Turing-Hopf Patterns near onset Mariano Rodríguez Ricard PDE Session X International Conference in Operation Research 2012 (Universidad de La Habana) 06/03/2012 1 / 48
Objectives Our concern in this presentation: pattern formation in reaction diffusion systems near a codimension two Turing-Hopf bifurcation point. The travelling wave initiation of time-oscillatory patterns. (Universidad de La Habana) 06/03/2012 2 / 48
Plan The normal mode approach in the study of Turing instabilities (Universidad de La Habana) 06/03/2012 3 / 48
Plan The normal mode approach in the study of Turing instabilities The Hopf bifurcation revisited (Universidad de La Habana) 06/03/2012 3 / 48
Plan The normal mode approach in the study of Turing instabilities The Hopf bifurcation revisited Turing-Hopf bifurcation: more than the overlap of TI and HB (Universidad de La Habana) 06/03/2012 3 / 48
Plan The normal mode approach in the study of Turing instabilities The Hopf bifurcation revisited Turing-Hopf bifurcation: more than the overlap of TI and HB Way of propagation of Turing-Hopf patterns (Universidad de La Habana) 06/03/2012 3 / 48
Turing instability in reaction diffusion systems D u ∆ u + f ( u , v ; a ) u t = (1) = D v ∆ v + g ( u , v ; a ) v t ∂ u ∂ n = ∂ v ∂ n = 0 on ∂ Ω (2) u , v - the profiles of reactant concentrations under diffusion, ( u 0 , v 0 ) spatially homogeneous steady solution (Universidad de La Habana) 06/03/2012 4 / 48
Normal modes Let us consider normal modes of the type Z ( x , t ) = exp ( σ t ) U k ( x ) R (3) as non-trivial solutions to the linearized equation ∂ Z ∂ t = D ∆ Z + J a Z (4) where − ∆ U k ( x ) = λ k U k ( x ) 0 on ∂ Ω ∂ n U k = spatial eigenfunctions associated to the spatial eigenvalues λ k ( k ∈ N ) (Universidad de La Habana) 06/03/2012 5 / 48
Normal modes Stability analysis by small disturbances with the form ∞ ∑ Z ( x , t ) = exp ( σ k t ) U k ( x ) R k (5) k = 1 � � j a J a = be the jacobian matrix ij δ a = det ( J a ) > 0, and τ a = trace ( J a ) for each k , σ k is an eigenvalue ; R k corresponding eigenvector of E k = ( J a − λ k D ) (Universidad de La Habana) 06/03/2012 6 / 48
conditions for diffusive instability linear stable steady state ( u 0 , v 0 ) in an activator-inhibitor (or positive feedback) system τ a < 0 , δ a > 0 so, τ T < 0 τ T = trace ( J a − λ k D ) = τ a − λ k ( D u + D v ) (6) and, follows the condition for instability δ T < 0 δ T = det ( J a − λ k D ) = δ a − λ k ( D u j a 22 + D v j a 11 ) + λ 2 k D u D v (7) (Universidad de La Habana) 06/03/2012 7 / 48
parameter space the boundary δ T = 0 d = D v / D u d j a 11 + j a 22 > 0 (8) so d � = 1 . (Universidad de La Habana) 06/03/2012 8 / 48
representation Remark Turing patterns might be represented by the set of positiveness of the dominant unstable spatial eigenfunction. steady spatially varying profiles in the reactant concentrations (Universidad de La Habana) 06/03/2012 9 / 48
Definition of pattern Turing showed that dissimilar diffusion coefficients of the participating reactants would destabilize the steady state of the reaction kinetics leading to pattern formation. The appearance of Turing instabilities about the stable steady state ( τ a < 0 ) is a consequence of algebraic inequalities between the ( reaction and diffusion ) parameters. These relations are builded from Fourier normal modes: Z ( x , t ) = exp ( σ t ) U k ( x ) R (Universidad de La Habana) 06/03/2012 10 / 48
Definition of pattern Turing showed that dissimilar diffusion coefficients of the participating reactants would destabilize the steady state of the reaction kinetics leading to pattern formation. The appearance of Turing instabilities about the stable steady state ( τ a < 0 ) is a consequence of algebraic inequalities between the ( reaction and diffusion ) parameters. These relations are builded from Fourier normal modes: Z ( x , t ) = exp ( σ t ) U k ( x ) R The ultimate pattern emerges (see Murray’s) due to the boundedness of the unstable modes by the nonlinear reaction terms in Eq.1 (Universidad de La Habana) 06/03/2012 10 / 48
Definition of pattern Turing showed that dissimilar diffusion coefficients of the participating reactants would destabilize the steady state of the reaction kinetics leading to pattern formation. The appearance of Turing instabilities about the stable steady state ( τ a < 0 ) is a consequence of algebraic inequalities between the ( reaction and diffusion ) parameters. These relations are builded from Fourier normal modes: Z ( x , t ) = exp ( σ t ) U k ( x ) R The ultimate pattern emerges (see Murray’s) due to the boundedness of the unstable modes by the nonlinear reaction terms in Eq.1 without a nonlinear theory, we have only a presumption about the ultimate pattern towards which the destabilized solution converges, which is connected with the dominant unstable mode. (Universidad de La Habana) 06/03/2012 10 / 48
