Time-periodic forcing of Turing patterns in the Brusselator model
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1 Time-periodic forcing of Turing patterns in the Brusselator model B. Peña and C. Pérez García Instituto de Física. Universidad de Navarra, Irunlarrea, Pamplona, Spain Abstract Experiments on temporal forcing of the CDIMA reaction reaction have revealed the oscillation of Turing patterns, before disappear for higher stress [? ]. We present a preliminar work on Turing patterns under time-periodic forcing in the Brusselator model. We have studied the effect of a periodic modulation in the control parameter on a parameter region where reentrant hexagons appear. Surprisingly, under certain conditions, the oscillating solution changes his symmetry for values high enough of forcing amplitude: competition between bands and hexagons can replace, for example, a stationary hexagonal pattern by oscillating squares, a kind of pattern still unobserved in chemical experiments. The theoretical mechanism underlying this change of symmetry is still under discussion. Key words: Chemical kinetics, Turing patterns 1 Introduction In recent years forcing of spatially extended systems has attracted the interest of many researchers in the framework of chemical reactions. Swinney et al. have investigated the effect of periodic forcing on a photosensitive form of the Belousov-Zhabotinsky reaction [? ]. The experiments have revealed that induced resonances transform an initial spiral pattern in labyrinths or domain (Ising and Bloch) walls. Although the dynamics of the Brusselator can not describe all the properties of experiments, it reproduces some important aspects as the transition to labyrinths [? ]. It was also shown that the Brusselator model with a parametric forcing exhibits similar pattern and reproduces the resonance regions ( Arnold s tongues) predicted theoretically. Some of this features can be explained in a general complex Ginzburg-Landau equation under a resonant forcing [? ]. Preprint submitted to Elsevier 1 June 2016
2 Other authors investigated the temporal forcing of Turing patterns arising in CDIMA reaction chlorine dioxide-iodine-cido malnico. For small amplitude of light intensity, they obtained oscillating Turing structures, while they dissapear for higher amplitudes [? ]. This is even quantitatively described by the Lengyel-Epstein model in which illumination effects are included in an additive term [? ]. Threshold of different solutions are strongly modified, but no new solutions were obtained. In the present paper we investigate the effect of time-periodic forcing on the symmetry of Turing patterns in the Brusselator model. In Section 2 we present the model and shortly explain the dynamics without forcing. In Section 3 time periodic forcing is introduced and the behavior is studied through numerical simulations. The Floquet analysis is carried out in Section 4 and finally we discuss the main results and summarize our conclusions in Section 5. 2 Stability of Turing patterns in the Brusselator Model Fig. 1. Reaction scheme of Brusselator model. The Brusselator model, is a very simple model which exhibits Hopf and Turing instability. For this reason has been used very often to understand the behaviour of CDIMA reaction [??? ]. This model corresponds to the reactive scheme (Fig.1) which, after suitable rescaling, results into the dimensionless equations: t X = A (B + 1)X + X 2 Y + 2 X (1) t Y = BX X 2 Y + D 2 Y (2) in which A and B represent the input reactants, which can be externally controlled, C and D are products continuously eliminated, and the dynamical species are U (activator) and V (substrate in autocatalytic reaction). We focus 2
3 on the parameter values for which stationary bands and two kind of hexagons (normal and reentrant )were obtained in experiments. We analysed in previous work the stability of these solutions in the framework of amplitude equations [?? ] which were found to agree with the stability in numerical simulations. The diagram of Fig. 2 summarizes these results for A = 4.5 and D = 8.0, and will be a reference diagram for the present study. Shaded and striped regions in Fig. 2 correspond to stable hexagons and bands, respectively. Near onset hexagons with total phase 0 (H 0 )appear, while for higher values of the control parameter π-hexagons are stabilized (H π ). Then, the maxima (white) or the minima (black) of activator U are located in a hexagonal lattice. The wavenumbers of numerical solutions are represented by different symbols: asterisk for hexagons, circles for bands and squares for mixtures of both patterns. The selected wavenumber increases with the control parameter, in agreement with the maximum linear growth of perturbations [? ]. Fig. 2. Stability diagram without external forcing for A = 4.5 and D = 8.0 [? ]. 3 Time periodic forcing The forcing that has been applied upon shining the CDIMA reaction was additive [? ]. Here we consider a parametric temporal forcing through the control parameter B, by replacing B B(1 + γsin(ωt) in Eqs.2. In a continuously feeded reactor this forcing could be performed by a periodic change in one of the input reactants, a procedure likely difficult to implement in experiments. We integrate the model Eq. (2) in 2D from random initial conditions. We use a pseudo spectral method [? ] with x = and t =, the laplacian operators are discretized and the non-linear part is solved in the Fourier space at lower order. In Fig. 3 the stability regions for different values of the supercriticality 3
