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 & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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. Google Scholar

[12]

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

[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.  Google Scholar

[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.   Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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. Google Scholar

[12]

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

[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.  Google Scholar

[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.   Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

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