I. I NTRODUCTION The use of the phase dynamics associated to nonlinear oscillators is a widely accepted tool to rigorously analyze complex collective phenomena like synchronization, pattern formation, and resonance. Examples of such behaviors are found, for instance, in biology [1], physics [2], and engineering [3], [4], [5], [6], [7], [8]. However, the reduction of periodic oscillatory dynamics to the associated phase model is relevant only if the inputs and disturbances are small compared to the attractivity of the limit cycle. This problem is of crucial importance in control engineering applications, where state-dependent inputs are used to achieve a desired objective. Motivated by neuroscience applications, we have recently developed a feedback control law that aims at altering synchronization in an interconnected neuronal population. Under some assumptions, the phase dynamics of the closed-loop system was analytically computed and sufficient conditions for different control objectives were derived [9], [10]. Nevertheless, the relevance of these results for the original ensemble of nonlinear oscillators (modeling the neuronal population) is not straightforward due to aforementioned intrinsic limitations of the phase reduction. In this paper, we generalize the oscillator model introduced in [9] and study its phase dynamics. This generalization permits to describe a wider range of coupling and feedback schemes and embraces in a unified model different interesting special cases, including the normal form (A.4) in [11], which is at the basis of several results in computational neuroscience. Based on classical results on normal hyperbolic invariant manifolds, we rigorously derive the closed-loop phase dynamics and individuate the parameters determining its validity, *The research leading to these results has received funding from the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n257462 HYCON2 Network of excellence, and by the French CNRS through the PEPS project TREMBATIC. 1 A. Franci is with Univ. Paris Sud 11 - L2S - Sup´ elec, Gif sur Yvette, 91192 France [email protected]s.supelec.fr 2 W. Pasillas-L´epine is with CNRS - L2S, same address

[email protected] 3 A.

Chaillet is with L2S - Univ. Paris Sud 11 - Sup´elec, same address

[email protected]

along with explicit bounds on its accuracy in reproducing the behavior of the original system. These results are used to study the existence of phase-locked solutions in the original nonlinear oscillator population. More precisely, we show that, if the coupling and feedback strengths are sufficiently small, then generically no phase-locked solutions oscillating with non-zero frequency can exist under proportional mean-field feedback. Beside the mathematical interest, these results find an application in neuroscience, in particular, Parkinson’s disease. Phase-locked solutions model pathological states and the therapeutic objective is to bring a synchronous neuronal population to a more disordered state, corresponding to a healthy condition. The use of electrodes implanted in the deep brain (deep brain stimulation) is a recent medical technology that might permit to overcome pharmacological limitations in achieving this goal. From a control theory point of view, it might also allow closed-loop stimulation of neuronal populations. The mathematical modelling of this control scheme was our starting point for the theoretical exploration presented in [9], [10]. However, from a medical point of view, the non-existence of phase-locked solutions is not sufficient to ensure the disappearance of a pathological state. The resulting network behaviour might indeed correspond to an “almost” phase-locked state [9]. On the contrary, specific medical control objectives (desynchronization and inhibition) were explored in [10]. Based on the results presented in the first part of this note, future work will aim at extending the results in [10] to the generalized oscillator dynamics (1) considered in this note. The paper is organized as follows. The oscillator network dynamics under analysis is presented in Section II. Section III contains a rigorous derivation of the associated phase dynamics. The existence of phase-locked solutions in the phase dynamics is studied in Section IV. A similar existence result is finally proved in Section V for the original oscillator network.

