Stability and Bifurcation of Mixing in the Kuramoto Model with Inertia

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System of ordinary differential equations (1.1) may be viewed as an extension of the classical Kuramoto model (KM), which is used to study collective behavior in large coupled systems [8]. In this context, it is called the KM with inertia, because the second order terms in (1.1) introduce inertial effects into the system’s dynamics. On the other hand, system (1.1) has a clear mechanical interpretation. For instance, it may be viewed as a system of coupled pendula or coupled power generators, as it is perhaps best known for [11,15]. Like its classical counterpart, the KM with inertia features transition to synchronization. For small positive values of the coupling strength, the system exhibits irregular dynamics, called mixing. Upon increasing the coupling strength K, mixing loses stability and a gradual (soft) transition to synchronization takes place. The onset of synchronization in the classical KM with all-to–all coupling was analyzed in [1,5] and in the model on graphs in [3,4]. The analysis of synchronization in the KM with inertia involves new technical challenges related to the dimensionality of the phase space, which make it an interesting stability problem. In addition, synchronization in the KM with inertia has important applications. First, synchronization provides a more efficient regime of operation of power networks. Therefore, it is important to know what factors determine stability of synchrony in the second order model on graphs. Previous studies of the onset of synchronization in the KM with inertia [13, 14] relied on asymptotic analysis and numerical simulations. In this work, for the first time we perform a rigorous linear stability analysis of mixing and identify the instability of mixing. Furthermore, we perform a formal center manifold reduction and identify the bifurcation, marking the onset of synchronization, as a pitchfork bifurcation. We verified these results numerically for the KM on various graphs including Erd˝os-Rényi (ER) and small-world (SW) graphs.

To proceed, we rewrite (1.1) as a system of coupled second order ordinary differential equations and add a separate equation for ωi: 9 θi “ ψi ` ωi, n 9 2K n ψi “´ψi ` a sinpθj ´ θiq, (1.2) n ij j“1 ÿ ω9 i “ 0, i P rns.

To study the dynamics of (1.2) for large n, we employ the following Vlasov equation

Btf ` Bθ ppψ ` ωqfq ` Bψ pp´ψ ` Nrfsqfq“ 0, f “ fpt,θ,ψ,ω,xq, (1.3) where K Nrfspt,θ,xq“ eíθhpt,xq´ eiθhpt,xq . i The justification of the Vlasov equation as a mean´ field limit for the original¯ KM on graphs can be found in [3, 7]. It extends verbatim to cover the model at hand.

The following local order parameter plays a key role in the analysis of synchronization in the KM model:

hpt,xq“ W px,yqeiθfpt,θ,ψ,ω,yqdθdψdωdy, (1.4) T R2 ż ˆ ˆI 2 where I “ r0, 1s and T “ R{2πZ. W is a square integrable function on I2, which describes the limit of the graph sequence tΓnu. W is called a graphon. For the details on graph limits and their applications to the continuum description of the dynamical networks, we refer the interested reader to [3, 10].

A weak (distribution–valued) solution fpt,θ,ψ,ω,xq of the initial value problem for the Vlasov equation (1.3) yields the probability distribution of particles in the phase space T ˆ R2 for each pt,xq P R` ˆ I (cf. [6, 7]). By mixing we call the following steady state solution of (1.3) gpωq f “ δpψq, (1.5) mix 2π where δ stands for Dirac’s delta function and gpωq is the probability density function characterizing the distribution of intrinsic frequencies. fmix corresponds to the stationary regime when the values of θi are distributed uniformly over T. Stability of mixing is the main theme of this paper.

