Frequency locking by external forcing in systems with rotational symmetry

Frequency locking by external forcing in systems with rotational symmetry

Lutz Recke222Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany 444This work was partially supported by the DFG Research Centre MATHEON ”Mathematics for key technologies” under the project D8    Anatoly Samoilenko333Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska St. 3, 01601 Kiev, Ukraine 555This work was partially supported by the DFG cooperation project between Germany and Ukraine WO 891/3-1    Viktor Tkachenko333Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska St. 3, 01601 Kiev, Ukraine 555This work was partially supported by the DFG cooperation project between Germany and Ukraine WO 891/3-1    Serhiy Yanchuk222Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany 666This work was partially supported by the DFG Research Centre MATHEON ”Mathematics for key technologies” under the project D21.
Abstract

We study locking of the modulation frequency of a relative periodic orbit in a general -equivariant system of ordinary differential equations under an external forcing of modulated wave type. Our main result describes the shape of the locking region in the three-dimensional space of the forcing parameters: intensity, wave frequency, and modulation frequency. The difference of the wave frequencies of the relative periodic orbit and the forcing is assumed to be large and differences of modulation frequencies to be small. The intensity of the forcing is small in the generic case and can be large in the degenerate case, when the first order averaging vanishes. Applications are external electrical and/or optical forcing of selfpulsating states of lasers.

Key words.    Frequency locking; rotational symmetry; relative periodic orbits; external force; averaging; local coordinates

AMS subject classifications.    34D10; 34C14; 34D06; 34D05; 34C29; 34C30

1 Introduction

This paper concerns systems of ordinary differential equations of the type

 dxdt=f(x)+γg(x,βt,αt). (1.1)

Here and are -smooth with , and are parameters. The vector field is supposed to be -equivariant

 f(eAξx)=eAξf(x) (1.2)

for all and , where is a non-zero real -matrix such that and . The forcing term is supposed to be -periodic with respect to the second and third arguments

 g(x,ψ+2π,φ)=g(x,ψ,φ)=g(x,ψ,φ+2π),

and to possess the following symmetry

 g(eAξx,ψ,φ+ξ)=eAξg(x,ψ,φ) (1.3)

for all and Finally we assume that the unperturbed system

 dxdt=f(x) (1.4)

has an exponentially orbitally stable quasiperiodic solution of the type

 x(t)=eAα0tx0(β0t). (1.5)

Here is -periodic, and , are the wave and modulation frequencies of the solution (LABEL:qp). The main property of quasiperiodic solutions of the type (LABEL:qp) is that they are motions on a 2-torus without frequency locking. Those solutions are sometimes called modulated waves [16] or modulated rotating waves [5] or relative periodic orbits [9]. We assume that the following nondegeneracy condition holds:

 The vectors Ax0(ψ) and dx0dψ(ψ) are linearly independent. (1.6)

It is easy to verify that (LABEL:eq:nondeg) is true for all if it holds for one . Condition (LABEL:eq:nondeg) implies that the set

 T2:={(eAφx0(ψ))∈Rn:φ,ψ∈R}, (1.7)

which is invariant with respect to the flow corresponding to (LABEL:02), is a two-dimensional torus.

Our main results describe open sets in the three-dimensional space of the control parameters , and with and , where stable locking of the modulation frequencies of the forcing and of the solution (LABEL:qp) occurs, i.e. where the following holds: For almost any solution to (LABEL:01), which is at a certain moment close to , there exists such that

 infψ∥x(t)−eAψx0(βt+σ)∥≈0 for large t. (1.8)

Our results essentially differ in the so-called non-degenerate and degenerate cases (see Section 2). The degenerate case includes all cases when the averaged (with respect to the fast wave oscillations) forcing

 g1(x,ψ):=12π∫2π0g(x,ψ,φ)dφ (1.9)

vanishes identically, for example, if is of the type . Roughly speaking, under additional generic conditions the following is true:

Non-degenerate case: If is sufficiently large and is sufficiently small and is of order , then locking occurs.

