The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability

Abstract

<p style='text-indent:20px;'>We consider reaction-diffusion systems for which the trivial solution simultaneously becomes unstable via a short-wave Turing and a long-wave Hopf instability. The Brusseletor, Gierer-Meinhardt system and Schnakenberg model are prototype biological pattern forming systems which show this kind of behavior for certain parameter regimes. In this paper we prove the validity of the amplitude system associated to this kind of instability. Our analytical approach is based on the use of mode filters and normal form transformations. The amplitude system allows us an efficient numerical simulation of the original multiple scaling problems close to the instability.</p>