Bifurcations in Twinkling Patterns for the Lengyel–Epstein Reaction–Diffusion Model

Abstract

Conditions for the emergence of strong Turing–Hopf instabilities in the Lengyel–Epstein CIMA reaction–diffusion model are found. Under these conditions, time periodic spatially inhomogeneous solutions can be induced by diffusive instability of the spatially homogeneous limit cycle emerging at a supercritical Bautin–Hopf bifurcation about the unstable steady state of the reaction system. We report numerical simulations by an Alternating Directions Implicit (ADI) method that show the formation of twinkling patterns for a chosen parameter value, thus confirming our theoretical results.

Turing‐Turing and Turing‐Hopf bifurcations in a general diffusive Brusselator model

“…Suppose that 𝑚 > 1 and 𝛽 ≤ 𝛽 𝐻 0 . Then there exists a suitable 𝓁 > 0 such that system (2) undergoes the Turing-Turing bifurcation at the point 𝑇 * = (𝛼 * , 𝛽 * ) = (𝛼(𝑘 1 , 𝑘 2 ), 𝛽(𝑘 1 , 𝑘 2 )) and the Turing-Hopf bifurcation at 𝐻 * = (𝛼 * , 𝛽 * ), where 𝛼(𝑘 1 , 𝑘 2 ), 𝛽(𝑘 1 , 𝑘 2 ), 𝛼 * and 𝛽 * are defined in (10) and (11), respectively, and we assume that 1 ≤ 𝑘 1 < 𝑘 2 ≤ 𝑘.…”

“…Therefore, by Equation (10) it is easy to check that 𝒞 𝑘 1 (𝜆) = 0 has a zero root 𝜆 = 0 and the other one with a negative real part 𝜆 = 𝑇 𝑘 1 (𝛼, 𝛽). Also, 𝒞 𝑘 2 (𝜆) = 0 has a zero root 𝜆 = 0 and the other one with a negative real part 𝜆 = 𝑇 𝑘 2 (𝛼, 𝛽), and for other wave numbers 𝑘 ≠ 𝑘 1 and 𝑘 ≠ 𝑘 2 , all roots of 𝒞 𝑘 (𝜆) = 0 have negative real parts.…”

“…Also, many chemical, biological, or other field models governed by coupled reaction-diffusion equations have been proposed. For instance, Gierer-Meinhardt model [2,3], FitzHugh-Nagumo model [4,5], Brusselator model [6,7], Gray-Scott model [8,9], Lengyel-Epstein model [10,11], as well as predator-prey models [12,13], and others. Among these, chemical models have genuinely exhibited two types of instability.…”