Degenerate case: If is sufficiently large and is sufficiently small and is of order , then locking occurs.

The present paper extends previous work on this topic for particular types of the vector field and the forces [23, 20, 19]. In particular, the considered special cases in [23, 20, 19] are doubly degenerate in the sense that not only the averaged forcing (LABEL:av) vanishes but also the second averaging term turns to zero. Remark that in [18] related results are described for the case that both differences of modulation and wave frequencies are small, and [17] concerns 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].

Systems of the type (LABEL:01) appear as models for the dynamical behavior of self-pulsating lasers under the influence of external periodically modulated optical and/or electrical signals, see, e.g. [15, 1, 12, 13, 14, 24, 26], and for related experimental results see [8, 22]. In (LABEL:01) the state variable describes the electron density and the optical field of the laser. In particular, the Euclidian norm describes the sum of the electron density and the intensity of the optical field. The -equivariance of (LABEL:02) is the result of the invariance of autonomous optical models with respect to shifts of optical phases. The solution (LABEL:qp) describes a so-called self-pulsating state of the laser in the case that 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.

External forces of the type appear for describing an external optical injection with optical frequency and modulation frequency . In spatially extended laser models those forces appear as inhomogeneities in the boundary conditions. After homogenization of those boundary conditions and finite dimensional mode approximations (or Galerkin schemes) one ends up with systems of type (LABEL:01) with general forces of the type with (LABEL:eq:symg). External forces of the type

 g(x,βt,αt)=g1(x,βt) with g1(eAξx,ψ)=eAξg1(x,ψ)

appear for describing an external electrical injection with modulation frequency .

The further structure of our paper is as follows. In Section LABEL:sec:Main-results, we present and discuss the main results. In Section LABEL:sec:Averaging, the averaging transformation is used to simplify system (LABEL:01). The necessary properties of the unperturbed system (LABEL:02) are discussed in Section LABEL:sec:Unperturbed-system. In Section LABEL:sec:Local-coordinates, we introduce local coordinates in the vicinity of the unperturbed invariant torus. Further, we show the existence of the perturbed integral manifold and study the dynamics on this manifold in Section LABEL:sec:Investigation-of-the. Remaining proofs of main results are given in Section LABEL:sec:Proofs-o-theorems. Finally, in Sections LABEL:sec:example1 and LABEL:sec:example2 we illustrate our theory by two examples.

2 Main results

In the co-rotating coordinates the unperturbed equation (LABEL:02) reads as

 β0dydψ=f(y)−α0Ay. (2.1)

The quasiperiodic solution (LABEL:qp) to (LABEL:02) now appears as -periodic solution to (LABEL:newcoord). The corresponding variational equation around this solution is

 β0dydψ=(f′(x0(ψ))−α0A)y. (2.2)

It is easy to verify (see Section LABEL:sec:Unperturbed-system), that (LABEL:0per1) has two linear independent (because of assumption (LABEL:eq:nondeg)) periodic solutions

 q1(ψ):=dx0dψ(ψ),q2(ψ):=Ax0(ψ), (2.3)

which correspond to the two trivial Floquet multipliers 1 of the periodic solution to (LABEL:newcoord). One of these Floquet multipliers appears because of the -equivariance of (LABEL:newcoord), and the other one because (LABEL:newcoord) is autonomous. From the exponential orbital stability of (LABEL:qp) it follows that the trivial Floquet multiplier 1 of the periodic solution to (LABEL:newcoord) has multiplicity two, and the absolute values of all other multipliers are less than one. Therefore, there exist two solutions and to the adjoint variational equation

 β0dzdψ=−(f′(x0(ψ))T+α0A)z (2.4)

with

 pTj(ψ)qk(ψ)=δjk (2.5)

for all and , where for and otherwise.

2.1 Non-degenerate case

Using notation (LABEL:av) we define a -periodic function by

 G1(ψ):=12π2π∫0pT1(ψ+θ)g1(x0(ψ+θ),θ)dθ (2.6)

and its maximum and minimum as

 G+1:=maxψ∈[0,2π]G1(ψ),G−1:=minψ∈[0,2π]G1(ψ).

