Journal of Nonlinear Science

, Volume 19, Issue 5, pp 467–496 | Cite as

Turing Instabilities at Hopf Bifurcation

  • M. R. RicardEmail author
  • S. Mischler


Turing–Hopf instabilities for reaction-diffusion systems provide spatially inhomogeneous time-periodic patterns of chemical concentrations. In this paper we suggest a way for deriving asymptotic expansions to the limit cycle solutions due to a Hopf bifurcation in two-dimensional reaction systems and we use them to build convenient normal modes for the analysis of Turing instabilities of the limit cycle. They extend the Fourier modes for the steady state in the classical Turing approach, as they include time-periodic fluctuations induced by the limit cycle. Diffusive instabilities can be properly considered because of the non-catastrophic loss of stability that the steady state shows while the limit cycle appears. Moreover, we shall see that instabilities may appear even though the diffusion coefficients are equal. The obtained normal modes suggest that there are two possible ways, one weak and the other strong, in which the limit cycle generates oscillatory Turing instabilities near a Turing–Hopf bifurcation point. In the first case slight oscillations superpose over a dominant steady inhomogeneous pattern. In the second, the unstable modes show an intermittent switching between complementary spatial patterns, producing the effect known as twinkling patterns.


Hopf bifurcation Turing instabilities Reaction-diffusion Averaging 

Mathematics Subject Classification (2000)

35K57 37G15 34C29 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidad de La HabanaHabanaCuba
  2. 2.CEREMADE, CNRS UMR 7534Université Paris IX-DauphineParis cedex 16France

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