# American Institute of Mathematical Sciences

January  2018, 23(1): 401-423. doi: 10.3934/dcdsb.2018028

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

 1 College of Inter-Faculty Individual Studies in Mathematics and Natural Sciences, University of Warsaw, Żwirki i Wigury 93, 02-089 Warsaw, Poland 2 Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

* Corresponding author: Monika Joanna Piotrowska

Received  September 2016 Revised  April 2017 Published  January 2018

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.

Citation: Monika Joanna Piotrowska, Joanna Górecka, Urszula Foryś. The role of optimism and pessimism in the dynamics of emotional states. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 401-423. doi: 10.3934/dcdsb.2018028
##### References:
 [1] N. Bielczyk, M. Bodnar and U. Foryś, Delay can stabilize: Love affairs dynamics, Applied Mathematics and Computation, 219 (2012), 3923-3937.  doi: 10.1016/j.amc.2012.10.028. [2] N. Bielczyk, U. Foryś and T. Płatkowski, Dynamical models of dyadic interactions with delay, J. Math. Soc., 37 (2013), 223-249.  doi: 10.1080/0022250X.2011.597279. [3] D. H. Felmlee and D. F. Greenberg, A dynamic systems model of dyadic interaction, Journal of Mathematical Sociology, 23 (1999), 155-180.  doi: 10.1080/0022250X.1999.9990218. [4] J. M. Gottman, J. D. Murray, C. C. Swanson, R. Tyson and K. R. Swanson, The Mathematics of Marriage: Dynamic Nonlinear Models, MIT Press, Cambridge, 2002. [5] D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.  doi: 10.21236/ADA045771. [6] L. Liebovitch, V. Naudot, R. Vallacher, A. Nowak, L. Biu-Wrzosinska and P. Coleman, Dynamics of two-actor cooperation-competition conflict models, Physica A, 387 (2008), 6360-6378. [7] J. D. Murray, Mathematical Biology. Ⅰ: An Introduction, Springer-Verlag, New York, 2002. [8] S. Rinaldi, Love dynamics: The case of linear couples, Applied Mathematics and Computation, 95 (1998), 181-192.  doi: 10.1016/S0096-3003(97)10081-9. [9] S. Rinaldi, F. Della Rosa and F. Dercole, Love and appeal in standard couples, International Journal of Bifurcation and Chaos, 20 (2010), 3473-3485.  doi: 10.1142/S0218127410027829. [10] S. Rinaldi and A. Gragnani, Love dynamics between secure individuals: A modeling approach, Nonlinear Dynamics, Psychology, and Life Sciences, 2 (1998), 283-301. [11] S. Rinaldi, P. Landi and F. Della Rosa, Small discoveries can have great consequences in love affairs: The case of beauty and the beast International Journal of Bifurcation and Chaos 23 (2013), 1330038, 8pp. doi: 10.1142/S0218127413300383. [12] S. Rinaldi, P. Landi and F. Della Rossa, Temporary bluffing can be rewarding in social systems: The case of romantic relationships, The Journal of Mathematical Sociology, 39 (2015), 203-220.  doi: 10.1080/0022250X.2015.1022280. [13] S. Rinaldi, F. D. Rossa and P. Landi, Landi, A mathematical model of pride and prejudice?, Nonlinear dynamics, psychology, and life sciences, 18 (2014), 199-211. [14] S. Rinaldi, F. Rossa Della, F. Dercole, A. Gragnani and P. Landi, Modeling Love Dynamics vol. 89 of World Scientific Series on Nonlinear Science Series A, World Scientific Publishing Co. Pte. Ltd., 2016. [15] S. Strogatz, Love affairs and differential equations, Math. Magazine, 65 (1988), p35. [16] S. Strogatz, Nonlinear Dynamics and Chaos, Westwiev Press, 1994.

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##### References:
 [1] N. Bielczyk, M. Bodnar and U. Foryś, Delay can stabilize: Love affairs dynamics, Applied Mathematics and Computation, 219 (2012), 3923-3937.  doi: 10.1016/j.amc.2012.10.028. [2] N. Bielczyk, U. Foryś and T. Płatkowski, Dynamical models of dyadic interactions with delay, J. Math. Soc., 37 (2013), 223-249.  doi: 10.1080/0022250X.2011.597279. [3] D. H. Felmlee and D. F. Greenberg, A dynamic systems model of dyadic interaction, Journal of Mathematical Sociology, 23 (1999), 155-180.  doi: 10.1080/0022250X.1999.9990218. [4] J. M. Gottman, J. D. Murray, C. C. Swanson, R. Tyson and K. R. Swanson, The Mathematics of Marriage: Dynamic Nonlinear Models, MIT Press, Cambridge, 2002. [5] D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.  doi: 10.21236/ADA045771. [6] L. Liebovitch, V. Naudot, R. Vallacher, A. Nowak, L. Biu-Wrzosinska and P. Coleman, Dynamics of two-actor cooperation-competition conflict models, Physica A, 387 (2008), 6360-6378. [7] J. D. Murray, Mathematical Biology. Ⅰ: An Introduction, Springer-Verlag, New York, 2002. [8] S. Rinaldi, Love dynamics: The case of linear couples, Applied Mathematics and Computation, 95 (1998), 181-192.  doi: 10.1016/S0096-3003(97)10081-9. [9] S. Rinaldi, F. Della Rosa and F. Dercole, Love and appeal in standard couples, International Journal of Bifurcation and Chaos, 20 (2010), 3473-3485.  doi: 10.1142/S0218127410027829. [10] S. Rinaldi and A. Gragnani, Love dynamics between secure individuals: A modeling approach, Nonlinear Dynamics, Psychology, and Life Sciences, 2 (1998), 283-301. [11] S. Rinaldi, P. Landi and F. Della Rosa, Small discoveries can have great consequences in love affairs: The case of beauty and the beast International Journal of Bifurcation and Chaos 23 (2013), 1330038, 8pp. doi: 10.1142/S0218127413300383. [12] S. Rinaldi, P. Landi and F. Della Rossa, Temporary bluffing can be rewarding in social systems: The case of romantic relationships, The Journal of Mathematical Sociology, 39 (2015), 203-220.  doi: 10.1080/0022250X.2015.1022280. [13] S. Rinaldi, F. D. Rossa and P. Landi, Landi, A mathematical model of pride and prejudice?, Nonlinear dynamics, psychology, and life sciences, 18 (2014), 199-211. [14] S. Rinaldi, F. Rossa Della, F. Dercole, A. Gragnani and P. Landi, Modeling Love Dynamics vol. 89 of World Scientific Series on Nonlinear Science Series A, World Scientific Publishing Co. Pte. Ltd., 2016. [15] S. Strogatz, Love affairs and differential equations, Math. Magazine, 65 (1988), p35. [16] S. Strogatz, Nonlinear Dynamics and Chaos, Westwiev Press, 1994.
Example of an influence function. We see that for $|\xi_2|=\xi_1$ we have $f(\xi_1)\leq -f(\xi_2)$
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
An example of the situation when two actors reach the same emotional state independently if they meet each other or not
An example of the change of emotional states of two friends with the opposite initial states
Emotional states of two strongly dependent on each other friends with the same parameters describing both of them but with the opposite initial states
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
En example of two actors for which different initial conditions yield different opposite final emotional states
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
An example of emotional states of two pessimists who have strong influence on each other
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
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
Possible changes of the dynamics of emotional states initiated by one of the interacting actors
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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