DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018028 DYNAMICAL SYSTEMS SERIES B Volume 23, Number 1, January 2018 pp. 401–423
THE ROLE OF OPTIMISM AND PESSIMISM IN THE DYNAMICS OF EMOTIONAL STATES
Monika Joanna Piotrowska∗ Institute of Applied Mathematics and Mechanics Faculty of Mathematics, Informatics and Mechanics, University of Warsaw Banacha 2, 02-097 Warsaw, Poland Joanna Gorecka´ 1 and Urszula Fory´s2 1College of Inter-Faculty Individual Studies in Mathematics and Natural Sciences University of Warsaw Zwirki˙ i Wigury 93, 02-089 Warsaw, Poland 2Institute of Applied Mathematics and Mechanics Faculty of Mathematics, Informatics and Mechanics, University of Warsaw Banacha 2, 02-097 Warsaw, Poland
Abstract. In this paper we make an attempt to study the influence of op- timism and pessimism into our social life. We base on the model considered earlier by Rinaldi and Gragnani (1998) and Rinaldi et al. (2010) in the con- text of romantic relationships. Liebovitch et al. (2008) used the same model to describe competition between communicating people or groups of people. Con- sidered system of non-linear differential equations assumes that the emotional state of an actor at any time is affected by the state of each actor alone, rate of return to that state, second actor’s emotional state and mutual sympathy. Using this model we describe the change of emotions of both actors as a result of a single meeting. We try to explain who wants to meet whom and why. Interpreting the results, we focus on the analysis of the impact of a person’s attitude to life (optimism or pessimism) on establishing emotional relations. It occurs that our conclusions are not always obvious from the psychological point of view. Moreover, using this model, we are able to explain such strange behavior as so-called Stockholm syndrom.
1. Introduction. People are social beings by nature. Almost all our life we talk, work or play with someone; contacts with other persons have a strong influence on our emotions. This impact can be positive or negative depending on various factors and it is most visible in the relationship during the meeting of two persons. People can like each other, and then a good mood of one person positively affects the mood of the other one, while a bad mood has a negative effect. Two persons may also not like each other, and then an emotional impact of one person to the other is opposite. Hence, a sadness of one person improves the mood of the other one and vice versa. The last possibility is that the relationship is mixed, and one person has a negative attitude to the second one, who is geared to their friendship. In such
2010 Mathematics Subject Classification. Primary: 91D99; Secondary: 37N99, 34D05, 34D23. Key words and phrases. Ordinary differential equations, steady states, stability, emotional state, communication, relationship. ∗ Corresponding author: Monika Joanna Piotrowska.
401 402 MONIKA JOANNA PIOTROWSKA, JOANNA GORECKA´ AND URSZULA FORYS´ a relationship the emotional impact of one person to the other is a compound of previously described influences. If we assume that our emotional state is influenced only by other people, then the obvious strategy is to be friendly for all the close-knit people and to avoid relationships with people who have negative attitude to us. In general, we hope to be like that, however if that were true, everyone would be friendly and happy, and it is not the case. Although one can try to explain this effect by saying that the feeling of pleasure is not the only desire of a man and various random events can destroy the balance, it would still not represent the full picture. On the other hand, individual factors also have an impact on our emotional state. These are certainly: the attitude to life (optimism and pessimism) and how strong is the influence of the current mood of a person on the change of her/his emotional state. In this paper, we make an attempt to analyse the influence of the level of opti- mism/pessimism on the profits from the meeting of people. We shall consider three main components which affect the changes of our emotions during the meeting with other person: the attitude to life, emotional inertia and the reaction to emotions of the person we communicate during the meeting. This suggests the usage of the model having similar components and well known in the description of dyadic in- teractions considered earlier by Rinaldi and Gragnani [10], Liebovitch et al. [6] and Rinaldi et al. [9]. Considered model is described by a system of non-linear differ- ential equations, while similar model of marital interactions reflected by discrete system was proposed by Gottman et al. [4,7]. To avoid misunderstanding, persons considered in this paper will be called actors since we do not wish to suggest that people communicating during the meeting must be involved in a romantic relationship, as it is interpreted by Rinaldi and his coauthors. Moreover, comparing to Liebovitch et al. or Rinaldi et al. we study considered model from a different point of view as we do not describe a very long time behavior but focus on a single meeting between two actors.
