Amplitude Equations

4.3 Amplitude Equations

As discussed in the previous chapters, a linear stability analysis reveals that the unselective state z(x) = 0 becomes unstable for r ≥ 0 and modes on the critical circle |k|= k_c start to grow. Directly after spontaneous symmetry breaking, when |z(x)|is still small compared to its asymptotic value reached at t→ ∞, the nonlinearity N₃[z]can be neglected and the dynamics is controlled by the linear terms. As shown in Chapter 3 the emerging pattern can then be approximated by a random superposition of modes and its statistics is expected to be Gaussian.

What can we tell about the dynamics at later times when the nonlinearities become important and the modes start to compete? In particular, what can we say about the attractors? For small values of the control parameterr �1, i.e. in theweakly nonlinear regime, solutions to the full dynamics can be approximated by planforms,

z(x) =^N−1�

j=0

A_je^ikjx, |kj|=k_c,

consisting of a linear superposition of discrete modes on the critical circle. Thereby the full dynamics Eq.(4.2) of the field z(x) is projected onto the finite dimensional subspace spanned by the amplitudes A_j resulting in amplitude equations, a set of coupled nonlinear differential equations describing the dynamics of the amplitudes Aj. Here we derive the amplitude equations for the dynamics Eq.(4.2). In the subsequent sections we then perform a stability analysis of their stationary solutions in order to identify sets of stable patterns. The perturbation theoretical analysis presented follows the treatise of [58] and [34].

The spectrum of L₀ is given by λ(k) =−(k_c²−k²)² and attains its maximum at k=|k_c|, where it is zero. Therefore, modes on the critical circle reside in the kernel of the linear operator L₀. Since one is interested in the dynamics in a small neighbourhood above the bifurcation point r = 0one introduces a small parameter γ ≥0 and assumes that the solutionz(x, t) andr can both be expanded into a power series in γ,

r = r₁γ+r₂γ²+r₃γ³+. . .

z(x, t) = z₁(x, t)γ+z₂(x, t)γ²+z₃(x, t)γ³+. . . . (4.7) In general this will be the case when the solution z(x, t) bifurcates from the homogeneous state in a continuous way. Note that the intrinsic time scale τ = r⁻¹ diverges at the bifurcation point, which means that forr→0 the dynamics ofz(x, t) is becoming arbitrarily slow, a phenomenon which is known as critical slowing down and which can be compensated by considering the dynamics on a slow time scale,

T =r t.

Expressed in rescaled time units the dynamics Eq.(4.2) becomes r ∂

∂Tz(x) =F[z(x)] (4.8)

and no longer exhibits any critical slowing down. We combine the ansatz (4.7) with the rescaled

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dynamics Eq.(4.8) and obtain

0 = L₀z−r∂_Tz+rz+r�Mz¯+N₃[z]

= γ(L0z₁) +γ²(r1z₁+L₀z₂−r₁∂Tz₁+r₁�Mz¯₁) + (4.9) +γ³(L₀z₃+r₁z₂+r₂z₁−r₁∂_Tz₂−r₂∂_Tz₁+r₁�Mz¯₂+r₂�Mz¯₁+N₃(z₁, z₁,z¯₁)) +

+γ⁴(L0z₄+r₁z₃+r₂z₂+r₃z₁−r₁∂_Tz₃−r₂∂_Tz₂−r₃∂_Tz₁+r₁�Mz¯₃+r₂�Mz¯₂+r₃�Mz¯₁ + +N3(z2, z₁,z¯₁) +N₃(z1, z₂,z¯₁) +N₃(z1, z₁,z¯₃))

+γ⁵(L0z₅+. . .) +. . .

Equation (4.9) can only be fulfilled when the terms inside the brackets vanish for every order in γ. This implies conditions of the form

L₀zi =r.h.s. (4.10)

wherei denotes the order of the term and the right hand side only depends on z_j with j < i.

The set of equations (4.10) can be solved in ascending order when the solvability conditions are met, i.e. when the right hand side is orthogonal to the kernel of the adjoint operator L^†₀ . In our case L₀ is formally self-adjoint and the kernels ofL^†₀ and L₀ are identical. For the first order in γ we have the condition

L₀z₁= 0

which implies thatz₁(x, T)∈kerL₀. The kernel ofL₀ is spanned by all Fourier modese^ikcx on the critical circle (and by the secular terms kcxe^ikcx, which are unbounded and thus irrelevant in our present context). The second order term yields to

L₀z₂ =r₁(−z₁+∂_Tz₁−�Mz¯₁)

from which we conclude that r₁ = 0, which is the only way to fulfill the compatibility condition, since the term in the brackets resides in the kernel of L₀. The compatibility condition applied to the third order term

L₀z₃=−r₂z₁+r₂∂_Tz₁−r₂�Mz¯₁+N₃(z1, z₁,z¯₁) yields a dynamical equation forz₁,

∂_Tz₁ =z₁+�Mz¯₁−P_cN₃(z1, z₁,z¯₁) (4.11) where we setr₂ = 1andP_c is the projection operator onto the kernel ofL₀. The planform ansatz

z₁(x) =^2n�−1

j=0

Aj(T)e^ikjx,

which consists of a superposition on 2n modes kj = k_c(cosα_j,sinα_j) on the critical circle, where we require that to each mode also its antiparallel mode is in the set, in combination with Eq.(4.11), yields a set ofamplitude equations

A˙_j =A_j+�e^4iαjA¯_j−+ ^�

k,l,m

A_kA_lA¯_me^−ikjxP_jN₃(e^ikkx, e^iklx, e^−ikmx) (4.12)

4.3 Amplitude Equations where P_j denotes the projection onto the Fourier modee^ikj and j⁻ denotes the index of the mode antiparallel to modej with the corresponding wavevectork_j−=−kj.

