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 & Continuous Dynamical Systems - B, 2018, 23 (1) : 401-423. doi: 10.3934/dcdsb.2018028
References:
[1]

N. BielczykM. 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.  Google Scholar

[2]

N. BielczykU. Foryś and T. Płatkowski, Dynamical models of dyadic interactions with delay, J. Math. Soc., 37 (2013), 223-249.  doi: 10.1080/0022250X.2011.597279.  Google Scholar

[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.  Google Scholar

[4] J. M. GottmanJ. D. MurrayC. C. SwansonR. Tyson and K. R. Swanson, The Mathematics of Marriage: Dynamic Nonlinear Models, MIT Press, Cambridge, 2002.   Google Scholar
[5]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.  doi: 10.21236/ADA045771.  Google Scholar

[6]

L. LiebovitchV. NaudotR. VallacherA. NowakL. Biu-Wrzosinska and P. Coleman, Dynamics of two-actor cooperation-competition conflict models, Physica A, 387 (2008), 6360-6378.   Google Scholar

[7] J. D. Murray, Mathematical Biology. Ⅰ: An Introduction, Springer-Verlag, New York, 2002.   Google Scholar
[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.  Google Scholar

[9]

S. RinaldiF. 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.  Google Scholar

[10]

S. Rinaldi and A. Gragnani, Love dynamics between secure individuals: A modeling approach, Nonlinear Dynamics, Psychology, and Life Sciences, 2 (1998), 283-301.   Google Scholar

[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.  Google Scholar

[12]

S. RinaldiP. 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.  Google Scholar

[13]

S. RinaldiF. D. Rossa and P. Landi, Landi, A mathematical model of pride and prejudice?, Nonlinear dynamics, psychology, and life sciences, 18 (2014), 199-211.   Google Scholar

[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.  Google Scholar

[15]

S. Strogatz, Love affairs and differential equations, Math. Magazine, 65 (1988), p35.   Google Scholar

[16] S. Strogatz, Nonlinear Dynamics and Chaos, Westwiev Press, 1994.   Google Scholar

show all references

References:
[1]

N. BielczykM. 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.  Google Scholar

[2]

N. BielczykU. Foryś and T. Płatkowski, Dynamical models of dyadic interactions with delay, J. Math. Soc., 37 (2013), 223-249.  doi: 10.1080/0022250X.2011.597279.  Google Scholar

[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.  Google Scholar

[4] J. M. GottmanJ. D. MurrayC. C. SwansonR. Tyson and K. R. Swanson, The Mathematics of Marriage: Dynamic Nonlinear Models, MIT Press, Cambridge, 2002.   Google Scholar
[5]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.  doi: 10.21236/ADA045771.  Google Scholar

[6]

L. LiebovitchV. NaudotR. VallacherA. NowakL. Biu-Wrzosinska and P. Coleman, Dynamics of two-actor cooperation-competition conflict models, Physica A, 387 (2008), 6360-6378.   Google Scholar

[7] J. D. Murray, Mathematical Biology. Ⅰ: An Introduction, Springer-Verlag, New York, 2002.   Google Scholar
[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.  Google Scholar

[9]

S. RinaldiF. 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.  Google Scholar

[10]

S. Rinaldi and A. Gragnani, Love dynamics between secure individuals: A modeling approach, Nonlinear Dynamics, Psychology, and Life Sciences, 2 (1998), 283-301.   Google Scholar

[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.  Google Scholar

[12]

S. RinaldiP. 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.  Google Scholar

[13]

S. RinaldiF. D. Rossa and P. Landi, Landi, A mathematical model of pride and prejudice?, Nonlinear dynamics, psychology, and life sciences, 18 (2014), 199-211.   Google Scholar

[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.  Google Scholar

[15]

S. Strogatz, Love affairs and differential equations, Math. Magazine, 65 (1988), p35.   Google Scholar

[16] S. Strogatz, Nonlinear Dynamics and Chaos, Westwiev Press, 1994.   Google Scholar
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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