 0, eqs. (1) give dx1/dt > 0 and dx2/dt > 0. Hence, trajectories starting from the boundary of the positive quadrant enter into the positive quadrant and remain there forever. Proof of Property 2 The isoclines and of the system are given by (see From properties (2) it follows that these isoclines intersect either at one point with positive coordinates (see Fig. 4a), or at three points, one with Love Dynamics Between Secure Individuals: A Modeling Approach 299 positive coordinates and two with negative coordinates (see Fig. 4b). These two cases correspond, by definition, to robust and fragile couples. On the first isocline, trajectories are vertical, while on the second they are horizontal. Studying the signs of dx1/dt and dx2/dt in the regions delimited by the isoclines, one can determine the direction of the trajectory of the system at any point (x 1 ,x 2 ). These directions, indicated with arrows in Fig. 4, allow one to conclude that the equilibrium points E+ and E- are stable nodes, while the equilibrium point 5 is a saddle. Moreover, if the couple is robust (Fig. 4a) the equilibrium E+ is a global attractor, i.e., all trajectories tend toward this point as time goes on. By contrast, if the couple is fragile (Fig. 4b), there are two attractors, name