Figure 1.
Example of an influence function. We see that for $|\xi_2|=\xi_1$ we have $f(\xi_1)\leq -f(\xi_2)$
-
Previous Article
Treatment of glioma with virotherapy and TNF-α inhibitors: Analysis as a dynamical system
- DCDS-B Home
- This Issue
-
Next Article
Numerical treatment of contact problems with thermal effect
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 |
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.
Keywords:
Ordinary differential equations,
steady states,
stability,
emotional state,
communication,
relationship.
Mathematics Subject Classification:
Primary: 91D99; Secondary: 37N99, 34D05, 34D23.
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.
|
show all references
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.
|
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 ($c_{1}>0$ , $c_{2} < 0$ )
| [1] |
Serge Nicaise. Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021057 |
| [2] |
La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981 |
| [3] |
Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070 |
| [4] |
Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 |
| [5] |
Yacine Chitour, Jean-Michel Coron, Mauro Garavello. On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 643-672. doi: 10.3934/dcds.2006.14.643 |
| [6] |
Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445 |
| [7] |
Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416 |
| [8] |
Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 |
| [9] |
Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87 |
| [10] |
Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353 |
| [11] |
Qian Xu. The stability of bifurcating steady states of several classes of chemotaxis systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 231-248. doi: 10.3934/dcdsb.2015.20.231 |
| [12] |
Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049 |
| [13] |
Yongli Cai, Yun Kang, Weiming Wang. Global stability of the steady states of an epidemic model incorporating intervention strategies. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1071-1089. doi: 10.3934/mbe.2017056 |
| [14] |
Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849 |
| [15] |
Chao Xing, Ping Zhou, Hong Luo. The steady state solutions to thermohaline circulation equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3709-3722. doi: 10.3934/dcdsb.2016117 |
| [16] |
Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57 |
| [17] |
Kousuke Kuto, Tohru Tsujikawa. Bifurcation structure of steady-states for bistable equations with nonlocal constraint. Conference Publications, 2013, 2013 (special) : 467-476. doi: 10.3934/proc.2013.2013.467 |
| [18] |
Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147 |
| [19] |
Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165 |
| [20] |
W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]