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The role of optimism and pessimism in the dynamics of emotional states
1.  College of InterFaculty Individual Studies in Mathematics and Natural Sciences, University of Warsaw, Żwirki i Wigury 93, 02089 Warsaw, Poland 
2.  Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02097 Warsaw, Poland 
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 nonlinear 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 socalled Stockholm syndrom.
References:
[1] 
N. Bielczyk, M. Bodnar and U. Foryś, Delay can stabilize: Love affairs dynamics, Applied Mathematics and Computation, 219 (2012), 39233937. 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), 223249. 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), 155180. 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), 263291. doi: 10.21236/ADA045771. 
[6] 
L. Liebovitch, V. Naudot, R. Vallacher, A. Nowak, L. BiuWrzosinska and P. Coleman, Dynamics of twoactor cooperationcompetition conflict models, Physica A, 387 (2008), 63606378. 
[7]  J. D. Murray, Mathematical Biology. Ⅰ: An Introduction, SpringerVerlag, New York, 2002. 
[8] 
S. Rinaldi, Love dynamics: The case of linear couples, Applied Mathematics and Computation, 95 (1998), 181192. doi: 10.1016/S00963003(97)100819. 
[9] 
S. Rinaldi, F. Della Rosa and F. Dercole, Love and appeal in standard couples, International Journal of Bifurcation and Chaos, 20 (2010), 34733485. 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), 283301. 
[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), 203220. 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), 199211. 
[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. 
show all references
References:
[1] 
N. Bielczyk, M. Bodnar and U. Foryś, Delay can stabilize: Love affairs dynamics, Applied Mathematics and Computation, 219 (2012), 39233937. 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), 223249. 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), 155180. 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), 263291. doi: 10.21236/ADA045771. 
[6] 
L. Liebovitch, V. Naudot, R. Vallacher, A. Nowak, L. BiuWrzosinska and P. Coleman, Dynamics of twoactor cooperationcompetition conflict models, Physica A, 387 (2008), 63606378. 
[7]  J. D. Murray, Mathematical Biology. Ⅰ: An Introduction, SpringerVerlag, New York, 2002. 
[8] 
S. Rinaldi, Love dynamics: The case of linear couples, Applied Mathematics and Computation, 95 (1998), 181192. doi: 10.1016/S00963003(97)100819. 
[9] 
S. Rinaldi, F. Della Rosa and F. Dercole, Love and appeal in standard couples, International Journal of Bifurcation and Chaos, 20 (2010), 34733485. 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), 283301. 
[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), 203220. 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), 199211. 
[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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