For the sake of simplicity we will suppose that all singular points of are non-degenerate, i.e.

 G′′1(ψ)≠0 for all ψ with G′1(ψ)=0.

This implies that the set of singular points of consists of an even number of different points. The set of singular values of will be denoted by

 S1:={G1(ψ1),…,G1(ψ2N)}, where {ψ1,…,ψ2N}={ψ∈[0,2π):G′1(ψ)=0}.

Our first result describes the behavior (under the perturbation of the forcing term in (LABEL:01)) of , which is an integral manifold to (LABEL:02), as well as the dynamics of (LABEL:01) on the perturbed integral manifold to the leading order.

Theorem 2.1

For all there exist positive and such that for all , and satisfying

 α≥α∗,β1≤β≤β2,0≤γ≤γ∗, (2.7)

system (LABEL:01) has a three-dimensional integral manifold , which can be parameterized by in the form

 x=eAφx0(ψ)+γeAφU1(ψ,βt,γ)+γαU2(ψ,φ,βt,αt,γ,α−1), (2.8)

where functions and are smooth and -periodic in , and

The manifold is orbitally asymptotically stable and solutions on this manifold are determined by the following system

 dψdt = β0+γpT1(ψ)g1(x0(ψ),βt)+γ2S11+γαS12, (2.9) dφdt = α0+γpT2(ψ)g1(x0(ψ),βt)+γ2S21+γαS22, (2.10)

where functions are smooth and -periodic in , and

In addition, for any there exist positive , and such that for all , and satisfying

 α≥α1, 0≤γ≤γ1, γG−1<β−β0<γG+1, dist(β−β0γ, S1)≥ϵ (2.11)

the following statements hold:

1. System (LABEL:01) has integral manifold of the form (LABEL:eq:IM0) and even number of two-dimensional integral submanifolds which are parameterized by in the form

 x = eAφx0(βt+ϑj)+γV1j(φ,βt,αt,γ,α−1)+1αV2j(φ,βt,αt,γ,α−1). (2.12)

The dynamic on the manifold is determined by a system of the type

 dφdt=α0+γWj(φ,βt,αt,γ,α−1).

Here are constants and functions , , and are smooth and -periodic in , and

2. Every solution of system (LABEL:01) that at a certain moment of time belongs to a -neighborhood of the torus tends to one of the manifolds as

One of the main statements of this theorem is that under conditions (LABEL:cond2), within the perturbed manifold , there exist lower-dimensional “resonant” manifolds corresponding to the frequency locking. These manifolds attract all solutions from the neighborhood of . Hence, the asymptotic behavior of solutions is described by (LABEL:xm) and has the modulation frequency . One can observe also from (LABEL:xm), that the perturbed dynamics is, in the leading order, a motion along with the new modulation frequency. The following theorem describes these locking properties more precisely.

Theorem 2.2

For any and there exist positive , and such that for all parameters satisfying the conditions (LABEL:cond2) and for any solution of system (LABEL:01) such that for certain there exist such that

 infϕ∥x(t)−eAϕx0(βt+σ)∥<ϵ1 for all % t>T. (2.13)

The conditions for the locking (LABEL:cond2) depend on the properties of the function . For the case that consists of two numbers and only, i.e. that has only two singular values, the set of parameters satisfying these conditions is illustrated in Fig. LABEL:fig:generic (left) for a fixed value of . The admissible values of the parameters and belong to a cone with linear boundaries

 β=β0+γ(G−1+ϵ) and β=β0+γ(G+1−ϵ),

bounded from above by .

If has more than two singular values, then the corresponding regions in the parameter space

 ∣∣β−β0−γG1(ψj)∣∣<γϵ

should be excluded. Here are the additional singular points of . An example is shown in Fig. LABEL:fig:generic (right) for the case with two additional singular points.

