January  2018, 23(1): 181-191. doi: 10.3934/dcdsb.2018012

Some remarks on the Gottman-Murray model of marital dissolution and time delays

Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Received  November 2016 Published  January 2018

In the paper we consider mathematical model proposed by Gottman, Murray and collaborators to describe marital dissolution. This model is described in the framework of discrete dynamical system reflecting emotional states of wife and husband during consecutive rounds of talks between spouses. The model is, however, non-symmetric. To make it symmetric, one need to assume that the husband reacts with delay. Following this idea we consider the influence of time delays in the reaction terms of wife or/and husband. The delay means that one or both of spouses split their attention between present and previous rounds of talks. We study possibility of the change of stability with increasing delay. Surprisingly, it occurs that the delay has no impact on the stability, that is the condition of stability proposed by Murray remains unchanged.

Citation: Urszula Foryś. Some remarks on the Gottman-Murray model of marital dissolution and time delays. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 181-191. doi: 10.3934/dcdsb.2018012
References:
[1]

R. M. Baron, P. G. Amazeen and P. J. Beek, Local and global dynamics of social interactions, In R. R. Vallacher & A. Nowak (Eds. ), Dynamical systems in social psychology, (1994), 111–138. San Diego, CA: Academic Press.

[2]

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.

[3]

N. BielczykU. Foryś and T. Płatkowski, Dynamical models of dyadic interactions with delay, Journal of Mathematical Sociology, 37 (2013), 223-249.  doi: 10.1080/0022250X.2011.597279.

[4]

K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. 

[5]

F. Dercole and S. Rinaldi, Love stories can be unpredictable: Juliet at Jim in the vortex of life Chaos 24 (2014), 023134, 9pp. doi: 10.1063/1.4882685.

[6]

D. H. Felmlee and D. F. Greenberg, A dynamic systems model of dyadic interaction, Journal of Mathematical Sociology, 23 (1999), 155-180. 

[7]

U. Foryś, Biological delay systems and the Mikhailov criterion of stability, Journal of Biological Systems, 12 (2004), 1-16. 

[8]

U. Foryś, Time delays and the Gottman, Murray et al. model of marital interactions, In Proceedings of XXII National Conference, 2016.

[9]

J. M. Gottman, J. D. Murray, C. C. Swanson, R. Tyson and K. R. Swanson, The Mathematics of Marriage: Dynamic Nonlinear Models, Cambridge, MA: MIT Press, 2002.

[10]

S. G. Krantz, Rouché's Theorem, In Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999.

[11]

B. Latane and A. Nowak, Attitudes as catastrophes: From dimensions to categories with increasing involvement, In R. R. Vallacher & A. Nowak (Eds. ), Dynamical systems in social psychology, (1994), 219–249. San Diego, CA: Academic Press.

[12]

R. Leek and B. Meeker, Exploring nonlinear path models via computer simulation, Social Science Computer Review, 14 (1996), 253-268. 

[13]

X. Liao and J. Ran, Hopf bifurcation in love dynamical models with nonlinear couples and time delays, Chaos, Solitons, Fract., 31 (2007), 853-865.  doi: 10.1016/j.chaos.2005.10.037.

[14]

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

[15]

J. D. Murray, Mathematical Biology: Vol. 1. An Introduction, New York, NY: Springer-Verlag, 2002.

[16]

A. Nowak and R. R. Vallacher, Dynamical Social Psychology, New York, NY: Guilford Press, 1998.

[17]

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

[18]

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.

[19]

S. RinaldiF. Della Rosa and F. Dercole, Love and appeal in standard couples, International Journal of Bifurcation and Chaos, 20 (2010), 2443-2451. 

[20]

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, 8 pp. doi: 10.1142/S0218127413300383.

[21]

C. Rusbult and P. van Lange, Interdependence, interaction and relationships, Annual Review of Psychology, 54 (2003), 351-375. 

[22]

J. Skonieczna and U. Foryś, Stability switches for some class of delayed population models, Applied Mathematics (Warsaw), 38 (2011), 51-66.  doi: 10.4064/am38-1-4.

[23]

S. Strogatz, Love affairs and differential equations, Mathematics Magazine, 65 (1988), p35.

[24]

S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, Reading, MA: Perseus Books, 1994.

[25]

A. Turowicz, Geometria zer wielomianów (in Polish), Warsaw, PWN, 1967.

show all references

References:
[1]

R. M. Baron, P. G. Amazeen and P. J. Beek, Local and global dynamics of social interactions, In R. R. Vallacher & A. Nowak (Eds. ), Dynamical systems in social psychology, (1994), 111–138. San Diego, CA: Academic Press.

[2]

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.

