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

  • * Corresponding author: Monika Joanna Piotrowska

    * Corresponding author: Monika Joanna Piotrowska 
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  • 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.

    Mathematics Subject Classification: Primary: 91D99; Secondary: 37N99, 34D05, 34D23.

    Citation:

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  • Figure 1.  Example of an influence function. We see that for $|\xi_2|=\xi_1$ we have $f(\xi_1)\leq -f(\xi_2)$

    Figure 2.  All possible types of the phase space portrait of System (1) illustrating Theorem 3.3. (a): unique globally attractive focus; (b): unique globally attractive node; (c): bifurcation from one stable steady state to bistability; (d): bistability

    Figure 3.  An example of the situation when two actors reach the same emotional state independently if they meet each other or not

    Figure 4.  An example of the change of emotional states of two friends with the opposite initial states

    Figure 5.  Emotional states of two strongly dependent on each other friends with the same parameters describing both of them but with the opposite initial states

    Figure 6.  Examples of changes in emotional states of two actors with the opposite attitude to each other. Top: the steady state is a stable focus and emotions oscillate. Bottom: the steady state is a stable node and emotions slowly approach the equilibrium

    Figure 7.  En example of two actors for which different initial conditions yield different opposite final emotional states

    Figure 8.  Scheme of six possible interactions of the pessimist ($y$) with the other actor ($x$) who could be a pessimist or optimist. (a): $y$ is negatively ($c_{1} < 0$) and $x$ is positively ($c_{2}>0$) oriented; (b): $y$ is positively ($c_{1}>0$) and $x$ is negatively ($c_{2} < 0$) oriented; (c): both $x$ and $y$ are positively oriented ($c_{1}, c_{2}>0$) and $m_{1}m_{2} > c_{1}c_{2}$ indicating the existence of exactly one steady state; (d): both $x$ and $y$ are negatively oriented ($c_{1}, c_{2} < 0$) and $m_{1}m_{2} > c_{1}c_{2}$ indicating the existence of exactly one steady state; (e): both $x$ and $y$ are positively oriented ($c_{1}, c_{2}>0$) and $m_{1}m_{2} < c_{1}c_{2}$ indicating that there might exist more than one steady state; (f): both $x$ and $y$ are negatively oriented ($c_{1}, c_{2}>0$) and $m_{1}m_{2} < c_{1}c_{2}$ indicating that there might exist more than one steady state

    Figure 9.  An example of emotional states of two pessimists who have strong influence on each other

    Figure 10.  Graph of emotional states of two pessimists who are enemies. Signs of the initial conditions are opposite. Horizontal doted line indicates the uninfluenced equilibriums of both of actors

    Figure 11.  Relationships between two optimists: (g) $y$ is positively while $x$ is negatively oriented ($c_1 < 0, c_2>0$); (h) $y$ is negatively while $x$ is positively oriented ($c_1>0, c_2 < 0$); (i) both actors are negatively oriented $(c_1, c_2 < 0)$ and $m_{1}m_{2} > c_{1}c_{2}$ indicating the existence of exactly one steady state; (j) both actors are negatively oriented $(c_1, c_2 < 0)$ and $m_{1}m_{2} < c_{1}c_{2}$ indicating the existence of up to three steady states; (k) both actors are positively oriented $(c_1, c_2>0)$ and $m_{1}m_{2} > c_{1}c_{2}$ indicating the existence of exactly one steady state; (l) both actors are positively oriented $(c_1, c_2>0)$ and $m_{1}m_{2} < c_{1}c_{2}$ indicating the existence of up to three steady states

    Figure 12.  Possible changes of the dynamics of emotional states initiated by one of the interacting actors

    Figure 13.  Comparison of the dynamics of emotional states of two pessimists when they are friends ($c_{1}, c_{2}>0$) and when they have inconsistent relations ($c_{1}>0$, $c_{2} < 0$)

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