Frequency locking of modulated waves

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.

(Communicated by the associate editor name) Abstract. We consider the behavior of a modulated wave solution to an S 1equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs.
Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.
1. Introduction. This paper investigates systems of differential equations of the type where x ∈ R n , y ∈ C, the functions f, g : R n → R n , h : R n → C, and a : R → C are sufficiently smooth of class C l with some positive integer l. The function a is 2π-periodic, and α > 0, β > 0 and γ ≥ 0 are parameters. We assume that for γ = 0 the unperturbed system dx dt = f (x) + g(x)|y| 2 , (1.3) has an exponentially orbitally stable quasi-periodic solution of modulated wave type x(t) = x 0 (β 0 t), y(t) = y 0 (β 0 t)e iα0t . (1.5) Here α 0 > 0 and β 0 > 0 are constants, while x 0 : R → R n and y 0 : R → C are smooth 2π-periodic functions. We assume that the following nondegeneracy condition holds: It is easy to verify that (1.6) is true for all ψ ∈ R if it is true for one ψ. Moreover, without loss of generality we assume that ψ → arg y 0 (ψ) is periodic, i.e. the curve y = y 0 (ψ) in C does not loop around the origin (otherwise we should replace y 0 (β 0 t) by y 0 (β 0 t)e ikβ0t and α 0 by α 0 − kβ 0 with an appropriate k ∈ Z). It follows from assumption (1.6) that the set where T 1 = R/(2πZ) is the unit circle, is diffeomorphic to a two-dimensional torus. Obviously, T 2 is invariant with respect to the flow of (1.3)-(1.4), and the solution (1.5) lies on T 2 . Roughly speaking, our main result describes the domain in the three-dimensional space of the control parameters α, β and γ with |α − α 0 | ≫ 1 and β ≈ β 0 such that the following holds: For almost any solution (x(t), y(t)) to (1.1)- (1.2), which is at a certain moment close to T 2 , there exists σ ∈ R such that x(t) − x 0 (βt + σ) + |y(t)| − |y 0 (βt + σ)| ≈ 0 for large t.
Let us reformulate our result in a more abstract language as well as in the language of a physical application.
Abstractly speaking, (1.3)-(1.4) is an autonomous system which is equivariant under the T 1 -action (x, y) → (x, e iϕ y), ϕ ∈ T 1 , on the phase space. The solution (1.5) is a so-called modulated wave solution or relative periodic orbit to the T 1equivariant system (1.3)-(1.4). It is well-known that generically those solutions are structurally stable under small perturbations that do not destroy the autonomy and the T 1 -equivariance of the system. Thus, our results describe the behavior of exponentially orbitally stable modulated wave solutions to T 1 -equivariant systems under external forcings of modulated wave type in the case when the difference between the internal and the external modulation frequencies β − β 0 is small while the difference between the internal and the external wave frequencies α−α 0 is large. Note that in [12] related results are described for the case when both differences of modulation and wave frequencies are small, and [11] considers the case when the internal state as well as the external forcing are not modulated. For an even more abstract setting of these results see [4].
System (1.1)-(1.2) is a paradigmatic model for the dynamical behavior of selfpulsating lasers under the influence of external periodically modulated optical signals. For more involved mathematical models see, e.g., [1,7,8,9,10,17,18] and FREQUENCY LOCKING OF MODULATED WAVES 3 for related experimental results see [6,15]. In (1.1)-(1.2), the state variables x and y describe the electron density and the optical field of the laser, respectively. In particular, the absolute value |y| describes the intensity of the optical field. The T 1 -equivariance of (1.3)-(1.4) is the result of the invariance of autonomous optical models with respect to shifts of optical phases. The solution (1.5) describes a so-called self-pulsating state of the laser in the case when the laser is driven by electric currents which are constant in time. In those states the electron density and the intensity of the optical field are time periodic with the same frequency. Self-pulsating states usually appear as a result of Hopf bifurcations from so-called continuous wave states, where the electron density and the intensity of the optical field are constant in time.