Notation and preliminaries Rn≥0 denotes the closed orthant {x ∈ Rn : xi ≥ 0, i = 1, . . . , n}, whereas Rn>0 denotes the open orthant {x ∈ Rn : xi > 0, i = 1, . . . , n}. T n denotes the n-torus. Norms: Given n, m ∈ N and A = {Aij }i=1,...,n ,j=1,...,mq ∈ Rn×m , we denote the Frobenius Pn Pm 2 norm of A as |A| := i=1 j=1 Aij . When either n = 1 or m = 1, | · | is the Euclidean norm. Splittings: Given a finite dimensional vector space V , a splitting of V is a collection of linear subspace Vi ⊂ V , i = 1, . . . , l, such that V = ⊕li=1 Vi , where ⊕ denotes the direct sum. Given a linear application A : V → V , an

A-invariant splitting is a splitting V = ⊕li=1 Vi such that AVi ⊂ Vi , i = 1, . . . , l. Tangent maps: Given a n-dimensional manifold M, we denote its tangent space at x ∈ M as Tx M, and similarly for submanifolds. Given a set W ⊂ M and a map f : M → Rm , we denote by f |W : W → Rm the restriction of f to W , i.e. f |W (x) = f (x) for all x ∈ W . The tangent application of a C 1 function f : M → Rm is denoted as Df , i.e. in coordinates Df (x) = ∂f ∂x (x). Riemannian metric: Given an n-dimensional smooth Riemannian manifold M with metric h·, ·iR , we let | · |R denote the induced norm. Measure: The Lebesgue measure on Rn is denoted by µ, and for almost all (∀a.a.) denotes the equivalence operation with respect to this measure. II. L ANDAU -S TUART OSCILLATORS WITH DIFFUSIVE AND FEEDBACK COUPLING

We start by introducing the coupled oscillator system under analysis. Given ρi > 0, i = 1, . . . , N , consider the following dynamics on CN

z˙i = (iωi + ρ2i − |zi |2 )zi +

N X

κij eiδij (eiηj zj − eiηi zi ) + ui ,

j=1

(1) where κ := [κij ]i,j=1,...,N ∈ RN ×N ,

ui :=

N X

γ˜ij eiφij cos ϕj Re(eiψj zj ) + i sin ϕj Im(eiψj zj ) ,

j=1

(2)

neuron is represented by the real part of the associated oscillator, whereas the imaginary part of the oscillation accounts for the effects of other physical variables. The introduction of the phases Φ in (1) accounts for possible imprecision in the association between physical (voltages, conductances, ion concentrations, etc.) and mathematical (real and imaginary parts) variables. For instance: •

•

•

The phases [ηi ]i=1,...,N rotate the oscillator contributions to the diffusive coupling, accounting for model reduction imprecision in the identification of the membrane potential (resp. other physical variables) with the real (resp. immaginary) part of the oscillator. The phases [δij ]i,j=1,...,N rotate the diffusive coupling terms in such a way that the imaginary part of the coupling influences the real one and vice-versa. Similarly, the phases [ϕi ]i=1,...,N , [φij ]i,j=1,...,N , and [ψi ]i=1,...,N ) accounts for the same type of inaccuracies in the feedback coupling.

Special case 2 Another interesting special case of (1) is given by the normal form (A.4) in [11], which constitutes the basis of many theoretical works on synchronization phenomena between coupled oscillators such as [13], [14], [15], just to name a few examples. This is obtained with the choice ϕi = π4 , ηi = ψi = 0, and φij = δij , for all i, j = 1, . . . , N . Detailed computations are provided in the online preprint [16].

and γ˜ := [˜ γij ]i,j=1,...,N ∈ RN ×N . We also define

III. F ORMAL REDUCTION TO THE PHASE DYNAMICS

Φ := ([δij ]i,j=1,...,N , [ηi ]i=1,...,N , [ϕi ]i=1,...,N , [φij ]i,j=1,...,N , [ψi ]i=1,...,N ) ∈ RN ×(2N +3) .

