Nonlinear coupling terms in Hopf normal forms*

Allowing for nonlinear coupling terms (2.30) in the network dynamics (2.28), the corresponding phase interaction function H of the reduced phase model in- cludes also higher harmonics, which may hint at richer collective behavior, see Section 2.1.5. However, only a few nonlinear terms gklmn contribute to the (av- eraged) phase interaction function H. To be more precise, only two terms, g0010 and g0120, are the dominant contributors to the first and second harmonics of H at leading order.

To demonstrate this result, we consider the dynamics (2.29) of two coupled oscillators in Hopf normal form of arbitrary order M ≥ 1. As mentioned in the previous sub-section, without coupling,κ= 0, we find a stable limit-cycle solution

wc_{(t) = R}c_eiθc_(t)

for the two oscillators; for the sake of legibility we will drop the c _{and refer to them as w, w}0_{. The resulting phase model takes then the form}

˙

θ =ω+κH(θ−θ0), where the phase interaction function H can be expanded in Fourier space as in (2.17). In the complex plane, we can compute H as

H(θ−θ0) = hZ(θ)·g(w,w, w¯ 0,w¯0)i , (2.32) wherea·b= (¯ab+a¯b)/2 with a, b∈_C is the complex dot product, the averaging can be expressed in compressed form ashf(θ, θ0)i= ₂1_π R2π

0 f(θ+ϑ, θ

0_{+ϑ)dϑ and}

[15] _{To be precise, (2.31) is the Hopf normal form of the full network (2.28) with S}

N ×S1- equivariance and for exclusively pairwise interactions and large network size N 1, see Section S.5.

the (complex-valued) phase sensitivity function is given by Z(θ) = −c2+i

R e iθ_{; see} also Section 2.2.4.

The assumption of the Hopf normal form implies that f(w,w¯) consists only of the resonant terms |w|n_{w with n= 0,1,2, ..., and that the dynamics ˙w=f(w,w¯)} is rotation invariant. Consequently, both w(θ) and the phase sensitivity function

Z(θ) are of the form w= w(0)eiθ _{and Z(θ) = Z(0)e}iθ_{. For direct linear coupling}

g(w,w, w¯ 0,w¯0) = g0010w0 the interaction function H(θ−θ0) = hZ(θ)·g0010w0(θ0)i thus contains only first harmonics.

Being near a supercritical Hopf bifurcation, the amplitude of the oscillations is R = |w| = O(√µ), where µ denotes the distance to the Hopf bifurcation in parameter space. Introducingε2 =µ, we haveR =O(ε) and Z(θ) = O−1_{(ε). Any} higher order term |w|n_{w in f(w,w¯) presents then corrections of order O}3_{(ε) and}

O1_{(ε) to w(θ) and Z(θ), respectively. In view of the expansion in Fourier space} (2.17), these terms lead to corrections of orderO2_{(ε) ina}

1 andb1, but they do not contribute to higher harmonicsan, bn6= 0 for n≥2.

If we want the phase interaction functionHto contain higher harmonics, we thus have to take higher-order terms gklmn in the coupling functiong(w,w, w¯ 0,w¯0) into account. For simplicity, we consider g(w,w, w¯ 0,w¯0) = wk_w¯l_w0m_w¯0n _{a single mono-} mial with k, l, m, n≥ 0 and gklmn =δklmn. Then we have Z ·g =Ok+m+n+l−1(ε). On the other hand, it is

Z·g(w, w0)∝e−iθ· eiθk e−iθl eiθ0m e−iθ0n= ei(k−l−1)θei(m−n)θ0 .

The latter term contributes to the amplitudes aj and bj of the j-th harmonic (j >0) when it is a function of only±j(θ−θ0). This means that the set (k, l, m, n) has to fulfill k−l−1 = ±j and m−n =∓j. In particular, the term ¯wj−1w0j =

O2j−1_{(ε) contributes significantly to a}

j and bj. Therefore, the amplitudes of the

j-th harmonic are of orderO(Z·g(w, w0)) =O(2j−1)−1_{(ε), that is,}

aj, bj =O2(j−1)(ε) . (2.33) Note that the reasoning above is in line with the coefficient a0 of the zeroth har- monic,j = 0, whose major contributions come from the monomialg(w,w, w¯ 0,w¯0) =

wand result into a constant increase or decrease of the natural frequency depend- ing on the sign ofa0. Forj > 0, the term wj+1w¯0j =O2j+1_{(ε) gives contributions} of order O2j_{(ε) to the j-th harmonic. However, these contributions present only} minor corrections as they are smaller than aj, bj of two orders of magnitude, and can therefore be neglected. Following this argumentation, we consider coupling terms of order 3, which yield contributions of order O2_{(ε) to the phase dynam-} ics. The terms w2_w¯0_,|w|2_w0_{, and |w}0_|2_w0 _{contribute to the first harmonics a}