In the next section, we recast the stability problem in the Fourier space and derive the linearized problem. The next step is to characterize the spectrum of the linearized operator. This is done in Section 3. We start by describing the setting for the spectral problem. To this end, we define function spaces and operators involved in the linear stability problem. After that we derive a transcendental equation for the eigenvalues of the linearized problem. The analysis of the eigenvalue problem shows that the bifurcating eigenvalue fails to cross the imaginary axis filled by the residual spectrum. To overcome this difficulty, following the analysis of the classical KM [1, 4], we use the weak formulation for the eigenvalue problem for the linearized operator based on a carefully selected Gelfand triplet and the analytic continuation of the resolvent operator past the imaginary axis. In this setting the generalized eigenvalues are defined as singular points of the generalized resolvent. The corresponding eigenfunction is used in the center manifold reduction of the system’s dynamics. This concludes a rigorous linear stability analysis of mixing and sets the scene for the bifurcation analysis. In Section 3.5, we prove that mixing is asymptotically stable in the subcritical regime. In Sec- tion 4, we show that mixing undergoes a pitchfork bifurcation marking the onset of synchronization. The center manifold reduction is done under the assumption of the existence of the center manifold. However, the formal center manifold reduction alone is a formidable problem, as the analysis in this section shows. The proof of existence of the center manifold requires new additional techniques and is beyond the scope of this paper. The analytical results are illustrated with numerical examples of the bifurcation in the KM on ER and SW graphs in Section 5.

2 The Fourier transform

As a probability density function, g is a nonnegative integrable function such that

gpsqds “ 1. (2.1) R ż Since ω does not change in time, f satisﬁes the following constraint

gpωq“ fpt,θ,ψ,ω,xqdθdψ, @pt,xq P R` ˆ I. (2.2) T R ż ˆ In addition, throughout this paper, we will assume the following.

3 iηω Assumption 2.1. Let g be a real analytic function. In addition, let the Fourier transform gˆpηq :“ R e gpωqdω be continuous on R and ş lim |gˆpηq|eaη “ 0 (2.3) ηÑ8 for some 0 ă a ă 1.

Remark 2.2. By the Paley-Wiener theorem, (2.3) implies that gpωq has an analytic continuation to the region 0 ă Impzqă a.

In preparation for the stability analysis of mixing, we rewrite (1.3) using Fourier variables:

ipjθ`ζψ`ηωq ujpt,ζ,η,xq“ e fpt,θ,ψ,ω,xqdθdψdω, j P Z. (2.4) R2 T ż ˆ

The Vlasov PDE is then rewritten as a system of differential equations for the Fourier coefﬁcients uj, j P Z:

ipjθ`ζψ`ηωq B Btuj “´ e pψ ` ωqfdθdψdω Bθ ż B ´ eipjθ`ζψ`ηωq p´ψ ` Nrfsqfdθdψdω Bψ ż “ ij eipjθ`ζψ`ηωqpψ ` ωqfdθdψdω (2.5) ż K ` iζ eipjθ`ζψ`ηωq ´ψ ` pe´iθh ´ eiθhq fdθdψdω i ż ˆ ˙ Buj Buj “ pj ´ ζq ` j ` Kζphpt,xqu 1 ´ hpt,xqu 1q, j P Z, Bζ Bη j´ j` where hpt,xq“ W px,yqu1pt, 0, 0,yqdy. (2.6) żI is the local order parameter. In addition, we have the following constraints

u0pt, 0, 0,xq “ 1, (2.7)

u´jpt, ´ζ, ´η,xq “ ujpt,ζ,ω,xq, j P N. (2.8)

Here, (2.7) follows from (2.2) and (2.8) follows from the fact that f is real. Thus, it is sufﬁcient to restrict to j P N t0u in (2.5).

By changingŤ from ζ to ξ given by the following relations:

ζ ´ j “´e´ξj , ζ ´ j ă 0, (2.9) ζ ´ j “ e´ξj , ζ ´ j ě 0, " and setting ´ξ ujpt, j ´ e j ,η,xq, ζ ´ j ă 0, vjpt,ξj,η,xq :“ (2.10) ´ξ # ujpt, j ` e j ,η,xq, ζ ´ j ě 0,

4 we obtain

´ξj Btvj “ Bξj vj ` jBηvj ` Kpj ´ e q hpt,xqvj´1 ´ hpt,xqvj`1 , ´ ¯ hpt,xq“ W px,yqv1pt, 0, 0,yqdy, żI subject to the constraint v0pt, 8,η,xq “ gˆpηq. For j ě 0, we adopt the ﬁrst line of (2.9) because in the deﬁnition of the local order parameter, we need u1pt, 0, 0,xq, for which ζ ´ j “ ´1 ă 0. By the same reason, we use the second line for j ď´1.

