Two distinct ways toward "oscillatory Turing patterns"

The term "oscillatory Turing patterns" (OTP) appeared the first time in ref.[1], where we wrote "Classical Turing patterns are stationary. However, the ''twinkling eye'' is an oscillatory Turing pattern."

"Twinkling-eye pattern is a mixed-mode pattern, which originates from the interaction between a subharmonic Turing mode and an oscillatory mode." We first reported this pattern in ref.[2] by a coupled Brusselator model, then, repeated in ref.[1] by a coupled Oregonator model.

The term OTPs refers to patterns in reaction-diffusion (RD) systems, which show periodicity both in space and time. At this sense, OTPs resemble "standing waves", but having different origin --- the standing waves in RD systems are from a "short wavelength instability". At the sense of the spatial periodicity, OTPs are the same as the classic stationary Turing patterns, but OTPs oscillate somehow (period-1, period-2, or chaotic), or having fine waves superposing (Isolated spirals or targets on hexagonal array of Turing spots, Traveling waves on labyrinthine stripelike Turing structure, or "Pinwheels'').

Where OTPs come from?
Two distinct ways toward OTPs have been reported in ref.[1] and ref.[3], respectively.

1. Road 1: Strong Turing instability drives the system away from the steady state to form Turing structures, then, the spatial periodic Turing structures lose stability to gain oscillation. This scenario was reported in Fig.2(c) of ref.[1], as an example.
   Clearly, this road to OTPs requires a Turing instability, but does not require Hopf instability as a necessary condition! Because the oscillation can be caused by loose of stability of the spatial periodic structure (a secondary instability!)
2. Road 2: Strong Hopf instability(but not limited by Hopf) drives the system away from the steady state to form a bulk oscillation (BO). Then, the BO loses stability toward spatial perturbation, characterized by Floquet multipliers. There are three subcases.
   1. Kuramoto phase instability. At least one multiplier cross 1 within k=0 to k_cut. (Sample)
   2. Period-doubling instability. At least one multiplier cross -1 at a non-zero k=k_0. (Sample)
   3. Period-1 instability. At least one multiplier cross 1 at a non-zero k=k_0. (Sample)
   Comments:
   1. We had found that ii) and iii) can give OTPs.
   2. A good question is where the spatial periodicity comes from by Road 2. One can see it may not related to any Turing instability at all!
      
   3. The key point here is the existence of a limit-cycle solution. It may from a Hopf bifurcation, but may not. Therefore, Hopf instability is not a necessary condition of this road to OTPs.

Basically, one should see from where the OTPs are developed. If they are from the steady state(SS), do linearly stability analysis about the SS. If not, for example, does Hopf frequency is the oscillation frequency of the OTPs by Road 1 above (Q1)?, another example, does the spatial periodicity in Road 2 from Turing instability (Q2)?, the answers are NOT, because the system does not stay near the SS any more. For Q1, periodic structure begins to oscillates, so one should analyze the periodic structure, not the SS. For Q2, a limit-cycle is away from the SS, analysis based on the cycle should be taken.

1. Lingfa Yang, and Irving R. Epstein, "Oscillatory Turing patterns in reaction-diffusion systems with two coupled layers", Phys. Rev. Lett. 90(17), 178303 (2003).
2. Lingfa Yang, Milos Dolnik, Anatol M. Zhabotinsky and Irving R. Epstein, "Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes", Phys. Rev. Lett. 88(20), 208303(2002).
3. Lingfa Yang, A. M. Zhabotinsky, and I. R. Epstein, "Stable squares and other oscillatory Turing patterns in a reaction-diffusion model", Phys. Rev. Lett. 92, 198303 (2004)