Turing Patterns Self-organization in physical systems: rhythms, patterns and chaos 24.11.2010 Turing Patterns by Miriam Daeubler 1 Existence of Turing Patterns and different types of patterns will be discussed 1 Short Introduction to Turing Patterns 2 2 Linear Stability Analysis 6 3 Existence of Turing Patterns 11 4 Types of Turing Patterns in a System 20 5 Summary and Outlook 26 6 Literature 29 24.11.2010 Turing Patterns by Miriam Daeubler 2 Turing patterns can be used to describe evolution of cat pelts Leopard (left) and jaguar (right) pelt patterns at different ages of the animal Model not proven experimentally Source: 24.11.2010 R. T. Liu, S. S. Liaw, and P. K. Maini, Physical Review E 74, 011914 (2006) Turing Patterns by Miriam Daeubler 3 Turing patterns are topologically different, within one morphology different typical length scales exist Experimentally observed patterns in chlorite-iodine-malonic acid reactions Source: 24.11.2010 Ouyang/Swinney, Transition from a uniform state hexagonal and striped Turing patterns, Nature, Vol. 352, 610-611, 1991 Turing Patterns by Miriam Daeubler 4 The Turing model originally derived to explain morphogenesis in biological systems Some facts about Turing patterns • Model systems of two reacting and diffusing substances with concentrations u, v = , + ∆ = , + ∆ • • • The above system describes an activator-inhibitor system • Activator: substance that stimulates the growth of the concentration of both chemicals • Inhibitor: substance that leads to a decrease in the concentrations In 1952, Turing predicted that such a system can produce a stationary pattern, if the inhibitor diffuses much faster than the activator (local activation with lateral inhibition). Turing tried to explain morphogenesis, i.e. the development of shape or form in plants and animals. 24.11.2010 Turing Patterns by Miriam Daeubler 5 Existence of Turing Patterns and different types of patterns will be discussed 1 Short Introduction to Turing Patterns 2 2 Linear Stability Analysis 6 3 Existence of Turing Patterns 11 4 Types of Turing Patterns in a System 20 5 Summary and Outlook 26 6 Literature 29 24.11.2010 Turing Patterns by Miriam Daeubler 6 A diagonalizable coefficient matrix makes system of linear differential equations easy to handle A homogeneous linear system of ODE’s with constant coefficients ′ = (), ∈ ℝ() • • • For further discussion, assume that is diagonalizable The eigenvalues are obtained by solving the secular equation det − = 0 In order to construct fundamental system choose eigenbasis of ∈ ℂ , thus = ( 1 1 , . . , ), ∈ ℂ eigenvalues • The solution of the homogeneous system is ℎ = ∙ = • , =1 ∈ ℂ Usually, one is only interested in real solutions. Complex eigenvalues and eigenvectors always come in pairs, therefore, both can be combined to obtain only real solutions. 24.11.2010 Turing Patterns by Miriam Daeubler 7 Thanks to coordinate transform stability analysis only needs to be derived once Stability analysis in the simplest linear system (1/2) Consider the following system with constant coefficients ′ = + , ∈ ℝ() • • • For further discussion, assume that is diagonalizable Let ∗ be a special solution of interest, usually a fixed point of the system Transform coordinates such that = − ∗ ′ = () • Thus, it is only necessary to assess the stability of ∗ = 0. 24.11.2010 Turing Patterns by Miriam Daeubler 8 Stability analysis reduces to an eigenvalue problem Stability analysis in the simplest linear system (2/2) The stability is described by the initial value problem ′ = and 0 = with the solution = (0 )−1 Theorem The solution ∗ = 0 is a) strictly stable, if and only if all eigenvalues of A have a real part smaller than zero: < 0 b) stable, if and only if all eigenvalues of A have a real part smaller or equal zero: ≤ 0 c) unstable, in every other case 24.11.2010 Turing Patterns by Miriam Daeubler 9 Under certain conditions even the stability analysis of a nonlinear system reduces to an eigenvalue problem Stability analysis in the nonlinear case Consider the following nonlinear system of differential equations with being a nonlinear function which can be continuously differentiated a sufficient number of times ′ = ( ) Assume ∗ = 0 is a fixed point with 0 = 0. Using Taylor expansion one obtains ′ = + ( ) = (0), 0 = 0 = ( ) It can be shown that if the shortened system of differential equations ′ = is either strictly stable or unstable so is the nonlinear system. 