The steady state of the Vlasov equation, fmix, in the Fourier space has the following form

v0 “ gˆpηq, vj “ 0, j P N. (2.11)

To investigate the stability of (2.11), let w0 “ v0 ´ gˆpηq and wj “ vj for j ‰ 0. Then we obtain the system

´ξ1 Btw1 “ Bξ1 w1 ` Bηw1 ` Kp1 ´ e q hpt,xqgˆpηq` hpt,xqw0 ´ hpt,xqw2 , $ ´ ¯ dwj Bwj Bwj ´ξ ’ j K j e j h t,x w 1 h t,x w 1 , j 0 & j 1 . ’ “ ` ` p ´ q p q j´ ´ p q j` p ě q p ‰ q (2.12) ’ dt Bξj Bη &’ ´ ¯ hpt,xq“ W px,yqw1pt, 0, 0,yqdy, ’ I ’ ż ’ and %w0pt, 8,η,xq “ 0. Our goal is to investigate the stability and bifurcations of the steady state (mixing) wj “ 0, j P Z of this system.

The linearized system has the following form

dw1 “ L1rw1s` KBrw1s“: Srw1s (2.13) dt dw j “ L rw s, pj ě 0q&pj ‰ 1q, (2.14) dt j j where

Ljrφspξ,η,xq “ pBξ ` jBηq φpξ,η,xq, j P Z (2.15) Brφspξ,η,xq “ p1 ´ e´ξqgˆpηqW rφp0, 0, ¨qspxq, (2.16) and W rfspxq“ W px,yqfpyqdy. (2.17) R ż

5 3 The spectral analysis

3.1 The setting

To deﬁne the operators used in the linearized system (2.13) and (2.14) formally, we introduce the following Banach spaces. For α P r0, 1q, let

` αη ´ αη ξ β1 pηq“ maxt1, e u, β1 pηq“ mint1, e u, and β2pξq“ minte , 1u, (3.1) and deﬁne ˘ R ˘ Xα “ tφ : continuous on , }φ}X ˘ “ sup β1 pηq|φpηq| ă 8u, α η ˘ 2 ˘ Y “ tφ : continuous on R , }φ} ˘ “ sup β pηqβ2pξq|φpξ, ηq| ă 8u, (3.2) α Yα 1 ξ,η ˘ 2 ˘ Hα “ L pI; Yα q.

˘ The norms on Hα are deﬁned by 2 2 ˘ }φ} ˘ “ sup β1 pηqβ2pξq|φpξ,η,xq| dx. (3.3) Hα żI ˜ ξ,η ¸ They form a Gelfand triplet ` ` ´ ´ Hα Ă H0 “ H0 Ă Hα , which will be used in Section 3.4. Below, the function spaces in (3.2) are used for α P t0, au, where a is the ` same as in (2.3). Recall that gˆ P Xa for certain 0 ă a ă 1 (cf. Assumption 2.1). Now we can formally discuss the operators involved in the linearized problem (2.13), (2.14). First, we note that W deﬁned in (2.17) can be viewed as the Fredholm integral operator on L2pIq. It is a compact self-adjoint operator. Therefore, the eigenvalues of W are real with the only accumulation point at 0. We denote the set of eigenvalues of W by σppW q.

` Next, we turn to operators Lj and B. It follows that they are densely deﬁned on H0 . In addition, B is a bounded operator as shown in the following lemma.

` Lemma 3.1. B is a bounded operator on H0 .

6 ˘ Proof. When α “ 0, β1 pηq“ 1. By the Cauchy–Schwarz inequality, we have

2 B 2 ´ξ || rφs||H` “ sup β2pξqp1 ´ e q|gˆpηq| ¨ W px,yqφp0, 0,yqdy dx 0 I ˜ ξ,η I ¸ ż ˇż ˇ 2 2ˇ 2 ´ξ 2ˇ 2 ď |W px,yq| dxdy ¨ sup |gˆpηq| ˇ ¨ sup β2pξq p1 ´ e q ˇ |φp0, 0,xq| dx 2 η żI ξ,η żI W 2 2 2 ´ξ 2 2 “ || ||L2 ¨ sup |gˆpηq| ¨ sup β2pξq p1 ´ e q pβ2p0q|φp0, 0,xq|q dx η ξ,η żI W 2 2 2 ´ξ 2 2 ď || ||L2 ¨ sup |gˆpηq| ¨ sup β2pξq p1 ´ e q pβ2pξq|φpξ,η,xq|q dx η ξ,η żI W 2 2 2 ´ξ 2 2 “ || ||L2 ¨ sup |gˆpηq| ¨ sup β2pξq p1 ´ e q ||φ||H` . η ξ 0