2.2 Degenerate case

In this subsection we suppose that

 12π∫2π0g(x,ψ,φ)dφ=0 for all x∈Rn and ψ∈R. (2.14)

In this case the functions and , which are defined in (LABEL:av) and (LABEL:0per3), are identically zero and, hence, cannot give any information about locking behavior like in Theorem LABEL:theorem01. Instead, the following functions and will define the locking:

 g2(x,ψ):=−12π∫2π0∂u1∂x(x,ψ,φ)g(x,ψ,φ)dφ

and

 G2(ψ):=12π2π∫0pT1(ψ+θ)g2(x0(ψ+θ),θ)dθ. (2.15)

Here

 u1(x,ψ,φ)=∫φg(x,ψ,θ)dθ

is the antiderivative of satisfying

 ∫2π0u1(x,ψ,φ)dφ=0.

Now we proceed as in the non-degenerate case: We denote

 G+2:=maxψ∈[0,2π]G2(ψ),G−2:=minψ∈[0,2π]G2(ψ)

and suppose that all singular points of are non-degenerate

 G′′2(ψ)≠0 for all ψ with G′2(ψ)=0.

This implies that the set of singular points of consists of an even number of different points. The set of singular values of will be denoted by

 S2:={G2(ψ1),…,G2(ψ2N)}, where {ψ1,…,ψ2N}={ψ∈[0,2π):G′2(ψ)=0}.

Our next result describes the locking behavior of the dynamics close to in the degenerate case.

Theorem 2.3

Suppose (LABEL:deg) holds. Then for all there exist positive constants and such that for all , and satisfying

 β1≤β≤β2, γ2/α≤μ∗, α≥α∗, (2.16)

system (LABEL:01) has a three-dimensional integral manifold , which can be parameterized by in the form

 x=eAφx0(ψ)+γα~U1(ψ,φ,βt,αt,γ2α,1α)+γ2α~U2(ψ,φ,βt,αt,γ2α,1α), (2.17)

where functions and are smooth and -periodic in , and

The manifold is orbitally asymptotically stable and dynamics on this manifold is determined by the following system

 dψdt = β0+γ2αpT1(ψ)g2(x0(ψ),βt)+γα2~S11+γ2α2~S12+γ4α2~S13, (2.18) dφdt = α0+γ2αpT2(ψ)g2(x0(ψ),βt)+γα2~S21+γ2α2~S22+γ4α2~S23, (2.19)

where functions are smooth and -periodic in , and

In addition, for any , there exists some positive , , and such that for all , and satisfying

 α≥α1, c1α≤γ≤c2√α, γ2αG−2<β−β0<γ2αG+2, dist(αγ2(β−β0),S2)≥ϵ, (2.20)

then the following statements hold:

1. System (LABEL:01) has integral manifold of the form (LABEL:eq:IM0-deg) and even number of two-dimensional integral submanifolds which are parametrized by in the form

 x=eAφx0(βt+ϑj)+γα~V1j+γ2α~V2j+1α~V3j+1αγ~V4j, (2.21)

where smooth functions are -periodic in , and The system on the manifold reduces to

 dφdt=α0+γ2α~Wj1+γα2~Wj2+1α3~Wj3,

where functions are smooth and -periodic in , and

2. Every solution of system (LABEL:01) that at a certain moment of time belongs to -neighborhood of the torus tends to one of the manifolds as

Theorem LABEL:theorem011 gives verifiable conditions on the parameters , for which the locking of modulation frequency to the modulation frequency of the perturbation takes place for the case when the degeneracy condition is fulfilled. These conditions are given by (LABEL:cond33) and differ from the conditions of locking in the generic case given by Theorem LABEL:theorem01. The locking phenomenon in the leading order looks similarly in both cases, i.e. the solutions tend asymptotically to and modulation frequency . More precisely, the following theorem holds.

