Modeling Love Dynamics

The following sections are included:

• Preface
• Contents

In this introductory chapter we first recall that the evolution over time of an interpersonal relationship can be described graphically. The idea is a relatively unsophisticated one because it assumes that the interest (feeling) of one person in another can be captured by a single variable. Thus, a love story is represented by two graphs showing the evolution over time of the feelings of each of the lovers. The same love story can be represented more compactly by a single curve showing the simultaneous evolution of the feelings in a two-dimensional space. In the jargon of dynamical systems theory, this curve is called the trajectory and starts from the point that represents the initial feelings of the two individuals. A set of trajectories starting from different initial conditions is called the state portrait and is a very effective tool for discussing the consequences of various factors influencing the couple (e.g., secret extramarital affairs)…

In this part, composed of nine chapters, we deal with models in which each individual is characterized by a single variable—the interest of the partner. This means that these models, called simple, cannot be used to mimic love stories between individuals who are stressed by their social environments, who have extra emotional dimensions, or who are involved in two or more romantic relationships. All theoretical results are discussed with reference to couples composed of hypothetical individuals, while the examples refer to very specific stories described in well known films—“Cyrano de Bergerac”, “Beauty and The Beast”, “Pride and Prejudice”, “Gone with the Wind”, and “Jules et Jim”. This gives the reader a chance to better evaluate our approach and propose alternative models for the interpretation of some of the stories.

In this chapter we analyze the most simple class of couples, namely, that composed of secure and unbiased individuals, sometimes called standard. We here consider the very special case of linear reaction functions, while the general case is studied in the next chapter. The linearity assumption simplifies the study that can be performed analytically using the powerful theory of linear dynamical systems…

We continue in this chapter with the analysis of couples composed of secure and unbiased individuals (standard couples) by relaxing, with respect to the previous chapter, the unrealistic assumption of linearity (hence, of unboundedness) of the reaction to love. Moreover, we no longer constrain the appeals to be positive. The main property that makes this model radically different from the linear one is that for suitable values of its parameters it has two alternative stable states. This property can easily be proved by looking at the shape of the two null-clines and by noticing that variations of her [his] appeal shift vertically [horizontally] his [her] null-cline. The result is that in some couples (called robust) the null-clines intersect at a single (globally stable) equilibrium point, while in other couples (called fragile) they intersect at three distinct points, one unstable and two stable. Moreover, one of the two stable states is better than the other, in the sense that its components (the feelings of the two individuals) are greater than the corresponding ones of the other state. This means that one of the alternative stable states is satisfactory while the other is not. A fragile couple can therefore be in a satisfactory or unsatisfactory romantic regime depending upon their past history…

In this chapter a famous love story is analyzed using a mathematical model, with the aim of showing the power of temporary bluffing pointed out in the previous chapter…

This chapter is also devoted to the study of standard couples, even if our message, namely, that “small discoveries can have great consequences in love affairs,” holds true for many other kinds of couples. From a formal point of view, these great consequences are the result of so-called catastrophic bifurcations. They are important because in general they are associated with great emotions that emerge when there are dramatic breakdowns or explosions of interest. Examples of the first kind are all relationships based mainly on sex. Indeed, sexual appetite inexorably decreases over time so that a point can easily be reached where separation is both unavoidable and unexpected if the individuals are mainly interested in sex. A well known example of the second type is playboys who systematically reinforce their appeal until their prey falls in love with them…

In this chapter we study couples composed of insecure and unbiased individuals. Thus, the only difference with respect to the case of standard couples considered in Chapters 3, 4, and 5 is that the reaction functions first increase and then decrease with the love of the partner. This implies that the geometry of the two null-clines is different from that in the previous chapters, but that variations of the appeals still shift (vertically or horizontally) the two null-clines. For this reason, the main properties of the couple can still be derived from simple geometric considerations based on the geometry of the null-clines. For simplicity, we study only the case in which the reaction to love increases up to a threshold value of the love of the partner and then declines but remains positive. This implies that the analysis can be performed by looking only at trajectories in the positive quadrant of the space of the feelings because initially non-antagonistic partners remain so forever. Moreover, the reaction to love is described by a function identified by two parameters interpretable as reactiveness to love and degree of insecurity (higher insecurities characterize reactions to love that start declining earlier)…

This chapter is devoted to the study of the love story described in “Gone with the Wind”, one of the most popular films of all times. The film, released in Atlanta in December 1939, starring Vivien Leigh as Scarlett and Clark Gable as Rhett (see Figure 7.1), was inspired by Margaret Mitchell's 1936 bestseller. It was one of the first films in color, was awarded eight Oscars, has been seen at least once by 90% of Americans, and was, financially, the most successful film ever, until a few years ago…

We have seen in the preceding chapters that two unbiased individuals cannot have recurrent ups and downs in their feelings if they are both secure or both insecure. Since turbulent love stories exist and have attracted a lot of attention (see, for example, Chapters 9, 10, and 13), we investigate in this chapter the existence of periodic romantic regimes using our models…

In this chapter we study the love story involving Helen Grund—a brilliant and charming journalist from Berlin—and Franz Hessel—a profound but shy German writer. Their relationship is characterized by ups and downs: they married and divorced twice and lived again together after the second divorce…