The structure of our paper is as follows. The main results are formulated in Sec. 2. The proof is splitted into four sections. In Sec. 3 we use averaging transformations [2] in order to eliminate the fast oscillating terms with the frequency α. It appears that the first non-vanishing terms after the averaging procedure are of order γ 2 /α 2 . Local coordinates in the vicinity of the stable invariant toroidal manifold are introduced in Sec. 4 and then in Sec. 5 the existence of perturbed manifold is proved. The global behavior of a system on the perturbed torus is described in Sec. 6. Among others, the methods of perturbation theory [13,14] are used in our analysis.
We assume that    the trivial multiplier 1 of the monodromy matrix of linear periodic system (2.4) has multiplicity one, and the absolute values of all other multipliers are less than 1. (2.5) The adjoint system dp dψ = −A T (ψ)p, has a nontrivial periodic solution p(ψ) (A T denotes the transpose of A), which can be normalized such that Let us define the function G : R n × T 1 → R n+1 as follows Our first result describes the behavior (under the perturbation by the forcing term with γ > 0) of T 2 × R, which is an integral manifold to (1.1)-(1.2) with γ = 0, as well as the dynamics of the system (1.1)-(1.2) on the perturbed manifold. Then for all β 1 < β 2 there exist positive constants µ * , α * , δ, L and κ such that for all (α, β, γ) with α > α * , β 1 < β < β 2 and 0 ≤ γ α < µ * (2.7) the following holds: (i) The system (1.1)-(1.2) has a three-dimensional integral manifold M(α, β, γ) which can be parametrized by ψ, ϕ, t ∈ R in the form Here X j : R 4 × U → R and Y j : R 4 × U → C are C l−4 smooth, 4π-periodic with respect to ψ and 2π-periodic with respect to ϕ, βt and αt and (ii) The dynamics of (1.1)-(1.2) on M(α, β, γ) in coordinates ψ, ϕ and t is determined by a system of the type where the functions Ψ 1 , Φ 1 : are C l−4 -smooth, 4π-periodic with respect to ψ and 2π-periodic with respect to ϕ, βt and αt.
(iii) Any solution (x(t), y(t)) to (1.1)-(1.2) such that dist((x(t 0 ), y(t 0 )), T 2 ) < δ for certain t 0 ∈ R tends to one of the manifolds N j (α, β, γ) as t → ∞. Then for any ε > 0 and ε 1 > 0 there exist positive µ * , µ * and δ such that for all parameters (α, β, γ) satisfying the conditions (2.11)-(2.13) and for any solution (x(t), y(t)) of system (1.1)-(1.2) such that dist((x(t 0 ), y(t 0 )), T 2 ) < δ for certain t 0 ∈ R there exist σ, T ∈ R such that x(t) − x 0 (βt + σ) + |y(t)| − |y 0 (βt + σ)| < ε 1 for all t > T.   In Fig. 1 we show two typical cases of graphs of the function G. In the case (I) there exist one positive and one negative local extremum and in the case (II) two positive and two negative local extrema, i.e., (I) : : In Fig. 2 we show α=const sections of the locking region. In the case (I) this section is It is bounded by two straight lines γ = µ * /α and γ = µ * α and by two square root like curves In the case (II) the α=const section is bounded by the same two horizontal straight lines and by six square root like curves Finally, in Fig. 3 we show β=const sections of the locking region in the (1/α, γ) plane. We consider the parameter α in the region α > α * with sufficiently large α * > 0 (2.14) If (2.14) is satisfied, consider the set of all β > β 0 such that For any fixed α > α * , where α * satisfies (2.14), and for any fixed β > β 0 with (2.15), the line {(α, β, γ) : γ ∈ R} crosses the boundary of the locking region in two points γ = µ 1 α and γ = µ * α in case (I) and in four points γ = µ 1 α, γ = µ 2 α, γ = µ 3 α and γ = µ * α in case (II) (see also Fig. 4). Here we denoted 3. Averaging. In this section we perform changes of variables with the aim to average the nonautonomous terms with fast oscillating arguments αt. As the result of these transformations, we obtain an equivalent system, where the fast oscillating terms have the order of magnitude of γ 2 /α 2 and smaller. The principles and details of the averaging procedure can be found e.g. in [2]. Performing the change of variables , we obtain the transformed system where * denotes complex conjugation. In system (3.1)-(3.2), the fast oscillatory terms with frequency αt are now proportional to γ/α. Since the first averaging has not produced any nontrivial contributions on the zeroth order, the second averaging transformation is necessary: which allows eliminating fast oscillating terms of order γ/α.
