Normal form of O ( 2 ) Hopf bifurcation in a model of a nonlinear optical system with diffraction and delay

In this paper we construct an O(2)-equivariant Hopf bifurcation normal form for a model of a nonlinear optical system with delay and diffraction in the feedback loop whose dynamics is governed by a system of coupled quasilinear diffusion equation and linear Schrödinger equation. The coefficients of the normal form are expressed explicitly in terms of the parameters of the model. This makes it possible to constructively analyze the phase portrait of the normal form and, based on the analysis, study the stability properties of the bifurcating rotating and standing waves.


Introduction
Nonlinear optical systems with nonlocal feedback often possess certain symmetries that -if carefully studied -can help one understand the typical pattern formation scenarios.For instance, Hopf bifurcation in the presence of SO(2) symmetry gives rise to rotating waves: one-dimensional waves on a circle [9] or two-dimensional waves on a disc [11].
In its simplest form, Hopf bifurcation appears when two simple complex-conjugate eigenvalues of the linearized operator cross the imaginary axis with nonzero speed as a certain parameter is varied [12].However, when the system is O(2)-symmetric, Hopf bifurcation becomes degenerate as the eigenvalues are double, each with a two-dimensional eigenspace.For nondelayed equations this situation was studied with the use of normal forms [4,7] and branching equations [8].Before applying these ideas to a partial differential equation, one usually conducts a center manifold reduction and then proceeds to construct a normal form on the center manifold.Even for nondelayed equations the procedure is rather tedious (see [13] for a reaction-diffusion equation).
Teresa Faria extended this methodology to quasilinear functional differential equations (FDE) in Banach spaces [6]: she proposed a way to construct a normal form on a center manifold bypassing the explicit construction of the manifold and illustrated the approach on model problems.The method was successfully applied to a delayed diffusion FDE with SO(2) symmetry to study the stability properties of one-dimensional rotating waves [9].We note that, in the cited paper, the model lacks reflectional symmetry due to a transformation of the spatial argument in the feedback loop.
In [1], a model of a nonlinear optical system with diffraction and delay was studied that, unlike [9], includes just local spatial interactions and hence enjoys O(2) symmetry.An O(2)equivariant Hopf bifurcation permits not only rotating waves (both clockwise and counterclockwise) but also standing waves.The present paper is devoted to the construction of an O(2)-equivariant Hopf bifurcation for a nonlinear optical system with diffraction and delay that makes it possible to analytically study the stability of the bifurcating rotating and standing waves [2].

Notation
By H 2 (C) we denote the standard Sobolev space of complex-valued functions on the interval (0, 2π) that are Lebesgue square-integrable with their second derivative.By H 2 2π (C) we denote the closed subspace of H 2 (C) of 2π-periodic functions.It itself becomes a Hilbert space once endowed with the suitable inner product and norm [10].
Given a unitary space X, we write •, • X and • X to denote its inner product and the corresponding norm.An inner product •, • with no subscript stands for the standard L 2 (0, 2π) inner product Given a Banach space X with the norm • X , by C k ([a, b]; X) we denote the Banach space of k times continuously differentiable X-valued functions with the norm Finally, given a function space X(C) of complex-valued functions, we denote its realvalued counterpart by X.For example, H 2 (C) and H 2 .

Main equation and auxiliary statements
We consider a one-dimensional model of a nonlinear optical system with a delayed feedback loop and diffraction therein (see [3] for the physical aspects of the problem) To describe the effects of diffraction in the paraxial approximation we employ a linear operator that treats its input as the initial condition of a periodic initial-boundary value problem for the linear Schrödinger equation and propagates it along a distance z = z 0 .The sought real-valued function u(x, t) represents the phase modulation of the light wave in the nonlinear Kerr slice.The parameters involved in the problem statement are: D > 0 is the effective diffusion coefficient (actually, D = D/r 2 , where r is the circle radius); K > 0 is the nonlinearity coefficient (it is positive as it corresponds to Kerr-induced self-focusing of the light field); T > 0 is the temporal delay in the feedback loop; z 0 > 0 is the distance traversed by the light wave in the feedback loop (here, z 0 = z0 /r 2 ).Lemma 3.1 ([1, 2]).The operator B has a complete orthogonal system of eigenfunctions exp(inx), n ∈ Z, in H 2 2π (C).The corresponding eigenvalues are λ n (B) = exp(in 2 z 0 ).
Boundary value problem (3.1) admits spatially homogeneous equilibria u(x, t) ≡ K. Fixing a value K for the nonlinearity parameter and considering its perturbations K(µ) = K + µ, we get a branch of constant solutions u(x, t) ≡ K(µ).
We set u(x, t) = K(µ) + v(x, t) to bring (3.1) to its local form in the vicinity of K(µ) Taking out the linear part, we rewrite (3.3) as Clearly, L(µ) can be expanded as follows: is analytic in the neighborhood of the origin.The operator F and its Fréchet derivatives F w n µ m vanish at the origin when n < 2 or m > 1.
Below are the (nonzero) quadratic and cubic Fréchet derivatives of F at the origin:

