The role of optimism and pessimism in the dynamics of emotional states

Abstract

In this paper we make an attempt to study the influence of optimism 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 context of romantic relationships. Liebovitch et al. (2008) used the same model to describe competition between communicating people or groups of people. Considered 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.

Time delays in dyadic interactions on the example of relations between optimists and pessimists

“…For the mathematical analysis of the model, we do not postulate any particular form of the influence functions; however, they are the same for both actors and could include asymmetry in the feeling of negative and positive emotions of actors. Thus, staying in line with the assumptions posed previously in Piotrowska et al, 1 we consider functions f i fulfilling the following assumptions:…”

“…Note that Assumptions A1 and A3 together with boundedness of f i correspond to the assumptions posed on the influence functions in Rinaldi and Gragnani 2 where relations between secure individuals are described. Detailed analysis of system (1) can be found in previous studies 1,10,13 thus, below, we only summarize results regarding the existence and stability of steady states. Let (x,̄) represent a steady state, that is, the right-hand side of system (1) is set to zero.…”

“…Proposition 1 (existence). Depending on the parameters, system (1) has from one to three steady states:…”

“…Proposition 2 (local stability). Let S = (x,̄) be a steady state of system (1) and C be given by (3) Moreover, using Dulac-Bendixson criterion, it can be shown that system (1) has no periodic solutions. In addition, Poincaré-Bendixson theorem implies that if in some bounded region, there is only one steady state, then it is globally stable, for details see Piotrowska et al 1 and Rinaldi et al 3 Alternatively, constructing Lyapunov functions, one can show global stability of steady states under some additional conditions, see Piotrowska et al 1 The existence and stability results are combined in the following corollary.…”

“…Theorem 2. Let S = (x,̄) be a steady state of system (4) with 2 = 4 = 0 and m 1 m 2 > C. 1. If C > − m 1 m 2 , then at least one stability switch with increasing 1 , 3 may occur.…”