Turing instabilities at Hopf bifurcation

Stéphane Mischler (1) , Mariano Rodriguez Ricard (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 FACULTAD DE MATEMATICA Y COMPUTACION - Facultad de Matemática y Computación

The paper considers Turing instabilities of a limit cycle generated by a Hopf bifurcation in 2D reaction-diffusion systems of the form ut = Du∆u + f (u, v; a), vt = Dv ∆v + g(u, v; a), over a domain in Rn with Neumann boundary conditions, where a is some vector of parameters. The spatially homogeneous time-periodic solution is denoted by (u(t), v(t)). The authors first review the Hopf bifurcation including some averaging results which lead to a normal form close to a second-order scalar oscillator. Then linearizing around (u(t), v(t)) they obtain extended normal modes for this linearization and in particular derive conditions for a Turing instability to arise. Furthermore, this leads to the distinction between “weak Turing-Hopf (TH) instabilities”, where small oscillations over time superpose a dominant stationary inhomogeneous pattern, and “strong TH instabilities”, which result in so-called twinkling patterns. The analysis is complemented by results for the Schnakenberg system, giving conditions under which weak, or respectively strong, TH instabilities are expected. It should be interesting to relate these conditions to numerical simulations of the Schnakenberg system. The paper is well written and contains many interesting results.

Domaines: Equations aux dérivées partielles [math.AP]

Stéphane Mischler, Mariano Rodriguez Ricard. Turing instabilities at Hopf bifurcation. J. Nonlinear Sci., 2009, 19 (5), pp.467-496. ⟨hal-00561934⟩