2. Model description. Many models describing interactions between people fol- low the idea of modeling romantic relationships proposed originally by Strogatz [15]. Such models can even be linear, as considered in [15, 16,8,3,1,2]. However, non- linear models seem to be more appropriate. In this paper we would like to use the model having the same structure as the one proposed by Rinaldi and Grag- nani [10], which was deeply exploited by Rinaldi and his coauthors in the series of papers [10,9, 11, 13, 12] in the context of the description of various types of ro- mantic relationships. The text-book of Rinaldi et al. [14] gives an excellent review of the model history, mathematical analysis, and various types of couples known from the literature and movies described by this model, as well as some extensions of it. Basing on the same idea, Gottman et al. [4,7] proposed discrete time model which was then used to predicting divorce of married couples taking part in the experimental study in Gottman’s clinic. Another interpretation of this model was given by Liebovitch et al. [6], where the authors tried to reflect emotional states of communicating people. In this context, the actors could be interpreted not only as two interacting persons, but it could be groups of persons as well. We would like to stress that in general love relationships develop in a few weeks or months, however in this paper we wish to focus on single meetings. Therefore, our interpretation of the model and its parameters is different from those proposed OPTIMISM AND PESSIMISM IN THE DYNAMICS OF EMOTIONAL STATES 403 by Rinaldi et al. [14] and Gottman et al. [4,7], as well as Liebovitch et al. [6], however is more close to the interpretation given in [6]. We consider continuous-time model reflecting the dynamics of emotional states of two actors at time t. Let x(t) and y(t) reflect these emotional states. The system of differential equations we consider reads
x˙(t) = −m1x(t) + b1 + c1f1(y(t)), (1) y˙(t) = −m2y(t) + b2 + c2f2(x(t)), where constants m1 and m2 describe the rate of change of the mood of each actor in solitude, which can be also referred as to forgetting coefficients, b1 and b2 reflect some “ideal/reference” mood of each actor, functions f1 and f2 describe the impact of the emotional state of an actor y or x, respectively, on the emotional state of the other actor, while constants c1 and c2 determine the strength and direction of these influences. Because mi, i = 1, 2, reflects forgetting coefficient, it must be positive as describ- ing some time-scale on which an actor is considered in solitude. Hence,
m1 > 0, m2 > 0. Therefore, if there is no influence, that is for c = c = 0, then after any deviation 1 2 System (1) returns to the steady state b1 , b2 , where in [4] the state bi is called m1 m2 mi uninfluenced equilibrium for the ith actor. An actor characterized by a positive parameter bi has a positive steady state and is called an optimist, while that with negative parameter – a pessimist. Ri- naldi et al. (cf. [9, 14]) gave completely different interpretation of this parameter. In this interpretation b1 reflects appeal of the actor y for x, so when the actor is in solitude, this parameter is just equal to 0, as well as uninfluenced steady state, because there is no love/hate whenever there is no object of these emotions. It should be marked that although Rinaldi et al. got interesting results using this in- terpretation (e.g. they were able to explain the case of Beauty and the Beast [11]), we shall not follow this idea, but use the interpretation of Gottman et al. [4] and then Liebovitch et al. [6]. When both c1 and c2 are positive, both actors have a positive attitude to each other, while for c1, c2 < 0 they have a negative attitude to each other. Clearly, for c1 > 0 and c2 < 0 the first actor has a positive attitude towards the second one, who has a negative attitude to the first. As we describe interactions between two actors, we mainly assume c1 · c2 6= 0. Various particular influence functions fi were considered in the literature, for details see e.g. [4,6,7, 11]. However, in this paper we propose fi in general form based on the prospect theory of decision making problems [5]. We should also mark that under our interpretation the model described by Sys- tem (1) reflects emotional states of actors during a single meeting, but not, for example, a series of meetings. This is because between two meetings considered actors meet other people or spend time in solitude, which affect their mood at the beginning of the meeting, and therefore also the final result. 