4 Fig. 3. Patterns oberved under forcing in the plane µ = B Bc B c and forcing intensity γ for a given frecuency ω = 0.5 are drawn. Near the Turing instability (µ 0.05) oscillating π-hexagons are stable for small γ, but they are replaced by a homogeneous state HO at higher values ofγ. As µ is increased the situation becomes more involved. There is a wide region of bistability between bands B and H π, which spreads from γ = 0 to increasing values of µ. For a forcing strong enough, patterns of squares (S) stabilize for smaller γ as µ is increased. In agreement with the analysis without forcing bands are the sole solution near the µ-axis, while they coexist with H 0 - hexagons if one increases µ. For higher values of γ, squares appear also and there is an extended region of bistability with H 0. In bistability regions, the final pattern depends on the initial condition showing hysteresis effects. A typical example of a front between hexagons and squares is shown inf Fig. (4) Fig. 4. Front between squares and hexagons for µ = 0.3, γ = 0.3, (ω = 4.5). In Fig. 5 some examples are shown 4
5 Fig. 5. Evolution of an irregular pattern of oscillating squares µ = 0.3, γ = 0.35 and ω = Floquet exponents Obviously, to deal with these oscillations a Floquet analysis has to be made. The Floquet method gives the stability of periodic solutions, considered as fixed points in the Poincar map. Now we sketch how it can be applied to perturbations with a finite wavenumber. The stability condition is obtained from the eigenvalues of the linear stability problem ( Floquet multipliers). Here c 0 (t) = (X 0 (t), V 0 (t), W 0 (t)) denotes the reference solution, and L(t) the linear matrix. Both are periodic and, therefore, invariant under a temporal translation T : t t + T 0. In absence of spatial effects, perturbations over a limit cycle w(t) = c(t) + c 0 (t) obey, at linear order, the equation w = L(t) w, which admits solutions in the form w = e λt v. In the simplest case of non degenerated eigenvalues λ, vectors v are periodic with period T 0 [? ]. With diffusion terms included, the linear problem becomes w = L(t) w + D 2 w, where D is a diagonal matrix which contains the diffusion coefficients and L(t) is the Jacobian matrix evaluated on the limit cycle. By expanding the perturbations in Fourier modes: w = q k e ikx, (3) one obtains q k = [ L(t) Dk 2] q k. (4) Instead of deriving directly the eigenvalues for this particular problem, also called Floquet exponents, we apply the Floquet theorem. It ensures the solutions of Eq. (4) to be of the form q k = e λ kt u k (t), where u k (t) is also periodic: u k (t + T 0 ) = u k (t) [? ]. Hence we obtain: q k (t + T 0 ) = e λ k(t+t 0 ) u k (t + T 0 ) = e λ kt 0 q k (t) M k q k (t) (5) where the Monodromy matrix M k, gives the time evolution after a time period. Consequently the linear stability analysis just involves the derivation of the 5
6 matrix M k because e λ kt 0 q(t) = M k q(t). (6) Then, the eigenvalues of Eq. (4) are calculated from the eigenvalues of M k, denoted by σ k (Floquet multipliers), which are linked by: R[λ k ] = 1 T 0 Ln σ k. (7) λ k stands for the Floquet exponents. If these are positive, i.e., if σ k > 1 a mode with wavenumber k grows. The Monodromy matrix can be built in a simple manner. One takes the canonical initial conditions q i (0) = (δ i1, δ i2,..., δ in ) T, con i = 1, 2,...N, (δ ij stands here for the Kronecker delta) and the vectors are calculated after a time period is elapsed: T0 [ q i (T 0 ) = ] L(t) Dk 2 q i (0)dt. (8) Hence: 0 M k = [ q 1 (T 0 ), q 2 (T 0 ),...]. (9) We have applied this method to determine the stability of limit cycles in the forced Brusselator model, Eqs. 2 with B(t) = B(1 + γsin(ωt). The calculation scheme is smilar to that used for Poincare sections. The evolution of three initial conditions are considered: one on the limit cycle, c 0 (t), and two trivial vectors, given by q i (0) = (δ i1, δ i2 ) T (i = 1, 2, 3 ). The first point evolves to the model without diffusive terms. At each computational time step the value c 0 (t i ) is taken to calculate the linear problem: L(t i ) Dk 2 = B(t i) X 0 (t i )Y 0 (t i ) Dk 2 X 0 (t i ) 2 (10) B(t i ) 2X 0 (t i )Y 0 (t i ) X 0 (t i ) 2 k 2 which is needed to integrate Eq. 8. After a period T 0, the Monodromy matrix is built from q i (T 0 ) and the Floquet exponents reached. Preliminary results show that for small µ (µ 0.05) the maximum of Floquet exponents is independent of γ, (within our numerical accuracy). Away from onset (µ 0.10) the wavenumber of maximum growth slightly decreases as forcing is intensified. This is confirmed in numerical simulations in which an initially stable pattern is replaced by another one with a different k as γ is increased. We hope to complete these comparison in the near future 6
7 5 Discussion We have shown that the symmetry of Turing patterns can be modified by an external time-periodic forcing. Though the mechanism through which it occurs is still unclear, we think that this change of symmetry is related to the multistability in the Brusselator model. As γ is rised, the parameter controlling the distance to onset, µ(t), varies in time. In such a way, unstable solutions in absence of forcing play now a rol. From the competition of hexagons (H π and H 0 ) and bands, squares are favoured as an intermediate state between previous mentioned. Acknowledgements The authors wish to thank Dr. A.P. Muñuzuri (Santiago de Compostela) for fruitful comments. This work has been supported by the MCyT (Spanish Government) under grant BFM2002-O1002 and by the PIUNA (Univ. Navarra). 7
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