Theorem 2.4

For any and there exist positive , , , and such that for all parameters satisfying the conditions (LABEL:cond33) and for any solution of system (LABEL:01) such that for certain there exist such that

 infϕ∥x(t)−eAψx0(βt+σ)∥<ϵ1 for all % t>T. (2.22)

Let us illustrate the set of parameters (LABEL:cond33) leading to the locking and compare it with the generic case. For a fixed large enough , the region in the parameter space has again a shape of a cone as in Fig. LABEL:fig:generic, but now the cone has tangent boundaries

 β=β0+γ2α(G∓2±ϵ) (2.23)

leading to a smaller synchronization domain for moderate values of , see Fig. LABEL:fig:deg. Small values of are even excluded . But, on the other hand, the synchronization is now allowed for large values of up to . This means practically, that the locking occurs not only for small but also for large amplitude perturbations, contrary to the case .

Again, when the singular set of contains more than 2 points, then the corresponding regions in the parameter space

 ∣∣∣β−β0−γ2αψj∣∣∣<γ2αϵ

should be excluded. Here are additional points from . Example in Fig. LABEL:fig:deg (right), shows the case with two additional singular points.

Finally note that the locking phenomenon described in [19] for a simple model with symmetry corresponds to the case when an additional degeneracy takes place with . As it is shown in [19], the locking cone becomes even more high () and thin with the slope proportional to instead of in (LABEL:eq:bouny).

3 Averaging

We perform changes of variables which depend on the average of the perturbation function with respect to fast oscillation argument As the result of these transformations we obtain an equivalent system, where the fast oscillation terms have the order of magnitude of and and smaller. The principles and details of the averaging procedure can be found e.g. in [2, 10, 21]. This transformation has the form

 x = x1+γαu1(x,βt,αt). (3.1)

Inserting (LABEL:change1) into (LABEL:01), we obtain

 dx1dt+γα(∂u1∂xdxdt+∂u1∂(βt)β)+γ∂u1∂(αt)=f(x1+γαu1)+γg(x,βt,αt).

Accordingly to the idea of the averaging method, the terms of order depending on the high frequency should vanish due to the choice of . This leads to the condition

 ∂u1∂(αt)=g(x,βt,αt)−g1(x,βt), (3.2)

where

 g1(x,βt)=12π∫2π0g(x,βt,φ)dφ.

Hence

 u1(x,βt,αt)=∫αtφ0(g(x,βt,φ)−g1(x,βt))dφ

is a periodic function in and . Strictly speaking, the above integral is considered componentwise, and the initial points can be different for each component of vector-function . Additionally, we choose the initial points in such a way that

 ∫2π0u1(x,βt,φ)dφ=0.

The resulting averaged system reads

 dx1dt = f(x1)+γg1(x1,βt)+γαA1(x1,βt,αt)+γ2αA2(x1,βt,αt) (3.3) +γ2α2X1(x1,βt,αt,γα)+γ3α2X2(x1,βt,αt,γα),

where

 A1(x1,βt,αt) = ∂f∂xu1−∂u1∂xf−∂u1∂(βt)β, A2(x1,βt,αt) = −∂u1∂xg+∂g1∂xu1.

Functions and are -smooth and -periodic in and Here we use the fact that for small the change of variables (LABEL:change1) is equivalent to

 x = x1+γα~u1(x1,βt,αt,γα) (3.4)

with smooth bounded and periodic with respect to and function

Note that the averaged term is equivariant with respect to the group action

 g1(eAξx,ψ) = 12π∫2π0g(eAξx,ψ,φ)dφ (3.5) = 12π∫2π0eAξg(x,ψ,φ−ξ)dφ=eAξg1(x,ψ). (3.6)

If we perform the second averaging change of variables

 x1 = x2+γα2u21(x1,βt,αt)+γ2α2u22(x1,βt,αt). (3.7)

The functions and are selected from the conditions that the terms of the orders and , which depend on high frequency , vanish. This leads to

 ∂u21∂(αt)=A1(x1,βt,αt),∂u22∂(αt)=A2(x1,βt,αt)−g2(x1,βt), (3.8)

and

 g2(x1,βt)=12π∫2π0A2(x1,βt,φ)dφ=−12π∫2π0∂u1(x1,βt,φ)∂xg(x1,βt,φ)dφ. (3.9)

We have used in (LABEL:eq:yyyttt) the following property of :

 12π∫2π0A1(x1,βt,φ)dφ=0.

