New Symmetric Periodic Solutions for the Maxwell-Bloch Differential System

We provide sufficient conditions for the existence of a pair of symmetric periodic solutions in the Maxwell-Bloch differential equations modeling laser systems. These periodic solutions come from a zero-Hopf bifurcation studied using recent results in averaging theory.


Introduction and Statement of the Main Result
In nonlinear optics the Maxwell-Bloch equations are used to describe laser systems. These equations were obtained by coupling the Maxwell equations with the Bloch equation ( resonantly coupled to the laser field), see [1]. Now in MathSciNet appear 265 articles related with these equations, see for instance [4][5][6][9][10][11].
Recently in [7] it was studied the weak foci and centers of the Maxwell-Bloch systemu For c = 0 the differential system (1) has a singular line {(u, v, w)|u = 0, v = 0}; for c = 0 and ac(δ − ab) < 0 the differential system (1) has one equilibrium p 0 = (0, 0, δ); and for c = 0 and ac(δ − ab) > 0 the differential system (1) has three equilibria p + = u * , v * , w * , p − = − u * , −v * , w * and p 0 , where For a = δ = 0 the differential system has the singular line L = {(u, v, w)|v = 0, w = 0}. The periodic orbits bifurcating from the equilibrium p 0 was studied in [2]. Here we complete this study analyzing the periodic orbits which bifurcate form the other two singularities. We define a zero-Hopf equilibrium of a 3-dimensional autonomous differential system as an equilibrium point having two purely conjugate imaginary eigenvalues and a zero eigenvalue. The next result characterizes the zero-Hopf equilibria of system (1) that lies over the singular line L.
The only zero-Hopf equilibria of system (1) in the singular line L are q ± .
Proposition (1) is proved in Section 2. Perturbing the condition a = δ = 0 the line of singularity L disappears. However the next result shows that there are two equilibrium points in L which produce an isolated periodic solution due to a zero-Hopf bifurcation.
and ε a small parameter. Then for |ε| = 0 sufficiently small the Maxwell-Bloch differential system (1) has two symmetric isolated periodic solutions bifurcating from the equilibrium points q ± ∈ L when ε = 0 and 2a 3 c 1 1 + Theorem 2 is proved in Section 2.

The Proofs
Proof of Proposition 1 Consider q = (ū, 0, 0) ∈ L. In the following we discuss the conditions for q being a zero-Hopf equilibrium point of system (1). The characteristic equation at q is given as It is easy to check that (2) has the pair of pure imaginary roots ±iω (ω > 0) if and only ifū = ± 1 2 √ b 2 + ω 2 and c = −b.
Since system (1) is invariant under the transformation (x, y, z) → (−x, −y, z) we proceed the proof only for the point q − .
Proof of Theorem 2 Assuming the conditions of Theorem 2 and translating q − to the origin of coordinates, the differential system (1) writeṡ In order to write the linear part of system (3) into its Jordan normal form, we do the linear change of variables (u, v, w) → (x, y, z) where The differential system (3) becomeṡ To study the periodic orbits of system (4) when 0 < |ε| 1, we introduce the cylindrical coordinates x = R cos θ , y = R sin θ and z = Z. Doing this transformation system (4) becomes Rescaling the variables (R, Z) of system (5) as R = ε 2 r, Z = εz,and taking θ as the new independent we obtain the equivalent differential system Applying Theorem 3 from Appendix A to system (6) we calculate the correspondent averaging functions Note that the averaged equation g 3 (r, z) vanishes over the graph . Furthermore the Jacobian matrix of g 3 at z α is From (7) we have that α = −4a 3 π b 2 0 + ω 2 /ω 3 = 0. This verifies the conditions (i) and (ii) of Theorem 3. Thus we calculate the function f (x) and we get It is easy to check that f (α) has the positive zero

Appendix A: Averaging Theory
We consider differential systems of the forṁ with x in some open subset of R n , t ∈ [0, ∞), ε ∈ [−ε 0 , ε 0 ]. We assume F i and F for all i = 1, 2, 3, 4 are T -periodic in the variable t. Let x(t, z, 0) be the solution of the unperturbed systemẋ = F 0 (t, x), such that x(0, z, 0) = z. We define M(t, z) the fundamental matrix of the linear differential systemẏ such that M(0, z) is the n × n identity matrix. The displacement map of system (8) is defined as d(z, ε) = x(T , z, ε) − z.
The functions g 1 , g 2 , g 3 and g 4 will be called here the averaged functions of order 1, 2, 3 and 4 respectively of system (8). We say that system (8) has a periodic solution bifurcating from the point z 0 if there exists a branch of solutions x(t, z(ε), ε) such that the displacement function satisfies d(z(ε), ε) = 0 and z(0) = z 0 .
Let π : R m ×R n−m → R m and π ⊥ : R m ×R n−m → R n−m denote the projections onto the first m coordinates and onto the last n − m coordinates, respectively. For a point z ∈ U we also consider z = (a, b) ∈ R m × R n−m . Consider the graph The next theorem provides sufficient conditions for the existence of periodic solutions of the differential system (8) when the set Z is a continuum of zeros to the first non vanishing averaged equation.

Theorem 3
Let r ∈ {0, 1, 2, 3} such that r is the first subindex such that g r ≡ 0. In addition to hypothesis (H ) assume that (i) the averaged function g r vanishes on Z. That is g r (z α ) = 0 for all α ∈ V , and (ii) the Jacobian matrix We define the function Then the following statements hold.