From FDE to ODE in Banach space
To rewrite boundary value problem (3.3) in the common FDE terms [6], we use a function space C = C([−T, 0]; X), X = H 2 2π , and a function v t ∈ C that acts according to where v ∈ X; we also extend L(µ) onto C by L(µ)ϕ = L(µ)ϕ(−T) so that L(µ) is linear and bounded in C. We are ready to write (3.3) in its abstract form Here where Fn are the n-th order terms in the expansion of F. Consider the linearization of (4.1) at v = 0 and µ = 0: The corresponding characteristic equation is We restrict our attention to y ∈ {1, sin(nx), cos(nx)} ⊂ D(A) as this is an orthogonal basis of eigenfunctions of both A and L 0 in X. Characteristic equation (4.3) is thus reduced to a countable family of equations For a Hopf bifurcation to occur we demand the following from the solutions λ ∈ C of (4.4): 1.For all solutions λ their real parts Re λ ≤ 0. The first part of (Hopf) is unnecessary for the bifurcation itself but it makes the center manifold asymptotically stable.We do not mention the transversality condition explicitly for it is met automatically since Consider the generator A 0 : C → C of the flow of equation (4.2): According to [6], the roots λ of characteristic equation (4.3) are the eigenvalues of A 0 .As long as equation (3.3) is O(2)-equivariant, a four-dimensional eigenspace P ⊂ C is associated with λ = ±iν * and is spanned by Remark 4.2.On introducing a real vector space we can represent P as {Φz : z ∈ E 4 }.This will allow us to facilitate computations as we will be working in X(C) while technically staying in the context of real-valued functions X.
To decompose C into a direct sum of A 0 -invariant subspaces, we introduce a space C * ≡ C([0, T]; X) and a bilinear form It readily extends to a form •, • : C(C) * × C(C) → C that is antilinear in the first argument and linear in the second one: A formal adjoint with respect to •, • operator A * 0 is defined as and has the same imaginary eigenvalues.In the corresponding eigenspace we choose a basis Ψ that is biorthogonal to Φ.To this end we introduce and evaluate the following: We note that and, according to (Hopf), and, according to (Hopf), and Ψ = (κ φ1 , κ φ2 , κ φ3 , κ φ4 ) T is biorthogonal to Φ, i.e.Ψ, Φ = I.As a result, Q = ϕ ∈ C : Ψ, ϕ = (0, 0, 0, 0) T is invariant under the action of A 0 and C = P ⊕ Q.
To relax the constraints D(A 0 ) we present an enlarged phase space BC [6] that is composed of functions of the form ψ = ϕ + X 0 α, ϕ ∈ C, α ∈ X, with a norm ψ BC = ϕ C + α X , where X 0 (τ) = 0, −T ≤ τ < 0, X 0 (0) = I.In other words, BC comprises functions [−T, 0] → X that are uniformly continuous on [−T, 0).The extension Ã0 : BC → BC of the operator A 0 onto BC is defined as follows: Finally, we can formulate equation (4.1) as an ordinary differential equation in BC: It is shown in [6] that π(ϕ + X 0 α) = Φ( Ψ, ϕ + α, Ψ(0) X ) is a continuous projection onto P, which commutes with Ã0 on C 1 0 ; hence BC is decomposed into a topological direct sum BC = P ⊕ N(π).Going back to (4.5), we express v(t) ∈ C 1 0 as a sum v(t) = Φz(t) + y(t), where This leads to an equivalent system of differential equations in E 4 × N(π), which we write down in a way that is suitable for the computation of the normal form: where , is the restriction of Ã0 and