Next we show that many terms in the triple sum of Eq.(4.12) do not contribute due to symmetry.

As a result, the general form of the amplitude equations can be reduced to A˙_j =A_j+�A¯_j−e^4iαj−

2n�−1 k=0

g_jk|A_k|²A_j−

2n�−1 k=0

f_jkA_kA_k−A¯_j− (4.13) with real valued and symmetric matricesg_jkandf_jk which determine the coupling and competition between modes. They can be expressed in terms of angle-dependent interaction functions g(α) and f(α), which are obtained from the nonlinearityN₃[z](cf.[58, 30, 22]).

Due to the projection operatorP_j only termsN₃(e^ikkx, e^iklx, e^−ikmx) are contributing in which the wave vectors add up to kj, i.e.

kk+kl−km =kj.

Since all wave vectors have the same length this condition requires kk=kj, kl=km

or

kk=km, kl =kj

or

kk=−kl:=k_l−, km =−kj :=kj−

such that Eq.(4.12) becomes

A˙_j = A_j +�e^4iαjA¯_j−+^�

k�=j

A_j|A_k|²e^−ikjxP_jN₃(e^ikjx, e^ikkx, e^−ikkx) +

+^�

k�=j

Aj|A_k|²e^−ikjxPjN₃(e^ikkx, e^ikjx, e^−ikkx) +

+ ^�

k�=j,j−

A_k−A_kA¯_j−e^−ikjxP_jN₃(e^−ikkx, e^ikkx, e^ikjx) +Aj|Aj|²e^−ikjxPjN₃(e^−ikjx, e^ikjx, e^ikjx)

which can be brought into the form of Eq.(4.13) by setting

g_jk = −^�e^−ikjxP_jN₃(e^ikjx, e^ikkx, e^−ikkx) +e^−ikjxP_jN₃(e^ikkx, e^ikjx, e^−ikkx)^� f_jk = −1

2

�e^−ikjxP_jN₃(e^−ikkx, e^ikkx, e^ikjx) +e^−ikjxP_jN₃(e^ikkx, e^−ikkx, e^ikjx)^�

forj �=kand

gjj = −e^−ikjxPjN₃(e^ikjx, e^ikjx, e^−ikjx) f_jj = 0

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for the diagonal elements. For an isotropic system, which we assume here, the matrix elements g_jk and f_jk only depend on the angle difference α = |α_k−α_j| of the Fourier modes kj = kc(cosαj,sinαj). Therefore they can be expressed in terms of the continuous functions

g(α) = −^�e^−ik0xP N₃(e^ik0x, e^ih(α)x, e^−ih(α)x) +e^−ik0xP N₃(e^ih(α)x, e^ik0x, e^−ih(α)x)^�(4.14) f(α) = −1

2

�e^−ik0xP N₃(e^−ih(α)x, e^ih(α)x, e^ik0x) +e^−ik0xP N₃(e^ih(α)x, e^−ih(α)x, e^ik0x)^�

wherek0=k_c(1,0)andh(α) =k_c(cosα,sinα). From the definition (4.14) follows thatg(α) = g(α+2π)andf(α) =f(α+π). However, for the particular nonlinearity considered here, Eq.(4.5),

g(α) =g(α+π), (4.15)

which is due to the fact thatN₃[z]belongs to the class ofpermutation symmetric models satisfying N₃(zj, z_k, z_l) =N₃(zl, zj, z_k) (see[22]). The coupling coefficients are given by

g_jk = (1−1

2δ_jk)g(|α_k−α_j|) (4.16) and

f_jk = (1−δ_jk−δ_jk−)f(|α_k−α_j|). (4.17) For the nonlinearity Eq.(4.5) one obtains

g(α) =g+ (2−g)e(α) (4.18)

and

f(α) = 1

2g(α) (4.19)

withe(α) = 2 exp(−σ²k_c²) cosh(σ²k²_ccosα).

The amplitude equations can be derived from the energy functional

E = −

2n�−1 j=0

�A_jA¯_j+�(AjA_j−e^−4iαj+ ¯A_jA¯_j−e^4iαj)^� (4.20)

+1 2

2n�−1 j=0

2n�−1 k=0

�g_jk|A_j|²|A_k|²+f_jkA_jA_j−A¯_k−A¯_k^�

and written as a gradient descent

A˙j =−δE δA¯_j

In the following sections we identify classes of stationary solutions of the amplitude equations (4.13) and determine their stability criteria.

Im Dokument Eine Symmetrie der visuellen Welt in der Architektur des visuellen Kortex. (Seite 61-65)