Difference between Hopf-Turing bifurcation and traveling-waves bifurcation in reaction-diffusion systems

Question

I am perturbing a dynamical system with a perturbation that looks like $$\vec u e^{-i k x} e^{\sigma(p;k)t}$$ where $\sigma$ is a function of the parameters of the dynamical system and the wavenumber, coming out from the diffusion part of the system. Hopf bifurcation will be when for a $k=0$ I will have a non zero $\Im(\sigma)$ together with a $\Re(\sigma)>0$. For Turing bifurcation I am looking for $\Im(\sigma)=0$ and $\Re(\sigma) >0$ for a $k \neq 0$. If I search for a bifurcation in which $\Im(\sigma) \neq 0$ and $\Re(\sigma) > 0$ while $k >0$ I get a travelling wave form for the perturbation.

On the other hand, in order to get a Hopf-Turing bifurcation, should I use a perturbation of the sort $$\vec u \left(e^{-i k x}+ e^{-i \omega t}\right)e^{\sigma(p;k)t}$$ ?

Answer 1

You want a codimension-two point in parameter space where a time-independent, spatially periodic (Turing) mode (wave vector $\vec{q}_T$) and a spatially homogeneous, time-periodic (Hopf) mode (frequency $\omega_H$) bifurcate simultaneously. These modes may have a different vector $\vec{u}_T$ and $\vec{u}_H$, and a different (slow) time dependence $f_T(t)$ and $f_H(t)$, so near the bifurcation point the solution would look like

$$\vec{u}(\vec{x},t)=\vec{u}_T f_T(t)e^{i\vec{q}_T \cdot \vec{x}}+\vec{u}_H f_H(t)e^{i\omega_H t}$$