2.1. Influence functions. The prospect theory proposed by Kahneman and Tver- sky [5] relates to the wider issues of risk assessment and an attitude of men to the risk. Here, we briefly introduce the assumptions, which are useful from our point of view. There are three main principles of profit and loss assessment by people. First, generally we experience losses much stronger than profits of the same value. Second, 404 MONIKA JOANNA PIOTROWSKA, JOANNA GORECKA´ AND URSZULA FORYS´ everyone defines their own criteria with respect to which results of the decision are evaluated as a gain or loss. Third, every unit of gain is enjoyed with diminishing efficiency, and each consecutive loss is less saddening. Although the prospect theory describes the relationship between profits or losses and satisfaction, we believe that it can be used to describe the mutual influence of actors’ emotional states. When we are alone our emotional state depends only on our character. When we meet a friend who is happy, we gain his emotions, but not in the literal sense. We react to ones smile, lively tone of voice, etc. When we get a more positive stimulus we enlarge our profit more. However, according to the prospect theory, each additional unit gives less and less profit. Therefore, influence functions fi are certainly non-linear. The first derivative of it should be positive, decreasing for positive variables and increasing for negative ones. Moreover, it should tend to 0 in ±∞, because otherwise fi become almost linear asymptotically. Another issue is that generally we feel losses much stronger than profits of the same value. Indeed, people recognize negative emotions faster, more accurately and more strongly than positive ones. It is an adaptive process, because when we talk about surviving, the ability to recognize and quickly respond to the feeling of fear or anger is more important than joy. Hence, the negative experience makes us more sad than the same weight positive experience makes us happy. Last feature of that theory stating that everyone assesses gains and losses from ones own point of view, actually, is not so important from our model point of view. Benchmark is always a state in solitude, that means a situation in which the environmental impact is equal to zero. However, if we do not like someone, then his negative emotions are a profit for us, and the positive emotions are our loss, while if we like someone, it is vice versa. Concluding, to address the issues described above we assume: 2 0 fi ∈ C , fi(0) = 0, fi (ξ) > 0,
fi(ξ) ≤ −fi(−ξ) for ξ > 0, and 0 00 lim fi (ξ) = 0, ξfi (ξ) < 0 for ξ 6= 0, for i = 1, 2. |ξ|→∞ Moreover, to more explicitly show that the force of impact of actors on themselves depends on the value of |ci| we also assume that 0 fi (0) = 1, for i = 1, 2.
Figure 1. Example of an influence function. We see that for |ξ2| = ξ1 we have f(ξ1) ≤ −f(ξ2). OPTIMISM AND PESSIMISM IN THE DYNAMICS OF EMOTIONAL STATES 405
Exemplary graph of an influence function is shown in Fig.1. Notice that ξ = y for the function f1, while ξ = x for f2. It is worth to notice that functions fi may have different shape because people differ in ability to distinguish emotions, and consequently – reactions to them. This 0 0 means that f1(y) and f2(x) can change with different rates. Clearly, proposed in- teraction functions are defined in the general form, and hence we actually consider the whole family of functions. On the other hand, the interaction functions con- sidered previously in the literature belong to this family. However, to the best of our knowledge, for the first time a specific psychological theory has been used to explain the form of these functions.