The second averaging function is also equivariant with respect to the symmetry group action

 g2(eAξx,ψ)=eAξg2(x,ψ).

This can be seen from the following calculations

 g2(eAξx,ψ)=−12π∫2π0(∫φφ0∂g∂x(eAξx,ψ,θ)dθ)g(eAξx,ψ,φ)dφ =−eAξ12π∫2π0(∫φ−ξφ0−ξ∂g∂x(x,ψ,θ)dθ)g(x,ψ,φ−ξ)dφ =−eAξ12π∫2π−ξ−ξ(∫φφ0∂g∂x(x,ψ,θ)dθ)g(x,ψ,φ)dφ −eAξ12π∫φ0φ0−ξ∂g∂x(x,ψ,θ)dθ∫2π−ξ−ξg(x,ψ,φ)dφ =−eAξ12π∫2π0(∫φφ0∂g∂x(x,ψ,θ)dθ)g(x,ψ,φ)dφ=eAξg2(x,ψ).

After the second averaging, in the case , the system admits the form

 dx2dt=f(x2)+γ2αg2(x2,βt)+γα2g3(x2,βt,αt,γ2α,1α)+γ2α2g4(x2,βt,αt,γ2α,1α), (3.10)

where -smooth functions and are -periodic in and and function is -periodic in and equivariant with respect to action Note that by (LABEL:change1) and (LABEL:change2) the variable is expressed by in a form

 x=x2+γα~u2(x2,βt,αt,γ2α,1α) (3.11)

with smooth bounded and periodic with respect to and function

4 Useful properties of the unperturbed system

In this section, we consider useful properties of the linearized unperturbed system and introduce an appropriate basis (matrix ), which locally splits the coordinates along the invariant torus (LABEL:eq:torus) and transverse to it. Further, this basis will be used in section LABEL:sec:Local-coordinates for the introduction of appropriate local coordinate system.

Since the unperturbed system (LABEL:02) has quasiperiodic solution then

 dx0(β0t)dt+α0Ax0(β0t)=f(x0(β0t)). (4.1)

The corresponding variational system

 dydt=∂f(~x)∂xy (4.2)

has two quasiperiodic solutions

 AeAα0tx0(β0t) and eAα0tdx0(β0t)dt. (4.3)

The following properties of the Jacobian follow from the equivariance conditions

 ∂f(x)∂x=e−Aξ∂f(eAξx)∂xeAξ, (4.4)
 Af(x)=∂f(x)∂xAx. (4.5)

The latter conditions and the change of variables in (LABEL:var) lead to

 dwdψ=B0(ψ)w,B0(ψ)=1β0(∂f(x0(ψ))∂x−Aα0), (4.6)

where . The linear periodic system (LABEL:var3) has two periodic solutions

 q1(ψ)=dx0(ψ)dψ,q2(ψ)=Ax0(ψ), (4.7)

see (LABEL:eq:2sol).

Let be the trivial vector bundle , where is the natural projection onto . Consider corresponding to (LABEL:var3) linear skew-product flow with time and ,

 π(ψ,x0,ψ0)=(Ω(ψ,ψ0)x0,ψ+ψ0), (4.8)

where is the fundamental solution of (LABEL:var3) such that . The vector bundle is the sum of two sub-bundles and , which are invariant with respect to the flow (LABEL:eq:*). The two-dimensional bundle consists of periodic solutions of (LABEL:var3) spanned by two linearly independent periodic solutions (LABEL:eq:qq). The solutions from the complementary bundle tend exponentially to zero as Since the bundle is trivial, the bundle is stably trivial. By [11, p. 117], any stably trivial vector bundle whose fiber dimension exceeds its base dimension is trivial. Therefore, if , the -dimensional bundle is trivial and there exists a smooth map , which is isomorphism between and In the case ,