[3]

N. BielczykU. Foryś and T. Płatkowski, Dynamical models of dyadic interactions with delay, Journal of Mathematical Sociology, 37 (2013), 223-249.  doi: 10.1080/0022250X.2011.597279.

[4]

K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. 

[5]

F. Dercole and S. Rinaldi, Love stories can be unpredictable: Juliet at Jim in the vortex of life Chaos 24 (2014), 023134, 9pp. doi: 10.1063/1.4882685.

[6]

D. H. Felmlee and D. F. Greenberg, A dynamic systems model of dyadic interaction, Journal of Mathematical Sociology, 23 (1999), 155-180. 

[7]

U. Foryś, Biological delay systems and the Mikhailov criterion of stability, Journal of Biological Systems, 12 (2004), 1-16. 

[8]

U. Foryś, Time delays and the Gottman, Murray et al. model of marital interactions, In Proceedings of XXII National Conference, 2016.

[9]

J. M. Gottman, J. D. Murray, C. C. Swanson, R. Tyson and K. R. Swanson, The Mathematics of Marriage: Dynamic Nonlinear Models, Cambridge, MA: MIT Press, 2002.

[10]

S. G. Krantz, Rouché's Theorem, In Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999.

[11]

B. Latane and A. Nowak, Attitudes as catastrophes: From dimensions to categories with increasing involvement, In R. R. Vallacher & A. Nowak (Eds. ), Dynamical systems in social psychology, (1994), 219–249. San Diego, CA: Academic Press.

[12]

R. Leek and B. Meeker, Exploring nonlinear path models via computer simulation, Social Science Computer Review, 14 (1996), 253-268. 

[13]

X. Liao and J. Ran, Hopf bifurcation in love dynamical models with nonlinear couples and time delays, Chaos, Solitons, Fract., 31 (2007), 853-865.  doi: 10.1016/j.chaos.2005.10.037.

[14]

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

[15]

J. D. Murray, Mathematical Biology: Vol. 1. An Introduction, New York, NY: Springer-Verlag, 2002.

[16]

A. Nowak and R. R. Vallacher, Dynamical Social Psychology, New York, NY: Guilford Press, 1998.

[17]

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

[18]

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.

[19]

S. RinaldiF. Della Rosa and F. Dercole, Love and appeal in standard couples, International Journal of Bifurcation and Chaos, 20 (2010), 2443-2451. 

[20]

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, 8 pp. doi: 10.1142/S0218127413300383.

[21]

C. Rusbult and P. van Lange, Interdependence, interaction and relationships, Annual Review of Psychology, 54 (2003), 351-375. 

[22]

J. Skonieczna and U. Foryś, Stability switches for some class of delayed population models, Applied Mathematics (Warsaw), 38 (2011), 51-66.  doi: 10.4064/am38-1-4.

[23]

S. Strogatz, Love affairs and differential equations, Mathematics Magazine, 65 (1988), p35.

[24]

S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, Reading, MA: Perseus Books, 1994.

[25]

A. Turowicz, Geometria zer wielomianów (in Polish), Warsaw, PWN, 1967.

[1]

Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101

[2]

Lixuan Zhang, Xuefei Yang. On pole assignment of high-order discrete-time linear systems with multiple state and input delays. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022022

[3]

Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264

[4]

David Schley, S.A. Gourley. Linear and nonlinear stability in a diffusional ecotoxicological model with time delays. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 575-590. doi: 10.3934/dcdsb.2002.2.575

[5]

Shihe Xu, Fangwei Zhang, Meng Bai. Stability of positive steady-state solutions to a time-delayed system with some applications. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021286

[6]

Ting Kang, Qimin Zhang, Haiyan Wang. Optimal control of an avian influenza model with multiple time delays in state and control variables. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4147-4171. doi: 10.3934/dcdsb.2020278

[7]

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

[8]

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

[9]

Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823

[10]

Şeyma Bılazeroğlu, Huseyin Merdan, Luca Guerrini. Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 535-554. doi: 10.3934/dcdss.2021150

[11]

Desheng Li, P.E. Kloeden. Robustness of asymptotic stability to small time delays. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1007-1034. doi: 10.3934/dcds.2005.13.1007

[12]

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

[13]

Tao Pang, Azmat Hussain. An infinite time horizon portfolio optimization model with delays. Mathematical Control and Related Fields, 2016, 6 (4) : 629-651. doi: 10.3934/mcrf.2016018

[14]

Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182

[15]

Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011

[16]

Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129

[17]

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

[18]

María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic and Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024

[19]

Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213

[20]

Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (205)
  • HTML views (141)
  • Cited by (0)

Other articles
by authors

[Back to Top]