On Turing-Hopf Instabilities in Reaction-Diffusion Systems

Abstract

We examine the appearance of Turing instabilities of spatially homogeneous periodic solutions in reaction-diffusion equations when such periodic solutions are consequence of Hopf bifurcations. First, we asymptotically develop limit cycle solutions associated to the appearance of Hopf bifurcations in reaction systems. Particularly, we will show conditions to the appearance of multiple limit cycles after Hopf bifurcation. Then, we propose expansions to normal modes associated with Turing instabilities from spatially homogeneous periodic solutions associated to limit cycles which appear as a consequence of a Hopf bifurcation. Finally, we discuss examples of reaction-diffusion systems arising in biology and chemistry in which can be observed spatial and time-periodic patterning.

Turing–Hopf patterns on growing domains: The torus and the sphere

“…Namely, this instability has been studied in continuous population dynamical models [1,54,67], in discrete time models [50], and in chemical reactions [22,46,51]. Other literature deals with various theoretical aspects of this bifurcation [11,48]. In the vicinity of a Turing-Hopf bifurcation, the competition between unstable modes coming from the Turing bifurcation and those coming from the Hopf can result in complex spatio-temporal patterns in reaction-diffusion systems [10,35,57].…”