2 ´ξ 2 For both cases ξ ě 0 and ξ ď 0, we have supξ β2pξq p1 ´ e q “ 1, which proves the lemma.

` Since Lj is closed and B is bounded, S “ L1 ` KB is also a closed operator on H0 .

3.2 The spectrum of S

In this section, we establish several basic facts about the spectra of Lj and S. In particular, we describe the residual spectrum and derive a transcendental equation for the eigenvalues of S. Throughout this section, ` we view Lj and S as operators densely deﬁned on H0 (cf. (3.2)-(3.3)).

Let j P N be ﬁxed and consider pλ ´ Ljqu “ v. (3.4) Applying the Fourier transform to both sides of (3.4) and using (2.15), we have

pλ ´ iζ ´ ijωq Frus“ Frvs, where 1 ´ipξζ`ηωq Frvspζ,ω,xq“ 2 e vpξ,η,xqdξdη. (3.5) p2πq R2 ż Thus, ipξζ`ηωq ´1 ´1 e u “ pλ ´ Ljq v “ F Frus“ Frvspζ,ω,xqdζdω. (3.6) R2 λ ´ iζ ´ ijω ż

Next we use (3.6) to locate the residual spectrum of Lj.

Lemma 3.2. For arbitrary j P N1, the residual spectrum of Lj is given by

σpLjq “ tz P C : ´1 ď ℜz ď 0u.

7 Proof. The proof is based on (3.6) and the following identity for ℜ λ ‰ 0 8 e´pλíζíjωqtdt, ℜ λ ą 0, 1 0 “ $ ż 0 (3.7) λ ´ iζ ´ ijω ’ ´pλíζíjωqt &’ ´ e dt, ℜ λ ă 0, ż´8 ’ By plugging (3.7) into (3.6), we have % 8 e´λtvpξ ` t, η ` jt,xqdt, ℜ λ ą 0, ´1 0 pλ ´ Ljq rvspξ,η,xq“ $ ż 0 (3.8) ’ ´λt &’ ´ e vpξ ` t, η ` jt,xqdt ℜ λ ă 0. ż´8 ’` ` The right–hand side of (3.8) belongs to H%’0 for any v P H0 only if ℜ λ ă ´1 or ℜ λ ą 0. For ´1 ď ` ℜ λ ď 0, the set of v such that the right-hand side exists is not dense in H0 .The statement of the lemma follows.

Next, we turn to the eigenvalue problem for S (cf. (2.13))

λv “ pL1 ` KBqv. (3.9) Lemma 3.3. Let µ be a nonzero eigenvalue of W and let V P L2pIq be a corresponding eigenfunction. Deﬁne 1 e´ξ Dpλ; ξ, ηq“ ´ eiηωgpωqdω, (3.10) R λ ´ iω λ ` 1 ´ iω ż ˆ ˙ and 1 1 Dpλq :“ Dpλ; 0, 0q“ ´ gpωqdω. (3.11) R λ ´ iω λ ` 1 ´ iω ż ˆ ˙ Then the root λ “ λpµq of the following equation 1 Dpλq“ , (3.12) Kµ ` not belonging to BσpL1q “ tz P C : ℜ z “´1 or ℜ z “ 0u, is an eigenvalue of S on H0 . For each such root λ “ λpµq the corresponding eigenfunction is given by vpξ,η,xq“ Dpλ; ξ, ηqV pxq. (3.13)

Proof. From (3.9) we obtain ´1 ´ξ v “ Kpλ ´ L1q p1 ´ e qgˆpηqW rvp0, 0, ¨qs. (3.14) (3.6) gives ipξζ`ηωq ´1 ´ξ e ´ξ pλ ´ L1q p1 ´ e qgˆpηq“ Frp1 ´ e qgˆpηqsdζdω R2 λ ´ ipζ ` ωq ż eipξζ`ηωq (3.15) “ pδpζq´ δpζ ´ iqqgpωqdζdω R2 λ ´ ipζ ` ωq ż “ Dpλ; ξ, ηq,