stationary chemical patterns P.K. Maini, K.J. Painter and H.N.P. Chau, (1997), Spatial Pattern Formation in chemical and biological systems, J.Chem.Soc., Faraday Trans., 93(20), 3601-3610 Figure: Stationary patterns in CIMA (Universidad de La Habana) 06/03/2012 11 / 48
patterns in chemical reactions P.K. Maini, K.J. Painter and H.N.P. Chau, (1997), Spatial Pattern Formation in chemical and biological systems, J.Chem.Soc., Faraday Trans., 93(20), 3601-3610 Figure: Black-eye pattern (Universidad de La Habana) 06/03/2012 12 / 48
patterns in morphogenesis P.K. Maini, R.E. Baker and Cheng-Ming Chuong, (2006), The Turing Model Comes of Molecular Age, Science 314, 1397-1398 (2006); published online 1 December 2006 (10.116/science.1136396) (Universidad de La Habana) 06/03/2012 13 / 48
patterns in morphogenesis P.K. Maini, R.E. Baker and Cheng-Ming Chuong, (2006), The Turing Model Comes of Molecular Age, Science 314, 1397-1398 (2006); published online 1 December 2006 (10.116/science.1136396) (Universidad de La Habana) 06/03/2012 14 / 48
observed and simulated patterns R.Erban, H.G. Othmer, (2005), From signal transduction to spatial pattern formation in E. Coli: a paradigm for multiscale modeling in Biology, Multiscale Model Simul., 3(2), 362-394 Figure: Spatial patterns arising in E.Coli (Universidad de La Habana) 06/03/2012 15 / 48
The Hopf bifurcation � · u = f ( u , v ; a ) (9) · v = g ( u , v ; a ) P a = ( u 0 ( a ) ; v 0 ( a )) (10) � � j a J a = be the jacobian matrix of Eq.9 ij τ 2 a − 4 δ a < 0 (11) δ a = det ( J a ) > 0, and τ a = trace ( J a ) . (Universidad de La Habana) 06/03/2012 16 / 48
reduction to a second order oscillator [M.R. Ricard, On degenerate planar Hopf bifurcations, J. Phys. A: Math. Theor. 44 (2011) 065202 (15pp) weakly nonlinear oscillator in normal form : � � ·· · · ς − τ a ς + δ a ς = ε G ς , ς ; ε . (12) ε - small parameter to be determined later (Universidad de La Habana) 06/03/2012 17 / 48
subsequent reduction via averaging (Krylov-Bogoliubov technique) r = r � 2 { τ a − p ( r ; ε ) } (13) � θ = q ( r ; ε ) (14) considering φ = ω a t + θ . 2 π � ε p ( r ; ε ) = sin φ G ( r cos φ , − r ω a sin φ ; ε ) d φ (15) πω a r 0 � 2 π ε q ( r ; ε ) = − cos φ G ( r cos φ , − r ω a sin φ ; ε ) d φ . (16) 2 πω a r 0 (Universidad de La Habana) 06/03/2012 18 / 48
properties of the discriminant p ( r ; ε ) / r 2 and q ( r ; ε ) / r 2 have a finite limit as r → 0. the Taylor expansions of p ( r ; ε ) and q ( r ; ε ) must not contain odd powers of r , and p ( r ; ε ) = p 3 ε 2 r 2 + p 5 ε 4 r 4 + · · · (17) in which p s = p s ( τ a ) . The classical perturbation theory gives a uniform O ( ε ) -estimation for the difference between the corresponding solutions to the given system and to the average systems, but only on the time scale 1 / ε . In our scenario, we have that the amplitude of any solution to the given system starting in the region of attraction of the limit cycle can be uniformly expanded by the average solution uniformly for t > 0. (Universidad de La Habana) 06/03/2012 19 / 48
negligible coefficients Definition Let p 2 s + 1 be a coefficient in the formal development Eq. 17, which is derived from the formal ∞ -jet of F . It shall be called negligible if satisfying | p 2 s + 1 | ≤ K s | τ a | (18) for a certain constant K s > 0 as τ a → 0. Definition The function p ( r ; ε ) in Eq.17 is said to be negligible if for all s ∈ N the coefficient p 2 s + 1 is negligible. (Universidad de La Habana) 06/03/2012 20 / 48
Theorem on non-negligible discriminant Theorem If the function p ( r ; ε ) is non-negligible, there must exist a positive integer N and a positive real value r 0 = r 0 ( τ a ) such that p ( r , ε ) has the non-trivial Taylor expansion: � ε 2 N + 2 r 2 N + 2 � p ( r ; ε ) = χ ε 2 N r − 2 N r 2 N + O (19) 0 where χ = + 1 or − 1 . In addition, the behavior of the factor r − 2 N as 0 τ a → 0 obeys the following alternative: either τ a → 0 r − 2 N = r − 2 N lim > 0 (20) ∗ 0 or , for a given γ , 0 < γ < 1 , � | τ a | γ � r − 2 N = O S as τ a → 0 . (21) 0 (Universidad de La Habana) 06/03/2012 21 / 48
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