Joshua Sin, John W. Bonnes, Luke C. Brown, David M. Ambrose
{"title":"Existence and computation of stationary solutions for congestion-type mean field games via bifurcation theory and forward-forward problems","authors":"Joshua Sin, John W. Bonnes, Luke C. Brown, David M. Ambrose","doi":"10.3934/jdg.2023014","DOIUrl":null,"url":null,"abstract":"Time-dependent mean field games are a coupled system of a forward parabolic and backward parabolic partial differential equation. Stationary solutions are of interest, and then naturally the forward-backward structure in time becomes irrelevant. Forward-forward mean field games have been introduced with the rationale that they may be used to straightforwardly compute such stationary solutions. We perform some numerical simulations to find that typically stationary solutions of mean field games are unstable to the forward-forward evolution, i.e. frequently only trivial solutions can be found in this way. We then ask whether there are situations in which one would have reason to believe that the stationary solutions would be stable, and we use the exchange-of-stability phenomenon in bifurcation theory to give a class of examples for which the forward-forward solutions do converge to nontrivial stationary solutions as time increases.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"28 1","pages":"0"},"PeriodicalIF":1.1000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Dynamics and Games","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/jdg.2023014","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
Time-dependent mean field games are a coupled system of a forward parabolic and backward parabolic partial differential equation. Stationary solutions are of interest, and then naturally the forward-backward structure in time becomes irrelevant. Forward-forward mean field games have been introduced with the rationale that they may be used to straightforwardly compute such stationary solutions. We perform some numerical simulations to find that typically stationary solutions of mean field games are unstable to the forward-forward evolution, i.e. frequently only trivial solutions can be found in this way. We then ask whether there are situations in which one would have reason to believe that the stationary solutions would be stable, and we use the exchange-of-stability phenomenon in bifurcation theory to give a class of examples for which the forward-forward solutions do converge to nontrivial stationary solutions as time increases.
期刊介绍:
The Journal of Dynamics and Games (JDG) is a pure and applied mathematical journal that publishes high quality peer-review and expository papers in all research areas of expertise of its editors. The main focus of JDG is in the interface of Dynamical Systems and Game Theory.