24.11.2010 Turing Patterns by Miriam Daeubler 10 Existence of Turing Patterns and different types of patterns will be discussed 1 Short Introduction to Turing Patterns 2 2 Linear Stability Analysis 6 3 Existence of Turing Patterns 11 4 Types of Turing Patterns in a System 20 5 Summary and Outlook 26 6 Literature 29 24.11.2010 Turing Patterns by Miriam Daeubler 11 The possible existence of Turing patterns can be derived using stability analysis Stability analysis of Turing model (1/7) Let there be a steady, spatially homogeneous solution = 0 (fixed point) = , + ∆ = , + ∆ Firstly, linearize the system of equations in the vicinity of the fixed point + ∆ = − − + ∆ With = c= 24.11.2010 0 0 , = − , = − 0 0 0 0 0 0 Turing Patterns by Miriam Daeubler Signs of a, b, c and d chooses in such way that u corresponds to the activator, v to the inhibitor 12 The instability shall be diffusion driven, the system must be stable in the absence of diffusion Stability analysis of Turing model (2/7) Neglecting the diffusion terms the linearized system becomes = − − In order to test stability, one must solve the eigenvalue problem − − =0 − 1 ⟺ ± = − ± + 2 det 2 − 4 For the fixed point to be strictly stable ± < 0. Thus, < 24.11.2010 ∧ < Turing Patterns by Miriam Daeubler 13 In order to reduce the complexity of the PDE system use Fourier integral transform Stability analysis of Turing model (3/7) Linearizing one obtained a system of PDEs + ∆ − = − + ∆ Integral transforms are standard way to reduce complex PDEs to simpler PDEs or even ODEs, here use Fourier transform = ( , ) −∙ 3 = ( , ) −∙ 3 Fourier transforms describe a change in basis, new basis consists of eigenfunctions of Laplace operator Fourier transforms of a function only exist if it is absolutely integrable ( , ) 3 < ∞ 24.11.2010 Turing Patterns by Miriam Daeubler 14 Properties of Fourier transforms needed to derive system of differential equations in Fourier space Stability analysis of Turing model (4/7) Fourier transform of gradient of a function , −∙ 3 = = , −∙ 3 − , −∙ 3 , −∙ 3 = (, ) Fourier transform of Laplace of a function ∆ , −∙ 3 = = ∙ , −∙ 3 − , −∙ 3 = ∙ , , ∙ −∙ 3 = − 2 (, ) Fourier transform of derivative of a function with respect to time , −∙ 3 1 = lim , + − , ℎ→0 1 ℎ→0 = lim 24.11.2010 , + − (, ) = −∙ 3 (, ) Turing Patterns by Miriam Daeubler 15 Fourier transforms reduce PDEs to ODEs and stability analysis becomes an eigenvalue problem Stability analysis of Turing model (5/7) Fourier transform the system of PDEs ⟺ −∙ 3 = + ∆ −∙ 3 = + ∆ − − + ∆ −∙ 3 − −∙ 3 − + ∆ − 2 − ⟺ = − − 2 − 2 − = − − 2 Fourier transforms reduce PDEs to ODEs and stability analysis becomes an eigenvalue problem in Fourier space 24.11.2010 Turing Patterns by Miriam Daeubler 16 For a pattern to evolve some spatial modes must be unstable Stability analysis of Turing model (6/7) Fourier transforming the linearized system of differential equations yields − 2 = − − − 2 To find a criterion for unstable spatial modes find eigenvalues − 2 − − =0 − − 2 ⟺ 2 + − + + 2 + 2 + 2 − 2 + + = 0 Compare with − − =0 ⟺ 2 − + + = 0 and 24.11.2010 2 2 + + = 0 ⟺ ± = − ± Turing Patterns by Miriam Daeubler 2 2 − 17 Condition for instability to occur linked with typical length scales Stability analysis of Turing model (7/7) To sum up, the only possibility for an instability is for the product of the roots to be negative ( 2 ) ≡ 2 − 2 + + < 0 The Turing instability occurs as long as > and the minimum of 2 is negative. First requirement can be rewritten > Source: 24.11.2010 ⟺ > Hoyle, Pattern Formation, Cambridge University Press, 2006, p.20 Turing Patterns by Miriam Daeubler 18 Nonlinear bifurcation theory has to be used to predict morphologies Results obtained from stability analysis Stability analysis the parameter range for which instabilities are present the characteristic length of the resulting patterns, which are independent of the initial conditions, therefore intrinsic to the system initial conditions only influence the phase of the pattern, i.e. position and alignment of pattern entities the different possible morphologies of the patterns the selection rules for multistable systems Apply nonlinear bifurcation theory to approximately predict the stability of different