(3)

The dynamics (1) can be split in three parts. The term (iωi + ρ2i − |zi |2 )zi is the oscillator internal dynamics. It corresponds to a stable oscillation of radius ρi and frequency ωi and is commonly referred to as Landau-Stuart oscillator, which represents a normal form of a supercritical Andronov-Hopf bifurcation [12]. The second term constitutes a linear coupling between the oscillators, where κij eiδij is the (complex) coupling gain and where the phases ηi rotate the oscillator contribution to the coupling. The last term ui constitutes a feedback couinjects the output of each oscillator yj := pling term which cos ϕj Re(eiψj zj ) + i sin ϕj Im(eiψj zj ) back in the network with complex gains Bij := γ˜ij eiφij . The analysis of (1) is motivated by the two following special cases:

Special case 1 The choice Φ = 0 and γ˜ij = βi αj , i, j = 1, . . . , N , for some α := [αi ]i=1,...,N ∈ RN and β := [βi ]i=1,...,N ∈ RN , reduces (1) to [9, Equation (3)]. Motivated by neuroscience applications, we recently introduced that model in [9], [10] to analyze the behavior of a network of diffusively coupled periodically spiking neurons under the effect of an electrical stimulation that is proportional to the ensemble mean membrane voltage. In those works, the membrane voltage of each

The goal of this section is to derive the phase dynamics of the closed-loop system (1)-(2). We start by writing the oscillator states in polar coordinates, that is zi = ri eiθi , for all i = 1, . . . , N , where ri := |zi | ∈ R≥0 and θi := arg(zi ) ∈ T 1 . We stress that the oscillator phases θi are defined only for |zi | = ri > 0. In these coordinates the dynamics (1)-(2) reads

r˙i eiθi + iri θ˙i eiθi = (iωi + ρ2i − ri2 )ri eiθi + N X κij eiδij (eiηj rj eiθj − eiηi ri eiθi ) + ui j=1

where N h i X ui = γ˜ij eiφij cos ϕj Re(eiψj rj eiθj)+i sin ϕj Im(eiψj rj eiθj) . j=1 −iθi

By multiplying both sides of this dynamics by e ri , extracting the real and imaginary part, and using some basic trigonometry, we get, for ri > 0, i = 1, . . . , N ,

θ˙i = ωi + fi (θ, r, κ, γ˜ , Φ) r˙i = ri (ρ2i − ri2 ) + gi (θ, r, κ, γ˜ , Φ),

(4a) (4b)

where, for all i = 1, . . . , N ,

fi (θ, r, κ, γ˜ , Φ) := N N X X κij rj κij sin(δij + ηi ) + sin(θj − θi + δij + ηj ) − ri j=1 j=1 N X γ˜ij rj sin ϕj + cos ϕj + sin(θj − θi + φij + ψj ) ri 2 j=1 sin ϕj − cos ϕj sin(θj + θi − φij + ψj ) + 2 gi (θ, r, κ, γ˜ , Φ) := N N X X κij rj cos(θj − θi + δij + ηj ) −ri κij cos(δij + ηi ) + j=1

j=1

+

N X

γ˜ij rj

j=1

sin ϕj + cos ϕj cos(θj − θi + φij + ψj ) 2 cos ϕj − sin ϕj cos(θj + θi − φij + ψj ) + 2

which defines the phase/radius dynamics of (1)-(2) on T N × RN >0 . Let f := [fi ]i=1,...,N and g := [gi ]i=1,...,N . When the diffusive coupling and the feedback are both zero, that is κ = γ˜ = 0, we have f ≡ g ≡ 0. Hence, in this case, equation (4) reduces to ω θ˙ = H(θ, r) := (5) r(ρ2 − r2 ) r˙