8 where we used (3.10) in the last line. The combination of (3.14) and (3.15) yields

v “ KDpλ; ξ, ηqW rvp0, 0, ¨qs. (3.16)

By plugging ξ “ η “ 0 in (3.16), we have

vp0, 0,xq“ KDpλqW rvp0, 0, ¨qspxq. (3.17)

If λ “ λpµq solves (3.12), then (3.17) holds for

vp0, 0,xq“ V pxq. (3.18)

This shows that the set of roots of (3.12) for µ P σppW q coincides with the set of eigenvalues of S. The corresponding eigenfunctions are found from (3.16):

v “ KµDpλpµq; ξ, ηqV, (3.19)

` which coincides with (3.13) up to a multiplicative constant. It remains to show that v P H0 when λ R BσpL1q. To this end, note that (3.7) is applicable to (3.10) when ℜ λ ‰ 0, 1 as

8 ´λt ´pξ`tq 0 e p1 ´ e qgˆpη ` tqdt, if ℜ λ ą 0, Dpλ; ξ, ηq“ ´8e´λtgˆpη ` tqdt ´ 8e´λte´pξ`tqgˆpη ` tqdt, if ´ 1 ă ℜ λ ă 0, $ ş0 0 ´8e´λtp1 ´ e´pξ`tqqgˆpη ` tqdt, if ℜ λ ă´1. & ş0 ş ` ` ` This shows that Dpλ; ξ,% ηqş P Y0 and v P H0 if gˆ P X0 and λ R BσpL1q.

3.3 The eigenvalue equation

In this subsection, we study equation (3.12), whose roots yield the eigenvalues of S.

Note that Dpλ; ξ, ηq is a holomorphic function on H0 :“ tλ P C : ℜ λ ą 0u. Using Assumption 2.1, Dpλ; ¨q can be extended analytically to Ha :“ tλ P C : ℜ λ ą áu, 0 ă a ă 1. Indeed, by rewriting the integrand in the definition of Dpλ; ξ, ηq 8 Dpλ; ξ, ηq“ e´λtp1 ´ e´pξ`tqqgˆpη ` tqdt 0 ż 8 “ e´pλàqteáηp1 ´ e´pξ`tqqeapη`tqgˆpη ` tqdt, 0 ż H ` one can see that this integral exists and defines an analytic function on a, because gˆ P Xa (cf. Assump- tion 2.1). Moreover, by applying Sokhotski–Plemelj formula [12] to (3.10), the analytic continuation is also written by 1 e´ξ ´ eiηωgpωqdω p“ Dpλ; ξ, ηqq, ℜλ ą 0, R λ ´ iω λ ` 1 ´ iω $ ż ˆ 1 ˙ e´ξ Dpλ; ξ, ηq“ ’ lim ´ eiηωgpωqdω, λ “ iy, (3.20) ’ xÑ0` R x ` ipy ´ ωq 1 ` ipy ´ ωq ’ & ż 1ˆ e´ξ ˙ ´ eiηωgpωqdω ` 2πeηλgpíλq, á ă ℜλ ă 0. ’ R λ ´ iω λ ` 1 ´ iω ’ ż ˆ ˙ ’ %’ 9 ´ ` From these expressions, it turns out that for á ă ℜλ ă 0, Dpλ; ¨q P Ya , while Dpλ; ¨q P Ya for ℜλ ą 0.

Denote Dpλq“ Dpλ; 0, 0q. Dpλq provides an analytic continuation of Dpλq to Ha. Deﬁnition 3.4. Let µ be a nonzero eigenvalue of W and let V denote the corresponding eigenfunction. Suppose λ P Ha is a root of the following equation 1 Dpλq“ . (3.21) Kµ Then λ is called a generalized eigenvalue of S. The corresponding generalized eigenfunction is given by

´ v “ Dpλ; ξ,µqV pxq P Ha .

