Turing-Hopf interaction

One may also observe a transition to a type of defect-mediated turbulence in this Turing system (see figure C3.6.12 (b). Here the defects divide the system into domains of spots and stripes. The defects move erratically and lead to a turbulent state characterized by exponential decay of correlations [59]. Turing bifurcations can interact with the Hopf bifurcations discussed above to give rise to very complicated spatio-temporal patterns [63, 64]. 

For/ = lando > / , the system exhibits a primary homogeneous Hopf bifurcation to an oscillatory state. This is a general feature systems which present Turing instabilities also present homogeneous Hopf instabilities. Interactions between these two instabilities can play an important role which will be discussed further. From the condition d > 1, it follows that the two diffusion coefficients must be different for a Turing instability to occur the activator must diffuse faster than the other species. 

The recent simulations of Pearson [44] on a fed autocatalator [104], on the other hand stages interactions between Turing and saddle-node bifurcations or even situations where a Hopf bifurcation interacts with the two previous ones. Then a plethora of scenarios may result. However, the Turing bifurcations are genuine but they occur on the stable branch of an isola of uniform states and may exist beyond the limits of the isola. 

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