The two subspaces Nxu and Nxs are the called the unstable and the stable space of Tx N with respect to the tangent application of (6), respectively. Conditions ii-a) (resp. ii-b)) means that, when Nxu 6= ∅ (resp. Nxs 6= ∅), the flow of (6) expands (resp. contracts) the unstable (resp. stable) space more sharply than its tangent space. Lemma 1. The dynamics (5) is normally hyperbolic at T0 . Proof. Denote Q := T N × RN ≥0 . Moreover, denote ϑ := (θ, ρ) ∈ T0 . Since r is constant n on T0 , the tangent spaceo Tϑ T0 ∂ at a point ϑ ∈ T0 is spanned by ∂θ , i = 1, . . . , N . Let i ϑ ∂ Nϑs := span , i = 1, . . . , N . ∂ri ϑ Fix coordinates in the tangent space in such a way that ∂ , i = 1, . . . , N , is a base of Tϑ T0 , where (ˆ ei , 0N ) = ∂θ i i eˆj = 0 if j 6= i and eˆii = 1, and, similarly, (0N , eˆi ) = ∂r∂ i , i = 1, . . . , N , is a base of Nϑs . In these bases, for all ϑ ∈ T0 the tangent application of (5) at ϑ reads " # ∂ θ˙ ∂ θ˙ ∂θ ∂r DHϑ = ∂ r˙ ∂ r˙ ∂θ ∂r ϑ 0N ×N 0N ×N = . (7) 0N ×N diag{−2ρ2i }i=1,...,N It follows from (7) that, for all ϑ ∈ T0 the splitting

Tϑ Q = Tϑ T0 ⊕ Nϑs

where ρ := [ρi ]i=1,...,N ∈ RN . It is obvious that T0 := T N × {ρ} ⊂ T N × RN >0 is invariant for (5), since all its points are fixed points of the radius dynamics in (5). Moreover, it is normally hyperbolic as defined and proved below.

is DHϑ -invariant, which verifies condition i) of Definition u 1 with N(θ,ρ) = ∅. In particular, in the bases (ˆ ei , 0N ) and i (0N , eˆ ), we have DH 0 := DH(θ,ρ) = 02N ×2N (8a)

Definition 1. Given an n-dimensional smooth Riemannian manifold M, let x(·, x0 ) denote the solution of an autonomous dynamical system

DHϑs := DHϑ |N s ϑ 0N ×N 0N ×N = . 0N ×N diag{−2ρ2i }i=1,...,N

x˙ = F (x),

x ∈ M,

(6)

starting at x0 ∈ M at t = 0, everywhere it exists. Let, moreover, N ⊂ M be a smooth compact m-dimensional submanifold. The dynamical system (6) is said to be normally hyperbolic at N if the two following conditions are satisfied: i) For all x ∈ N , there exists a DFx -invariant splitting

Tx M = Nxu ⊕ Tx N ⊕ Nxs of Tx M over N . In this case, for all x ∈ N , denote DFxu := DF (x)|N u , DFx0 := DF (x)|Tx N , and x DFxs := DF (x)|N s . x ii) We have either Nxr = ∅, for all x ∈ N , r = s, u, or: DF u DFx0 v e e x v R R > max 1, sup sup ii-a) inf inf u x∈N v∈Tx N |v|R x∈N v∈Nx |v|R

ii-b) sup sup x∈N

v∈Nxs

DF s e x v

|v|R

R

DFx0 v e R < min 1, inf inf x∈N v∈Tx N |v|R

ϑ

Tϑ T0

(8b)

We now endow Q with a suitable Riemannian metric. To this aim note that since Dρ := diag{−2ρ2i }i=1,...,N is Hurwitz, there exists a positive definite matrix P ∈ RN ×N > 0 such that√the exponential application eDρ contracts the norm |x|P := xT P x induced by P , that is |eDρ x|P < |x|P for all x ∈ RN . Since Dρ does not depend on ϑ, so does P . Let IN 0N ×N P˜ := . 0N ×N P We endow Q with the constant Riemannian metric < ·, · >P˜ defined by < v, u > ˜ = v T P˜ u, v, u ∈ T(θ,r) Q. P