Remark 3.5. Generalized eigenvalues coincide with regular eigenvalues on H0. The reason why the roots of (3.21) are called generalized eigenvalues of S will become clear in Section 3.4. See [2] for the generalized spectral theory.

The following theorem describes solutions of (3.21) for even unimodal g.

Theorem 3.6. Suppose g is an even unimodal probability density function and denote

´1 Kc “ pµmaxg0q , (3.22) where gpsq g0 :“ πgp0q´ 2 ds (3.23) R 1 ` s ż and µmax is the largest (positive) eigenvalue of W .

Under Assumption 2.1, the following holds.

1. For K P r0, Kcq there are no generalized eigenvalues with with positive real part.

2. For sufﬁciently small ǫ ą 0, there is a real generalized eigenvalue λ “ λpKq for K P pKc ´ ǫ, 8q. In addition, 1 λpKcq“ 0 and λ pKcqą 0.

´1 2 ´1 Remark 3.7. Using the identity π “ Rp1 ` s q ds, one can see that g0 deﬁned in (3.23) is positive:

ş gp0q´ gpsq g0 “ 2 ds ą 0, (3.24) R 1 ` s ż Remark 3.8. The positive generalized eigenvalue λpKq, K ą Kc, identiﬁed in the theorem is an eigenvalue of S. At K “ Kc`0 it hits the residual spectrum of S. There are no eigenvalues of S for K ď Kc. However, as a generalized eigenvalue, λpKq is well deﬁned for Kc ´ ǫ ă K ď Kc. It crosses the imaginary axis transversally at K “ Kc.

The proof of the theorem relies on the following observations.

10 Im z 2 3

1 Re z -10 1.5 4

0 -2 0 2 a b -3

Figure 1: A unimodal probability density function g (a) and the corresponding critical curve (b).

Lemma 3.9. For Dpλq, we have gpsq gpsq lim Dpx ` iyq “ πgpyq´ i pv ds ´ ds, (3.25) xÑ0` R y ´ s R 1 ` ipy ´ sq ż ż lim lim Dpx ` iyq “ 0, (3.26) yÑ˘8 xÑ0` D1p0q ă 0. (3.27) In addition, C “ DpiRq is a bounded closed curve which intersects positive real semiaxis at a unique point pg0, 0q (see Figure 1(b)).

We postpone the proof of Lemma 3.9 until after the proof of the theorem.

Proof of Theorem 3.6. Let µ ą 0 be an eigenvalue of W . Note that C “ DpiRq is a bounded closed curve intersecting positive real semiaxis at a single point pg0, 0q by Lemma 3.9. Since Dpλq is holomorphic in H0, by the Argument Principle, the eigenequation 1 Dpλq “ pKµq´ (3.28) 1 has a root in H0 if the winding number of D about pKµq´ is positive. Therefore, (3.28) has no roots in H0 ´1 ´1 for 0 ă Kµ ă g0 for any µ P σppW qzt0u, i.e., for K P r0, Kcq with Kc “ pµmaxg0q . Since gpωq is an even function, DpiRq intersects positive real semiaxis when λ “ 0. This implies that g0 “ λp0q, which is obtained from (3.25) as gpsq 1 ` is lim lim Dpx ` iyq “ πgp0q` i pv ds ´ 2 gpsqds yÑ0 xÑ0` R s R 1 ` s ż ż 1 “ πgp0q´ 2 gpsqds. R 1 ` s ż This proves the ﬁrst statement of the theorem with formulae (3.22) and (3.23).

Since D1p0q ‰ 0 by Lemma 3.9 and D is holomorphic at 0, it is locally invertible, and the inverse is a holomorphic function. By differentiating both sides of (3.28) with respect to K, we have ´1 1 0 λ pKcq“ 2 1 ą . µmaxKc D p0q

Since µmax is an isolated eigenvalue of W , there are no other roots of (3.28) for K P pKc ´ ǫ, Kc ` ǫq.