Turing patterns 24.11.2010 Turing Patterns by Miriam Daeubler 19 Existence of Turing Patterns and different types of patterns will be discussed 1 Short Introduction to Turing Patterns 2 2 Linear Stability Analysis 6 3 Existence of Turing Patterns 11 4 Types of Turing Patterns in a System 20 5 Summary and Outlook 26 6 Literature 29 24.11.2010 Turing Patterns by Miriam Daeubler 20 Bifurcations describe sudden qualitative changes in the set of solutions of a system as parameters are varied Bifurcation theory in a nutshell (1/2) Consider the following nonlinear system of differential equations with being a nonlinear function which can be continuously differentiated a sufficient number of times ′ = ( ) The solutions of the system define a flow , with 0 , 0 = 0 . Definition bifurcation At a bifurcation there is a sudden qualitative change in the flow in response to infinitesimal changes in one or more parameters of the system of differential equations. The phase portrait, the number and stability of fixed points or periodic orbits is usually affected. Local vs. global bifurcation Local bifurcation: changes of the flow in the vicinity of a fixed point or periodic orbit Global bifurcation: changes that affect the large scale property of the flow 24.11.2010 Turing Patterns by Miriam Daeubler 21 For Turing patterns local bifurcations are of interest Bifurcation theory in a nutshell (2/2) Local bifurcations A local bifurcation occurs when a parameter change causes the stability of a fixed point or periodic orbit to change. This corresponds to the real part of an eigenvalue of a fixed point or PO passing through zero. Goal of bifurcation theory Bifurcation theory tries to produce parameter space maps or bifurcation diagrams that divide the parameter space into regions of topologically equivalent systems, in our case equivalent Turing pattern morphologies. Source: 24.11.2010 Ott, Chaos in Dynamical Systems, Second Edition, Cambridge University Press, 2002, p. 46 Turing Patterns by Miriam Daeubler 22 Stripes and hexagonal spotty pattern typical for 2D reaction-diffusion systems Typical patterns of 2D reaction-diffusion systems Source: 24.11.2010 Leppänen, The theory of Turing pattern formation, Helsinki University of Technology, 2004 Turing Patterns by Miriam Daeubler 23 Bifurcation theory able to predict stable morphologies found in numerical simulations of 2D system Bifurcation theory applied to Turing patterns: 2D hexagonal lattice 2D hexagonal lattice is a bistable system Source: 24.11.2010 Leppänen, The theory of Turing pattern formation, Helsinki University of Technology, 2004 Turing Patterns by Miriam Daeubler 24 Turing patterns discussed theoretically beforehand have been observed experimentally Experimentally observed patterns in chlorite-iodine-malonic acid reactions Source: 24.11.2010 Ouyang/Swinney, Transition from a uniform state hexagonal and striped Turing patterns, Nature, Vol. 352, 610-611, 1991 Turing Patterns by Miriam Daeubler 25 Existence of Turing Patterns and different types of patterns will be discussed 1 Short Introduction to Turing Patterns 2 2 Linear Stability Analysis 6 3 Existence of Turing Patterns 11 4 Types of Turing Patterns in a System 20 5 Summary and Outlook 26 6 Literature 29 24.11.2010 Turing Patterns by Miriam Daeubler 26 There is no general way to determine which state a morphologically multistable system will choose Results obtained from stability analysis and nonlinear bifurcation theory Stability analysis and nonlinear bifurcation theory the parameter range for which instabilities are present the characteristic length of the resulting patterns, which are independent of the initial conditions, therefore intrinsic to the system initial conditions only influence the phase of the pattern, i.e. position and alignment of pattern entities the different possible morphologies of the patterns the selection rules for multistable systems Formal theory of self-organization based on non-equilibrium thermodynamics, however, this theory is incomplete There is no general way to figure out the selection rules for multistable systems 24.11.2010 Turing Patterns by Miriam Daeubler 27 Besides missing selection rules there are many more topics concerning Turing patterns currently analyzed Further topics concerning Turing patterns • • • • • Turing patterns also appear in 