In the base (ON , eˆi ), i = 1, . . . , N , a generic vector v ∈ Nϑs is represented by (0N , v r ), v r ∈ RN . Therefore, we have s

sup sup s ϑ∈T0 v∈Nϑ

|eDHϑ v|P˜ |v|P˜

s

=

sup sup ϑ∈T0 v r ∈RN

=

sup sup ϑ∈T0 v r ∈RN

|eDHϑ (0N , v r )|P˜ |(0, v r )|P˜ |eDρ v r |P < 1. |v r |P

Condition ii-b) of Definition 1 follows by noticing that, since DHϑ0 = 02N ×2N , DHϑ0 v e P˜ = 1. min 1, inf inf ϑ∈T0 v∈Tϑ T0 |v|P˜

where

f¯i (θ, κ, γ˜ , Φ) := (12) N N X X kij sin(θj − θi + δij + ηj ) κij sin(δij + ηi ) + − j=1

j=1

+

N X

γij

j=1

We are now going to apply a classical result of Hirsch et al. [17, Theorem 4.1] to show that, if κ, γ˜ are sufficiently small, then (4) still has an attractive normally hyperbolic invariant manifold in a neighborhood of T0 . Theorem 1. Given ρi > 0, i = 1, . . . , N , there exist constants δh , Ch > 0 depending only on ρi , i = 1, . . . , N , such that, if

|(κ, γ˜ )| < δh

(9)

then there exists an attractive invariant manifold Tp ⊂ T N × RN >0 normally hyperbolic for (4) and satisfying

|r − ρ| ≤ Ch |(κ, γ˜ )|,

∀(θ, r) ∈ Tp .

(10)

Remark 1. Note that the constants Ch and δh , and thus the small coupling condition (9), depend only on the oscillator natural radius ρi . In particular, they are independent of the natural frequencies ω . Local input-to-state stability analysis applied to the linearized dynamics r˙i = −2ρ2i (ri − ρi ) + 1 gi (θ, r, κ, γ˜ , Φ) suggest that Ch ≈ mini=1,...,N {ρi } , whereas δh ≈ mini=1,...,N {ρi }. Theorem 1 states that, if the coupling and the feedback strengths are smaller then a constant δh depending only on the natural radius ρi , then the network dynamics (1) evolves on a normally hyperbolic invariant manifold Tp . Moreover, the distance between the sets Tp and T0 is less than Ch |(κ, γ˜ )|, where again Ch depends only on the natural radius. We refer to condition (9) as the small coupling condition. If the small coupling condition is satisfied, then Theorem 1 has two important consequences: 1. On the attractive normally hyperbolic invariant torus Tp , the oscillator radius variations around their natural radius are bounded by |r(t) − ρ| ≤ Ch |(κ, γ˜ )|, for all t ≥ 0. In particular, they are small, provided that |(κ, γ˜ )| is small. 2. To the first order in |(κ, γ˜ )| the phase dynamics does not depend on the radius dynamics. Indeed from (10) and (16) it follows that ∂f (θ, r, κ, γ˜ , Φ)(r − ρ) ≤ ∂f (θ, r, κ, γ˜ , Φ) |r − ρ| ∂r ∂r ≤ Cf |(κ, γ˜ )|Ch |(κ, γ˜ )|

= Cf Ch |(κ, γ˜ )|2 . Hence, to the first order in |(κ, γ˜ )|, i.e. to the first order in the coupling and feedback strength, if the small coupling condition (9) holds true, then (4) boils down to the phase dynamics equation