In this chapter we study the love story between Helen Grund and Henry-Pierre Roché, the friend of her husband. As in the previous chapter, we use the novel “Jules et Jim” as a reliable source of information to understand the characters of the two lovers (Kathe and Jim, see Figure 10.1). Jim is insecure, as all “Don Juans” are to avoid deep involvements and unbiased. Kathe is secure and synergic. Thus, insecurity and synergism—the ingredients that have been shown to be necessary for the emergence of romantic turbulence—are present in the couple. With the parameters of the model fixed at realistic reference values, simulations show that the couple is characterized by a periodic romantic regime. The period of the cycle, as well as other details, are in agreement with the novel. Moreover, the results of a simple bifurcation analysis support the credibility of the identified romantic cycle…

In this part, composed of five chapters, we describe complex models obtained by relaxing the simplifying assumptions we made in the first part. We show that with the addition of one or few variables it is virtually possible to determine the impact of the social environment on romantic relationships, identify the potential consequences of extra emotional dimensions, like artistic inspiration, and extend the analysis to triangular love stories. The analysis from the models suggests a number of interesting conclusions concerning the emergence of turbulence and unpredictability in the evolution of love affairs. These theoretical results are reinforced through the analysis of two famous love stories. The first is described by Francesco Petrarch in his “Canzoniere”, the most celebrated collection of love poems in the Western world, and the second is the triangular love story described by Henry-Pierre Roché in his novel “Jules et Jim”, popularized by the homonymous film of François Truffaut.

In this chapter we remove one of the main simplifying assumptions made until now, namely, the absence of exogenous events influencing the life of the couple. In the proposed model all the exogenous factors are lumped together and captured by a single variable, called environmental stress. Some stresses are strong but cease after a short time (shocks), while others maintain their strength for a long time or have periodic ups and downs. In the model these stresses can be represented with impulse, step, and sinusoidal functions of time. The responses to these particular environmental stresses, called canonical responses, are interesting because they mimic a number of real or realistic cases. But they are also interesting because they can be fully derived analytically in the case of linear models (see Chapter 2). This is of great value in all cases in which the stresses are not too heavy because in such cases the couple can be approximately described with a linear model (derived by linearizing the nonlinear model). In the cases of linear or linearized models, the asymptotic patterns of the responses are simply replicates of the patterns of the stresses. For example, if the stress is periodic, the feelings vary in the same way and at the same frequency, in agreement with the theory of linear systems. But if there are nonlinearities and the stresses are quite heavy, the couple can be more complex than a simple replicator. For example, the romantic regime can be chaotic even if the stress is periodic. An example is dedicated to this case and a number of interesting properties are pointed out. In particular, it is shown that romantic chaos is more easily promoted if the frequency of the stress (environmental clock) is tuned with that of the couple (romantic clock)…

Until now, individuals have been assumed to be endowed with a single emotional dimension—the romantic sphere. In reality, almost all individuals have other emotional dimensions some of which interfere with the romantic dimension. For example, involvement in professional activity can strongly influence the quality of romantic relationships…

In this chapter we study the 21-year love story between Laura, a beautiful but married lady, and Francesco Petrarch a famous Italian poet of the XIV century (see Figure 13.1). Their relationship is described in the “Canzoniere,” one of the most celebrated collection of love poems in the Western world. She is strongly insecure; he is secure but with a reaction to appeal inhibited by his poetic inspiration, described by the equation used in the previous chapter. Thus, only one of the two individuals has an extra emotional dimension and the model is three-dimensional…

In this chapter we focus on triangular relationships and show that conflict and jealousy can easily trigger unpredictability. For this, we first identify the structure of all possible triangles with one central individual involved in two romantic relationships. The two non-central individuals can be aware or not of the existence of the triangular relationship, and hence be jealous or not, while the central individual can or cannot feel the conflict between the involvements for the two lovers. Combining these features in all significant ways, we obtain six structurally different triangles that are analyzed separately…

In this chapter we study the triangular love story involving Helen Grund, a brilliant and charming journalist, her husband Franz Hessel, a profound but shy German writer, and his best friend Henry-Pierre Roché. The story is described in detail in the autobiographic novel “Jules et Jim”, written in 1953 by Roché, and popularized in 1961 by the homonymous film of François Truffaut—one of the prominent directors of the “Nouvelle Vague”. In the novel, and in the film, Helen, Franz, and Henry-Pierre are Kathe, Jules, and Jim, respectively (see Figure 15.1). The story has two distinctive features. First, the two friends are introduced to Kathe practically at the same time, so that all initial feelings can be set to zero. This simplifies the analysis and eliminates the ambiguities concerning possible alternative attractors. Second, although she is permanently involved with both friends, Kathe is always in a monogamic relationship and changes partner each time, there is an inversion in her preference ranking…

This Appendix is devoted to readers who are not yet familiar with mathematical modeling and, in particular, with dynamical systems and their bifurcations. The readers who have never worked with mathematical models could read the Appendix entirely, starting with the first section, where the sense of developing a model is explained through a simple example concerning the rate of sexual intercourse in a couple. In contrast, it is suggested that readers who already have some experience in mathematical modeling but are not familiar with the theory of dynamical systems and its jargon (trajectories, equilibria, limit cycles and their stability) start with Section A.2. Finally, those who only lack the notion of bifurcations and structural stability can start from Section A.3.

The following sections are included:

• Bibliography
• Index