@ARTICLE{aminzare_sontag_loglipchitz2012,
   AUTHOR       = {Z. Aminzare and E.D. Sontag},
   JOURNAL      = {Nonlinear Analysis: Theory, Methods & Applications},
   TITLE        = {Logarithmic Lipschitz norms and diffusion-induced 
      instability},
   YEAR         = {2013},
   OPTMONTH     = {},
   OPTNOTE      = {},
   OPTNUMBER    = {},
   PAGES        = {31-49},
   VOLUME       = {83},
   KEYWORDS     = {contractions, contractive systems, matrix measures, 
      logarithmic norms, Turing instabilities, diffusion, 
      partial differential equations, synchronization},
   PDF          = {../../FTPDIR/aminzare_sontag_contractions_j_nonlinear_analysis_2013.pdf},
   ABSTRACT     = {This paper proves that ordinary differential equation 
      systems that are contractive with respect to $L^p$ norms remain so 
      when diffusion is added. Thus, diffusive instabilities, in the sense 
      of the Turing phenomenon, cannot arise for such systems, and in fact 
      any two solutions converge exponentially to each other. The key tools 
      are semi-inner products and logarithmic Lipschitz constants in Banach 
      spaces. An example from biochemistry is discussed, which shows the 
      necessity of considering non-Hilbert spaces. An analogous result for 
      graph-defined interconnections of systems defined by ordinary 
      differential equations is given as well.}
}

{"_id":"saebXWxQmMpnyKn34","bibbaseid":"aminzare-sontag-logarithmiclipschitznormsanddiffusioninducedinstability-2013","downloads":0,"creationDate":"2018-10-18T05:07:05.937Z","title":"Logarithmic Lipschitz norms and diffusion-induced instability","author_short":["Aminzare, Z.","Sontag, E."],"year":2013,"bibtype":"article","biburl":"http://www.sontaglab.org/PUBDIR/Biblio/complete-bibliography.bib","bibdata":{"bibtype":"article","type":"article","author":[{"firstnames":["Z."],"propositions":[],"lastnames":["Aminzare"],"suffixes":[]},{"firstnames":["E.D."],"propositions":[],"lastnames":["Sontag"],"suffixes":[]}],"journal":"Nonlinear Analysis: Theory, Methods & Applications","title":"Logarithmic Lipschitz norms and diffusion-induced instability","year":"2013","optmonth":"","optnote":"","optnumber":"","pages":"31-49","volume":"83","keywords":"contractions, contractive systems, matrix measures, logarithmic norms, Turing instabilities, diffusion, partial differential equations, synchronization","pdf":"../../FTPDIR/aminzare_sontag_contractions_j_nonlinear_analysis_2013.pdf","abstract":"This paper proves that ordinary differential equation systems that are contractive with respect to $L^p$ norms remain so when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems, and in fact any two solutions converge exponentially to each other. The key tools are semi-inner products and logarithmic Lipschitz constants in Banach spaces. An example from biochemistry is discussed, which shows the necessity of considering non-Hilbert spaces. An analogous result for graph-defined interconnections of systems defined by ordinary differential equations is given as well.","bibtex":"@ARTICLE{aminzare_sontag_loglipchitz2012,\n AUTHOR = {Z. Aminzare and E.D. Sontag},\n JOURNAL = {Nonlinear Analysis: Theory, Methods & Applications},\n TITLE = {Logarithmic Lipschitz norms and diffusion-induced \n instability},\n YEAR = {2013},\n OPTMONTH = {},\n OPTNOTE = {},\n OPTNUMBER = {},\n PAGES = {31-49},\n VOLUME = {83},\n KEYWORDS = {contractions, contractive systems, matrix measures, \n logarithmic norms, Turing instabilities, diffusion, \n partial differential equations, synchronization},\n PDF = {../../FTPDIR/aminzare_sontag_contractions_j_nonlinear_analysis_2013.pdf},\n ABSTRACT = {This paper proves that ordinary differential equation \n systems that are contractive with respect to $L^p$ norms remain so \n when diffusion is added. Thus, diffusive instabilities, in the sense \n of the Turing phenomenon, cannot arise for such systems, and in fact \n any two solutions converge exponentially to each other. The key tools \n are semi-inner products and logarithmic Lipschitz constants in Banach \n spaces. An example from biochemistry is discussed, which shows the \n necessity of considering non-Hilbert spaces. An analogous result for \n graph-defined interconnections of systems defined by ordinary \n differential equations is given as well.}\n}\n\n","author_short":["Aminzare, Z.","Sontag, E."],"key":"aminzare_sontag_loglipchitz2012","id":"aminzare_sontag_loglipchitz2012","bibbaseid":"aminzare-sontag-logarithmiclipschitznormsanddiffusioninducedinstability-2013","role":"author","urls":{},"keyword":["contractions","contractive systems","matrix measures","logarithmic norms","Turing instabilities","diffusion","partial differential equations","synchronization"],"downloads":0,"html":""},"search_terms":["logarithmic","lipschitz","norms","diffusion","induced","instability","aminzare","sontag"],"keywords":["contractions","contractive systems","matrix measures","logarithmic norms","turing instabilities","diffusion","partial differential equations","synchronization"],"authorIDs":["5bc814f9db768e100000015a"],"dataSources":["DKqZbTmd7peqE4THw"]}