θ˙i = ωi + f¯i (θ, κ, γ˜ , Φ),

(11)

sin ϕj + cos ϕj sin(θj − θi + φij + ψj ) 2

sin ϕj − cos ϕj + sin(θj + θi − φij + ψj ) , 2 i h κij ρj and γ := with k := [kij ]ij=1,...,N := ρ i i h γ ˜ij ρj , and the radius dynamics can be [γij ]ij=1,...,N := ρi neglected, as it was done, for instance, in [18], [9]. Remark 2. We stress that, if the small coupling condition (9) is satisfied, the error between the nominal dynamics (4) and its phase dynamics (11) is of the same order as |(κ, γ˜ )|2 . Proof of Theorem 1. Even though for κ = γ˜ = 0 it holds that f ≡ g ≡ 0, as soon as (κ, γ˜ ) 6= 0, f and g are unbounded, due to singularities at ri = 0 and ri = ∞, i = 1, . . . , N . However, since the persistence of the normally hyperbolic invariant torus solely relies on local arguments, we can construct a locally defined auxiliary smooth dynamical system, which is identical to (4) near T0 . The auxiliary system possesses a normally hyperbolic invariant manifold Tp near T0 if and only if the same holds for the original dynamics (4). S TEP 1: Compactification. The result of [17, Theorem 4.1] applies for dynamical systems defined on compact manifolds. Thus, we construct our auxiliary dynamics on a compact manifold containing T0 . To this end, we consider some smooth functions Gi : R≥0 → [0, 1] such that (see [19, page 54]) ρi 0 if ri ∈ 0, 2 Gi (ri ) = 1 if ri ∈ 3ρ4 i , 5ρ4 i (13) 0 if ri ≥ 3ρ2 i . By denoting ρi 3ρi : r ∈ , , i = 1, . . . , N , P := r ∈ RN i >0 2 2 we let M be the compact submanifold

M := T N × P . We define our auxiliary dynamics as a dynamical system on the compact submanifold M as follows:

θ˙i = ωi + fi (θ, r, κ, γ˜ , Φ), θ ∈ T N(14a) r˙i = Gi (ri ) ri (ρ2i − ri2 ) + gi (θ, r, κ, γ˜ , Φ) , r ∈ P . (14b) Note that, by definition, the two dynamics (4) and (14) coincide on (15) M := T N × P , where

P :=

3ρi 5ρi r ∈ RN : r ∈ , , i = 1, . . . , N . i >0 4 4

Hence, (4) has an attractive normally hyperbolic invariant manifold Tp ⊂ M if and only if (14) does. S TEP 2: Invariance. In order to apply [17, Theorem 4.1] to (14) with κ and γ˜ as the perturbation parameters, we also have to show that the compact manifold M is invariant with respect to the flow of (14) independently of (κ, γ˜ ) ∈ RN ×2N . In this case, for all perturbation parameters, the flow associated to (14) defines a diffeomorphism of M, as required by [17, Theorem 4.1]. By construction of the functions Gi in (13), the border of M, i.e. n ρ o 3ρ N N ∂M := T × ∪ T × 2 2 is made of fixed points of the radius dynamics (14b), independently of the value of the parameters κ, γ˜ , Φ. In other words, for all (κ, γ˜ , Φ) ∈ RN ×(4N +3) , the border of M ρ is N given by the union of the two invariant torus T × 2 and T N × 3ρ 2 . This in turn ensures that M is invariant for (14). To see this, suppose M is not invariant. Then, by continuity of the solutions of (14) there must exists some initial conditions (θ0 , r0 ) ∈ M, an instant t¯ ∈ R, some > 0, and a trajectory θ(·, (θ0 , r0 )),r(·, (θ0 , r0 )) of (14), such that θ(t¯, (θ0 , r0 )), r(t¯,(θ0 , r0 )) ∈ ∂M and θ(t¯ + , (θ0 , r0 )), r(t¯ + , (θ0 , r0 )) 6∈ M, which violates the invariance of ∂M. S TEP 3: The nominal invariant manifold and construction of the perturbed one. For κ = γ˜ = 0, the N-torus T0 = T N × {ρ} is attractive normally hyperbolic invariant for (14), since T0 ⊂ M and the same holds for (4). It remains to show that, if |(κ, γ˜ )| is small, then the C 1 -norm1 of the functions

