Logarithmic Lipschitz norms and diffusion-induced instability. Aminzare, Z. & Sontag, E. *Nonlinear Analysis: Theory, Methods & Applications*, 83:31-49, 2013. abstract bibtex 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.

@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.}
}

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