11 Proof of Lemma 3.9. Equation (3.25) follows from the Sokhotski–Plemelj formula [12]. Equation (3.26) follows from (3.25), because g P L1pRq. To show (3.27), note ﬁrst

1 1 1 D pλq“´ 2 ´ 2 gpsqds R pλ ´ isq pλ ` 1 ´ isq ż ˆ ˙ (3.29) 1 1 “´i ´ g1psqds. R λ ´ is λ ` 1 ´ is ż ˆ ˙ Using the Sokhotski–Plemelj formula again, from (3.29) we get g1psq 1 lim D1pλq“ pv ` i g1psqds ´ πg1p0q . (3.30) λÑ0` R s R 1 ´ is ż ˆż ˙ Since g1 is an odd function, we have 8 1 1 1 1 g psq´ g p´sq sg psq lim D pλq“ lim ds ´ 2 ds λÑ0` εÑ0` s R 1 ` s żε ż 8 g1psq 8 sg1psq “ 2 lim ds ´ 2 2 ds εÑ0` s 0 1 ` s żε ż 8 g1psq “ 2 lim 2 ds ă 0, εÑ0` sp1 ` s q żε where we used that g1psqď 0 for s ą 0 and is not everywhere zero on R` to obtain the last inequality.

Finally, (3.26) implies that C is a bounded closed curve. From (3.25), for Gpyq “ limxÑ0` Dpx ` iyq we have gpsq ℜGpyq “ πgpyq´ 2 ds, (3.31) R 1 ` py ´ sq ż py ´ sqgpsq gpsq ℑGpyq “ 2 ds ´ pv ds. (3.32) R 1 ` py ´ sq R y ´ s ż ż Using even symmetry of g, from (3.31) and (3.32) it follows that pg0, 0q is a point of intersection of C with the positive semiaxis (Figure 1(b)). Even symmetry and unimodality of g provides the uniqueness: y ´ ω y ´ ω ´ℑDpx ` iyq“ 2 2 ´ 2 2 gpωqdω R x ` py ´ ωq px ` 1q ` py ´ ωq ż ˆ ˙ ω ω “´ 2 2 ´ 2 2 gpy ` ωqdω R x ` ω px ` 1q ` ω ż ˆ ˙ 0 8 ω ω “´ ` 2 2 ´ 2 2 gpy ` ωqdω 0 x ` ω px ` 1q ` ω "ż´8 ż *ˆ ˙ (3.33) 8 ω ω “´ 2 2 ´ 2 2 gpy ` ωqdω 0 x ` ω px ` 1q ` ω ż ˆ ˙ 8 ω ω ` 2 2 ´ 2 2 gpy ´ ωqdω 0 x ` ω px ` 1q ` ω ż ˆ ˙ 8 ω px ` 1q2 ´ x2 “´ 2 2 2 2 pgpy ` ωq´ gpy ´ ωqq dω. 0 px ` ω q ppx ` 1q ` ω q ż ` ˘ The expression in the last line of (3.33) is 0 only when y “ 0.

12 3.4 The resolvent and the Riesz projector

In this section, we compute the resolvent operator

´1 Rλ “ pλ ´ Sq and the Riesz projector for S.

` Lemma 3.10. For φ P H0 ,

´1 ´1 ´1 Rλφ “ pλ ´ L1q φ ` KDpλ; ξ, ηq pI ´ KDpλqW q W pλ ´ L1q φ p0, 0, ¨q . (3.34) ”´ ¯ ı

Proof. From the deﬁnitions of Rλ and S we have

` pλ ´ L1 ´ KBq Rλφ “ φ @φ P H0 . (3.35)

We rewrite (3.35) as ´1 pλ ´ L1q I ´ K pλ ´ L1q B Rλφ “ φ. Thus, (3.15) gives ´ ¯

´1 ´1 Rλφ “ pλ ´ L1q φ ` K pλ ´ L1q BRλφ ´1 “ pλ ´ L1q φ ` KDpλ; ξ, ηqW rpRλφqp0, 0, ¨qs . (3.36)

By setting ξ “ η “ 0 and applying W to both sides of (3.36), after some algebra we obtain

´1 ´1 W rpRλφqp0, 0, ¨qs “ pI ´ KDpλqW q W pλ ´ L1q φ p0, 0, ¨q . (3.37) ”´ ¯ ı By plugging (3.37) into (3.36), we obtain (3.34).