3. Model analysis. Properties of solutions of System (1) are known, and well described in [14]. However, for the purpose of this paper, we need to know the dynamics in the phase space for various parameter values. Therefore, we present the analysis of the phase space below. Moreover, to the best of our knowledge, the method of Lyapunov functions was not applied to this model before, so we present the results of global stability obtained by us on the basis of this method. Clearly, without particular forms of the influence functions we are not able to determine steady states explicitly. Hence, assume (xs, ys) to be a solution of the system b + c f (y) x = 1 1 1 , m 1 (2) b + c f (x) y = 2 2 2 , m2 i.e. (xs, ys) is a steady state. The first equation of (2) describes null-cline I1 for the first variable x, while the second one – I2 for y. Positions of I1 and I2 in the phase space (x, y) depend on the model parameters and specific forms of fi. In the analysis presented below we treat both null-clines as functions gi(x). Therefore, g2 0 is defined for all x ∈ R with g2(x) → 0 as |x| → ∞, while g1 takes all values from 0 R and g1(x) → ∞ as x tends to the end of its domain (either R or some bounded interval). Clearly, if c1c2 < 0, then one of the null-clines is increasing, while the other is decreasing, and therefore they intersect at exactly one point. If c1c2 > 0, then System (2) has always from one to three solutions depending on the parameter values. We briefly discuss the case c1, c2 > 0, as the case c1, c2 < 0 is symmetric. Consider first b1 = b2 = 0. Then null-clines always intersect at x = 0 and other 0 0 intersection points appear when g2(0) > g1(0), that is c2/m2 > m1/c1, while for 0 0 0 0 g2(0) ≤ g1(0) there is only one intersection. For g2(0) ≤ g1(0) shifting null-clines 0 0 does not change the situation, while for g2(0) > g1(0), if we shift at least one of the null-clines sufficiently far from the origin, then additional intersection points disappear. 1. System (1) has one steady state if • either c1c2 ≤ m1m2, • or c1c2 > m1m2 and there is no solution of System (2) satisfying
0 0 m1m2 f1(ys)f2(xs) ≥ . c1c2
2. System (1) has two steady states if c1c2 > m1m2, b1 6= 0 or b2 6= 0, and solutions of System (2) fulfill the conditions 406 MONIKA JOANNA PIOTROWSKA, JOANNA GORECKA´ AND URSZULA FORYS´
0 0 m1m2 0 0 m1m2 f1(y1)f2(x1) = and f1(y2)f2(x2) < . c1c2 c1c2 3. System (1) has three steady states if c1c2 > m1m2 and there exists a solution 0 0 m1m2 of System (2) fulfilling f (ys)f (xs) > . 1 2 c1c2 0 0 m1m2 Two additional steady states (xi, yi), i ∈ {1, 2} fulfill f (yi)f (xi) < . 1 2 c1c2 To study local stability of steady states we calculate the Jacobi matrix at a steady state (xs, ys), 0 −m1 c1f1(ys) J(xs, ys) = 0 , c2f2(xs) −m2 and the corresponding characteristic polynomial W (λ) = λ2 − Aλ + B, with 0 0 A = −(m1 + m2) < 0,B = m1m2 − c1c2f2(xs)f1(ys). √ A ± A2−4B Consequently, eigenvalues are equal to λ1,2 = 2 . From the assumptions we have A < 0. Therefore, stability depends only on the value of B. We see that if B is positive, then the steady state (xs, ys) is stable, and it is either focus (for A2 − 4B < 0) or node (for A2 − 4B > 0), while if B is negative, (xs, ys) is a saddle. Moreover, whenever there exists the unique steady state (xs, ys) and the impact of actors to each other is not higher than the inner dynamics of them, then (xs, ys) is globally stable, while if there are three steady states, then the dynamics changes to bistable, which means that we have two locally stable steady states.
Theorem 3.1. If m1m2 > |c1c2| and System (1) has exactly one steady state (xs, ys) satisfying System (2), then this state is globally stable. Proof. Proving global stability we use the method of Lyapunov functions. Let us define 1 C V (x, y) = (x − x )2 + (y − y )2 , 2 s 2 s where C > 0 is a constant, which should be chosen in appropriate way. Calculating the derivative along trajectories of System (1) we get V˙ (x, y) = (x − xs) − m1x + b1 + c1f1(y) + C (y − ys) − m2y + b2 + c2f2(x) .
Using the relations b1 = m1xs −c1f1(ys), b2 = m2ys −c2f2(xs), and the mean value theorem we obtain 2 2 V˙ (x, y) = − m1 (x − xs) + Cm2 (y − ys)