M −→ RN (θ, r) 7−→ f M (θ, r, κ, γ˜ , Φ) and

then (14) still has an attractive normally hyperbolic invariant N-torus Tp ⊂ M, which is |(κ, γ˜ )|-near in the C 1 -norm to T0 . In particular, there exists Ch > 0 such that, if |(κ, γ˜ )| < δh0 , then |r − ρ| ≤ Ch |(κ, γ˜ )|, ∀(θ, r) ∈ Tp , (18) where again Ch is independent of κ, γ˜ , ω , and Φ. The fact that δh0 and Ch depends only on the natural radius ρi , comes from the fact that the linearization (7) of the unperturbed dynamics (5) solely depends on ρi , i = 1, . . . , N . To prove the theorem, it remains to pick |(κ, γ˜ )| sufficiently small that Tp ⊂ (M\∂M). Indeed, the compact manifold M, defined in (15), is the region where the compactified (14) and the original (4) dynamics coincide. Since normal hyperbolicity is a local concept, (14) is normally hyperbolic at a manifold Tp ⊂ (M \ ∂M) if and only if is (4). To this aim, by picking

δh :=

mini=1,...,N {ρi } , 4Ch

and

|(κ, γ˜ )| < δh , it follows from (17), (18), and the definition (15) of M, that for all (θ, r) ∈ Tp , and all i = 1, . . . , N

|ri −ρi | ≤ |r−ρ| ≤ Ch |(κ, γ˜ )| < Ch

ρi mini=1,...,N {ρi } ≤ , 4Ch 4

which ensures that Tp ⊂ M.

IV. E XISTENCE OF PHASE - LOCKED SOLUTIONS IN THE PHASE DYNAMICS

M −→ RN (θ, r) 7−→ G M (r)g M (θ, r, κ, γ˜ , Φ) where G := [Gi ]i=1,...,N , is small in the C 1 -norm as well. ∂f To this aim, note that f and its derivative ∂(θ,r) are linear in the entries (κ, γ˜ )ij , i ∈ {1, . . . , N }, j ∈ {1, . . . , 2N } of the matrix (κ, γ˜ ). Furthermore, the coefficients multiplying (κ, γ˜ )ij , are smooth functions of (θ, r, Φ) and are uniformly bounded on M × RN ×(2N +3) . This also holds for the product Gg and its derivative ∂(Gg) ∂(θ,r) . It thus follows that there exists Cf , Cg > 0 such that, for all κ, γ˜ , ω , and Φ,

¯ kf |M (·, ·, κ, γ˜ , Φ)¯ k1 ≤ Cf |(κ, γ˜ )| (16a) ¯ kG|M (·)g|M (·, ·, κ, γ˜ , Φ)|M ¯ k1 ≤ Cg |(κ, γ˜ )|. (16b) That is, both f and g are |(κ, γ˜ )|-small in the C 1 -norm. Note that the constants Cf , Cg solely depend on the natural radius ρi . 1 The

We can finally apply [17, Theorem 4.1] to conclude the existence of δh0 > 0, independent of κ, γ˜ , ω , and Φ, such that, if |(κ, γ˜ )| ≤ δh0 , (17)

C 1 -norm

C1

of a bounded function with bounded derivatives F : M → RN is defined as ¯ kF ¯ k1 := max{supx∈M |F (x)|, supx∈M |∂F/∂x(x)|}.

Based on the analysis in Section III we formulate the following assumption, which, in view of Theorem 1 and Remark 2, is verified to the first order in the coupling and feedback strengths, provided that the small coupling condition (9) is satisfied. Assumption 1. For all i = 1, . . . , N , the solution of (1) satisfies |zi (t)| = ρi , for all t ≥ 0. This assumption is commonly made in synchronization studies: see e.g. [8], [7], [6], [20], [9]. The analysis in Section III provides a rigorous justification to it. We rely on Assumption 1 and study the existence of phaselocked solutions in (11)-(12), where phase-locked solutions are defined as follows. Definition 2. A solution {θi∗ }i=1,...,N of (11) or (4) is said to be phase-locked if it satisfies

θ˙j∗ (t) − θ˙i∗ (t) = 0,

∀ i, j = 1, . . . , N, ∀t ≥ 0.