where the remainder terms r 1 , r 2 are C l−2 smooth functions in all arguments and 2π-periodic in αt and in βt. Again, the second transformation has not produced any nontrivial contributions of the order 1/α. Let us perform the third change of variables which transforms the system to the following form: where the remainder terms r 3 , ..., r 6 are 2π-periodic in αt and in βt, of class C l−3 in all variables. The obtained system (3.3)-(3.4) contains a nontrivial contribution of the order γ 2 /α 2 and all fast oscillatory terms of the orders γ/α 3 , γ 2 /α 3 and smaller. The next section proceeds with the analysis of the averaged system (3.3)-(3.4).
It can be verified that . (4.14) We introduce new coordinates ψ and h instead of z in the neighborhood of the periodic solution z 0 by the formula With regard for (4.9) and (4.14), the relation (4.16) yields Since by our construction det dz0(ψ) dψ , Φ(ψ) = det Φ 1 (ψ) = 0 for all ψ, the matrix is invertible for sufficiently small h. Therefore taking into account the expansion we obtain for sufficiently small h : Hence, the equation (4.17) can be solved with respect to the derivatives dψ/dt and dh/dt : We supplement this system with equation (4.8): Using the equality 1 2π we replace the angular variable θ in system (4.18) -(4.20) by ϕ accordingly to the formula where ψ is a certain antiderivative of the function h 2 (z 0 (ξ)) − α 0 .

5.
Existence of the perturbed manifold. Using the local coordinates introduced in the previous section, we investigate here the existence and properties of the perturbed manifold. In addition to the circle T 1 = R/(2πZ) we will use the notation T ′ 1 = R/(4πZ) for the circle of length 4π and T k = T 1 × · · · × T 1 k times for k−dimensional torus.
In order to show this, for λ ∈ I λ0 the mapping T λ : F ρ → F ρ has been used, where ζ τ is solution of (5.8) for h = w(ζ, λ) with initial conditions ζ 0 = ζ. F ρ is the space of Lipschitz continuous functions w : T ′ 1 × T 3 → R n such that w C ≤ ρ, Lip w ≤ ρ, Lip w is Lipschitz constant of w with respect to ζ.
We consider the subset F ηa0 of F ρ0 which consists of functions w with w C ≤ ηa 0 , Lip ζ w ≤ ηa 0 , where a 0 is some positive constant.
Note that by [20], for sufficiently small λ the map is well defined. For proving C l−4 smoothness of integral manifold w 0 (ζ, λ) we use the fiber contraction theorem [3], p. 127. At first we show that invariant manifold is C 1 with respect to ζ. The continuous differentiability with respect to λ is proved analogously. The smoothness up to C l−4 can be improved inductively.
Following [3], p. 336, we introduce the set F 1 of all bounded continuous functions Φ that map T ′ 1 × T 3 into the set of all n × 4 matrices. Let F 1 ρ denote the closed ball in F 1 with radius ρ.
Analogously to the proof of Lemma 5.1, we apply the fiber contraction theorem and show that there exists a unique fixed point of T (w) in the neighborhood of (0, 0) ∈ C l−4 (T ′ 1 × T 3 ) × I λ0 . Functions in right-hand side of (6.21) are C l−4 smooth and 2π-periodic in ϕ 2 , ζ 1 , ζ 2 , such that v kj C l−4 ≤ M 2 , where positive constant M 2 does not depend on λ, χ and β k .
Proof of Theorem 2.2. In Lemma 6.1, the existence and local stability properties of the integral manifolds Π j , j = 1, ..., 2N have been proved. The integral manifolds N j correspond to the manifolds Π j after the averaging and transformations (4.3) and (4.15).
It has been proved in Theorem 2.1 that all solutions from some neighborhood of the torus T 2 are approaching the perturbed integral manifold M(α, β, γ). Therefore, for the proof of the statement 2 of Theorem 2.2 it is enough to show that the solutions on this manifold are approaching the solutions on one of the manifolds N j .
Next, by Lemma 6.1, as t further increases, the solution is attracted to one of the stable integral manifolds Π 2k or Π 2k+2 .