Delay can stabilize: Love affairs dynamics

• Published in 2012
• Added on 2013-01-19

We discuss two models of interpersonal interactions with delay. The first model is linear, and allows the presentation of a rigorous mathematical analysis of stability, while the second is nonlinear and a typical local stability analysis is thus performed. The linear model is a direct extension of the classic Strogatz model. On the other hand, as interpersonal relations are nonlinear dynamical processes, the nonlinear model should better reflect real interactions. Both models involve immediate reaction on partner's state and a correction of the reaction after some time. The models we discuss belong to the class of two-variable systems with one delay for which appropriate delay stabilizes an unstable steady state. We formulate a theorem and prove that stabilization takes place in our case. We conclude that considerable (meaning large enough, but not too large) values of time delay involved in the model can stabilize love affairs dynamics.

Links

• http://dx.doi.org/10.1016/j.amc.2012.10.028

Other information

BibTeX entry

@article{Bielczyk2012,
	key = {Bielczyk2012},
	type = {article},
	title = {Delay can stabilize: Love affairs dynamics},
	author = {Bielczyk, Natalia and Bodnar, Marek and Fory{\'{s}}, Urszula},
	abstract = {We discuss two models of interpersonal interactions with delay. The first model is linear, and allows the presentation of a rigorous mathematical analysis of stability, while the second is nonlinear and a typical local stability analysis is thus performed. The linear model is a direct extension of the classic Strogatz model. On the other hand, as interpersonal relations are nonlinear dynamical processes, the nonlinear model should better reflect real interactions. Both models involve immediate reaction on partner's state and a correction of the reaction after some time. The models we discuss belong to the class of two-variable systems with one delay for which appropriate delay stabilizes an unstable steady state. We formulate a theorem and prove that stabilization takes place in our case. We conclude that considerable (meaning large enough, but not too large) values of time delay involved in the model can stabilize love affairs dynamics.},
	comment = {},
	date_added = {2013-01-19},
	date_published = {2012-12-01},
	urls = {http://dx.doi.org/10.1016/j.amc.2012.10.028},
	collections = {Easily explained,Modelling},
	issn = 00963003,
	journal = {Applied Mathematics and Computation},
	keywords = {dyadic interactions,stability switches,stabilization of steady state,the hopf bifurcation,time delay},
	month = {dec},
	number = 8,
	pages = {3923--3937},
	url = {http://dx.doi.org/10.1016/j.amc.2012.10.028},
	volume = 219,
	year = 2012,
	urldate = {2013-01-19}
}