3D systems Turing patterns can be generalized to allow for transport mechanisms other than diffusion Interaction between Turing instability (constant in time) and other instabilities such as Hopf instability (constant in space) Spatially correlated noise can induce Turing patterns in systems that under normal conditions would not exhibit any … 24.11.2010 Turing Patterns by Miriam Daeubler 28 Existence of Turing Patterns and different types of patterns will be discussed 1 Short Introduction to Turing Patterns 2 2 Linear Stability Analysis 6 3 Existence of Turing Patterns 11 4 Types of Turing Patterns in a System 20 5 Summary and Outlook 26 6 Literature 29 24.11.2010 Turing Patterns by Miriam Daeubler 29 Literature used 1) Hoyle, Pattern Formation: An Introduction to methods, Cambridge University Press, 2006 2) Ott, Chaos in Dynamical Systems, Second Edition, Cambridge University Press, 2002 3) Leppänen et al., Spatio-temporal dynamics in a Turing model, submitted to InterJournal 2004 4) Sanz-Anchelergues et al., Turing pattern formation induced by spatially correlated noise, Phys. Rev. E, vol. 63, 056124 5) Guckenheimer, Bifurcation, doi: 10.4249/scholarpedia.1517 6) Leppänen et al., A new dimension to Turing patterns, Physica D, 168-169 (2002) 35-44 7) Leppänen et al., Dimensionality effects in Turing pattern formation, arXiv:condmat/0306121v1, 2003 8) Nakao/Mikhailov, Turing patterns in network-organized activator-inhibitor systems, doi: 10.1038/NPHYS1651 9) Gierer/Meinhardt, A Theory of Biological Pattern Formation, Kybernetik 12, 30-39 (1972) 10) Leppänen et al., Morphological transitions and bistability in Turing systems, Phys. Rev. E, vol. 70, 066202 (2004) 11) Leppänen, The theory of Turing pattern formation, Helsinki University of Technology, 2004 24.11.2010 Turing Patterns by Miriam Daeubler 30 Literature used 12) Ouyang/Swinney, Transition from a uniform state hexagonal and striped Turing patterns, Nature, Vol. 352, 610-611, 1991 13) Leppänen et al., Turing Systems as Models of Complex Pattern Formation, Brazil Journal of Physics, vol. 34, No. 2A, June 2004 14) Movie: Turing-Hopf bifurcation in monostable system downloaded from http://www.apmaths.uwo.ca/~mkarttu/turing.shtml 15) Movie: Self-organization of spherical structures under Gaussian noise in 3D downloaded from http://www.apmaths.uwo.ca/~mkarttu/turing.shtml 24.11.2010 Turing Patterns by Miriam Daeubler 31 Backup 24.11.2010 Turing Patterns by Miriam Daeubler 32 In order to solve a system of linear differential equations one needs to find the homogeneous solution first General remarks on systems of linear differential equations Definition of an explicit system of linear first order differential equations ′ = + with ∈ ℝ Systems of higher order can always be reduced to a system of first order and () ∈ ℝ continuous functions of time ∈ ℝ. General solution of an explicit system of linear first order differential equations = + ℎ being a particular solution of the inhomogeneous differential equation ℎ being any solution of the homogeneous differential equation Study solutions of homogenous system of differential equations 24.11.2010 Turing Patterns by Miriam Daeubler 33 One needs to construct a basis of the vector space spanned by the solutions Solutions of a system of homogeneous linear differential equations The solutions of the homogeneous linear system of differential equations form a subspace of 1 ℝ, ℝ , the real vector space of all continuously differentiable functions ℝ → ℝ , with finite dimension. Receipe to construct a basis of this subspace for an initial value problem • • Choose 0 ∈ ℝ and a basis , 1, . . , of ℝ Solve ′ = (), 0 = ( 1 , . . , ) • The matrix () is called a fundamental system. Then the general solution of the homogeneous equation is ℎ = ∙ = 24.11.2010 =1 (), ∈ ℝ Turing Patterns by Miriam Daeubler 34 Many methods enable one to obtain a particular solution of the inhomogeneous system Solutions of a system of inhomogeneous linear differential equations • • First a construct a basis of the solutions of the homogeneous differential equations () Use one of the abundant methods to obtain a solution for the inhomogeneous system of differential equations for initial value problems, such as • Variation of constants • Green’s functions … 24.11.2010 Turing Patterns by Miriam Daeubler 35

© Copyright 2018