(19)

A phase-locked solution is oscillating if, in addition, θ˙i∗ (t) 6= 0, for almost all t ≥ 0 and all i = 1, . . . , N .

Clearly, for a phase-locked solution, phase differences between pairs of oscillators remain constant. We also introduce the matrix Υ ∈ RN ×(4N +3) , which is defined as

Υ := (κ, γ˜ , Φ),

(20)

where Φ is defined in (3). In the following theorem, we show that, for a generic choice of the parameters, no oscillating phase-locked solution exists in the phase dynamics (11). ∗

N

Theorem 2. For all initial conditions θ (0) ∈ T , and for almost all ω ∈ RN and Υ ∈ RN ×(4N +3) , as defined in (20), the dynamics (11)-(12) admits no oscillating phaselocked solution starting at θ∗ (0). The proof, which can be found on the online pre-print [16], is based on two lemmas that generalize [9, Lemmas 1 and 2] to the phase dynamics (11). It mainly relies on arguments from analytic functions and measure theories.

V. E XISTENCE OF PHASE - LOCKED SOLUTIONS IN THE ORIGINAL DYNAMICS

We can readily apply Theorem 2 and the perturbation analysis of Section III to study the existence of oscillating phase-locked solutions in the full dynamics (1). The following corollary generalizes the result in [9] to the complex oscillator dynamics (1), including Special cases 1 and 2 presented in Section II. Corollary 1. For all θ(0) ∈ T N , for almost all ω ◦ ∈ [−1, 1]N , almost all κ◦ , γ˜ ◦ ∈ [−1, 1]N ×N \ {0}, and almost all Φ ∈ RN ×(2N +3) , such that N ◦ X |˜ γij |ρj |κ◦ij |ρj ◦ ◦ +2 > 0, (21) max |ωi |− |κij | + i=1,...,N ρi ρi j=1 there exists ε¯ > 0 such that, for all ε ∈ (0, ε¯], system (4) with natural frequencies ω = ε ω ◦ , coupling matrix κ = ε κ◦ , and feedback gain γ˜ = ε γ˜ ◦ , is normally hyperbolic at an invariant manifold Tp ⊂ T N × RN >0 such that no oscillating phaselocked solution exists starting in (θ(0), r(0)) ∈ Tp . Corollary 1 states that, for almost all natural frequency dispersion (i.e. ω ◦ ) and coupling and feedback topologies (i.e. κ◦ , γ˜ ◦ ), the full dynamics (1) admits no oscillating phaselocked solutions on the attractive normally hyperbolic invariant manifold Tp , provided that the absolute magnitude of the natural frequencies and the coupling and feedback strengths are small (i.e. ε ≤ ε¯) and that the reduced phase dynamics with the same parameters is oscillating, i.e. θ˙ 6≡ 0, as implied by (21). Note that there exist cases in which phase-locking is actually admitted, for instance, the case of zero meanfield feedback. However, such cases form a set of zero Lebesgue measure in the space of coupling and feedback topologies.

VI. C ONCLUSION AND PERSPECTIVE We have shown that a formal derivation of the phase dynamics associated to a network of coupled oscillators permits to derive some bounds on the reliability of the phase dynamics in

reproducing the behaviour of the original dynamics. By generalizing results in [9] and using the aforementioned bounds, we were able to show the generic non-existence of phase-locked solution in a network of Landau-Stuart oscillators under rather general coupling and feedback schemes. Future works, to be presented in a forthcoming journal publication, will aim at generalizing the results in [10] to the generalized oscillator dynamics (1).

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