However, their values differ at one order of magnitude from a1, b1, so that their contributions can be neglected. Likewise, |w|2_{w and w|w}0_|2 _{contribute negligibly} to the zeroth harmonic, namely by less than two orders of magnitude. The only cubic resonant term that affects the phase dynamics is ¯ww02_{, which contributes to} the second harmonicsa2, b2 at the same order of magnitudeO2(ε).

Moreover, we can show that no monomial in g(w,w, w¯ 0,w¯0) of even order will contribute to H. Indeed, employing the inner product in complex form (2.32) for

g(w, w0)∝exp i(k−l)θ+i(m−n)θ0, we have H(θ−θ0)∝ 1 2π Z 2π 0 αklmne−i(θ+ϑ)ei((k−l)θ+(m−n)θ 0_{+(k−l+m−n)ϑ)} + βklmnei(θ+ϑ)e−i((k−l)θ+(m−n)θ 0_{+(k−l+m−n)ϑ)} dϑ ∝ 1 2π Z 2π 0 αklmnei((k−l+m−n−1)ϑ)+βklmne−i((k−l+m−n−1)ϑ)dϑ , (2.34)

whereαklmn, βklmn ∈Care constants. Due to the inherent averaging in (2.34) and as exp(inϑ) is 2π-periodic, H(θ−θ0) will vanish if (k−l+m−n) is even. This means that only monomials of odd order will contribute to the phase interaction functionH.

2.2.2 Identifying the Hopf normal form

The starting point for all the Hopf normal form reductions is the oscillator network (2.1), where each node xk ∈ Rn is close to a supercritical Hopf bifurcation. More

specifically and given the main assumptions on weak coupling, (nearly) identical oscillators and the pairwise coupling structure, we reconsider (2.15),

˙ xk =f(xk;µ) + κ N N X j=1 Ckjg(xk,xj)

with vector functions f = (f1, . . . , fn) : Rn → Rn and g = (g1, . . . , gn) : RN×n →

Rn and small coupling strength |κ| 1. Without coupling, κ = 0, each node has

a stable fixed point solution ˜xk. Then f(x= ˜xk;µ) =0 for all k = 1, . . . , N and small values of the bifurcation parameterµ∈_R.

The essential assumption for the supercritical Hopf bifurcation is that there is a parameter value µ=µk such that each fixed point ˜xk undergoes a supercritical Hopf bifurcation in the absence of coupling: For µ < µk, the dynamics (2.15) has a stable fixed point, which loses stability at µ = µk, and stable oscillations emerge forµ > µk. Without loss of generality we translate the fixed point to the origin, i.e. ˜xk = 0 for all k, and assume that µk = 0. Then for µ > 0, each node exhibits stable limit cycle oscillations with natural frequency ωk 6= 0 and

amplitudeRk =O(ε) whereε =

√

µ.

In the following sub-sections we will illustrate three different ways how to reduce (2.15) to the Hopf normal form network (2.31),

˙ wk=αwk−β|wk|2wk+ κ N N X j=1 Ckj γwj +δw¯kwj2 .

Note that all four parametersα, β, γ, δ ∈_Cwill depend on the functionsf andgas well as on the bifurcation parameterµ. The different Hopf normal form reductions to be presented below vary not only in their methodical approach, but also in their accuracy. To be precise, while, e.g., the reductive perturbation approach in Sec- tion 2.2.2.1 discards any higher order dependence on µ, the nonlinear transforms approach,Section 2.2.2.2, respects thisµ-dependence at all times. The differences between the reduction techniques[16] may be negligible for small-amplitude oscil- lations, that is, close to the Hopf bifurcation point with 0 < µ µ0 1. But the resulting normal form techniques will diverge drastically when the amplitudes of oscillation become larger. These differences eventually become evident in the reduced phase dynamics and may cause qualitatively different collective dynamics. In anticipation of Section 2.2.3, the subsequent phase reductions of the reduced Hopf normal form dynamics (2.31) are all identical within the theory of weakly coupled oscillators. Hence, possible inconsistencies between the resulting phase models are solely due to the different levels of accuracy of the normal form reduc- tions. For this reason, we will refer to the analytic phase reduction techniques as their underlying normal form reductions,reductive perturbation reduction and the nonlinear transform reduction, respectively.