Normal form construction in the presence of O(2) symmetry
To construct a normal form, one has to simplify the power series expansion of the vector field term by term: on the j-th step, the j-th order non-resonant terms are canceled out via a change of variables.For a fixed j ∈ N and a Banach space Y consider a space V p j (Y) of homogeneous polynomials of degree j in p variables with coefficients from Y: We seek changes of the form (z, y) = ( z, ỹ) ), and U 2 j ∈ V 5 j (Q 1 0 ).Suppose we have already conducted the procedure for 1 ≤ l ≤ k − 1. Denote by fj = ( f 1 j , f 2 j ) the j-th order(in (z, y, µ)) terms we have obtained after the (k − 1)-th step; denote by g j = (g 1 j , g 2 j ) the j-th order terms after the k-th step.Then equations (4.6) take the form Here g j ( z, ỹ, µ) = fj ( z, ỹ, µ), 2 ≤ j ≤ k − 1, and where the operators M 1 k and M 2 k are defined as The terms we can cancel out are precisely the ones that lie in the images of M 1 k and M 2 k .In [14] a center manifold that satisfies ỹ = 0 is proved to exist.The flow on this center manifold is given by an ordinary differential equation in E 4   d dt To proceed we need to prescribe complementary subspaces to the images R(M 1 k ).

The operator M
) is well-defined.
3. The kernel N(M 1 2 ) has the following form 4. The kernel N(M 1 3 ) has the following form

Every V 5
k (E 4 ) can be decomposed as a direct sum The first 4 statements are straightforward to verify.The last assertion follows from the fact that the adjoint -with respect to a suitable inner product in V 5 k (E 4 ) -operator (M 1 k ) * has the same form as M 1 k but is associated with the matrix J * [5].

Computation of the normal form coefficients
We will construct the normal form up to the cubic terms.According to Lemma 5.1, we need to compute the following expressions: where P V is the projection onto V and f2 = f 2 .For the sake of brevity we will abuse some notation: ae 1 + be 3 + c.c. ≡ ae 1 + be 2 + be 3 + āe 4 ∈ E 4 , a, b ∈ C Using (4.7) we can evaluate f 1 2 (z, 0, µ) = f 1 2 (z, 0, µ): We will assume that Re A 1 = 0. This, actually, follows directly from (Hopf) if we impose a constraint n 2 * z 0 < π that is well-aligned with the applicability of the paraxial approximation of light propagation.
After we have dealt with the quadratic terms, f 1 3 becomes Hence it remains to project f 1 3 (z, 0, µ), U 1 2 (z, µ), U 2 2 (z, µ), and g 1 2 (z, 0, µ) onto the kernel N(M 1 3 ).Since Re A 1 = 0, we only need to compute the terms that are at most linear in µ as higher order terms do not affect the qualitative behavior of the trajectories.So we set µ = 0 to calculate the cubic terms.

.2)
Set h(z) ≡ U 2 2 (z, 0).We use (4.7) and the definition of M 2 2 to decipher equation (6.2): Since Φ is continuous and h ∈ V 5 2 (Q 1 0 ), we can pass to a limit in the first equation of (6.3) as τ → −0 and subtract the result from the second equation.Note that f 1 2 (z, 0, 0) vanishes; then (6.3) transforms into The right hand side of the second equation of (6.4) evaluates as We now solve equations (6.4).To this end we express h ∈ V 5 2 (Q 1 0 ) as a linear combination of monomials , and we deduce that h 2000 = h 0002 , h 0200 = h 0020 , h 1010 = h 0101 , h 1100 = h 0011 , and h 1001 , h 0110 ∈ Q 1 0 .On grouping the monomials, we obtain the following list of differential problems: where 2 , and K (2) 2 .

Conclusion
In this paper we constructed an O(2)-equivariant Hopf bifurcation normal form for a model of a nonlinear optical system with delay and diffraction in the feedback loop.The coefficients were expressed explicitly in terms of the parameters of the model.This makes it possible to constructively analyze the phase portrait of the normal form and, based on the analysis, study the stability properties of the bifurcating rotating and standing waves (see [2]).