` From (3.34) one can see that the spectrum of S, as an operator on H0 , is the union of the residual ´1 spectrum of L1 and point spectrum of S coming from the factor pI ´ KDpλqW q ;

σpSq“ σpL1q σppSq. ď Although tz P C : ´1 ď ℜz ď 0u is a residual spectrum of L1, as shown in Section 3.3, Dpλ; ξ, ηq and Dpλq have the analytic continuations from H0 to Ha denoted by Dpλ; ξ, ηq and Dpλq, respectively. Recall ´ ` L ´1 that Dpλ; ξ, ηq P Ya when gˆ P Xa . In a similar manner, we can verify that pλ ´ 1q has an analytic H H ` ´ continuation from 0 to a as an operator from Ha to Ha . Using them, we deﬁne the analytic continuation of Rλ from H0 to Ha by

´1 ´1 ´1 Rλφ “ pλ ´ L1q φ ` KDpλ; ξ, ηq pI ´ KDpλqW q W pλ ´ L1q φ p0, 0, ¨q . (3.38) ”´ ¯ ı ` ´ This is an operator from Ha into Ha called the generalized resolvent. Note that the generalized eigenvalues ˘ ˘ in Deﬁnition 3.4 are the poles of Rλ. Recall the deﬁnitions of the Banach spaces Ha and H0 (cf. (3.2)) and note that they form Gelfand triplet: ` ` ´ ´ Ha Ă H0 “ H0 Ă Ha .

13 ` ´ For the analytic continuation, the domain of Rλ is restricted to Ha and the range is extended to Ha .

` ´ The corresponding generalized Riesz projection Pλ : Ha Ñ Ha is given by 1 P φ “ R φdz, (3.39) λ 2πi z ¿c where c is a positively oriented closed curve, which encircles an eigenvalue λ but no other generalized eigenvalues. Note that the generalized eigenfunction corresponding to λ in Deﬁnition 3.4 is an element of the image Pλ. We conclude this section with the computation of the generalized Riesz projection for the bifurcating eigenvalue λ “ 0: 1 P0rφs“ R φdz, (3.40) 2πi z ¿c where c is a positively oriented contour around the origin, which does not contain any other generalized eigenvalues of S.

Here and in the remainder of this paper, we suppress the subscript of P0 since from now on we will only be interested in the bifurcating eigenvalue. Further, suppose that µmax, the largest positive eigenvalue of W , is simple. The Riesz projection for W corresponding to µmax is denoted by

1 1 P rΦs“ pz ´ W q´ Φdz, (3.41) 2πi C¿ where C is a positively oriented contour around µmax, which does not contain any other eigenvalues of W . Lemma 3.11. Prφspξ,η,xq “ ρ1Dp0; ξ, ηqP rΦsp0, 0,xq, (3.42) ´1 where Φ :“ limλÑ0` pλ ´ L1q φ and ρ1 ą 0.

Proof. First, note

´1 Rλrφspξ,η,xq“ Φλpξ,η,xq` KDpλ; ξ, ηq pI ´ KDpλqW q W rΦλp0, 0, ¨qs pxq, (3.43) where ´1 Φλ :“ pλ ´ L1q φ. (3.44)

Using (3.43), we have

´1 Rzrφspξ,η,xqdz “ K Dpz; ξ, ηq pI ´ KDpzqW q W rΦzp0, 0, ¨qspxqdz, (3.45) ¿c ¿c D1 where we used c Φzpξ,η,xqdz “ 0, because Φzpξ,η,xq is regular inside c. By continuity pzq ‰ 0 on c provided c is sufﬁciently small. Change variable in (3.45) to z :“ zpαq such that ű 1 α “ pKDpzqq´ . (3.46)

14 Note that ´1 µmax “ pKcDp0qq (3.47) holds (3.21). We have

D pzpαq; ξ, ηq ´1 ´1 Rzrφspξ,η,xqdz “´ W pI ´ α W q rΦ sp0, 0,xqdα α2D1 pzpαqq zpαq ¿c ¿ C (3.48) Dpzpαq; ξ, ηq 1 “´ W pα ´ W q´ Φ p0, 0,xqdα, αD1 pzpαqq zpαq ¿ C “ ‰ ´1 where C stands for the image of c under (3.46). Since µmax is a simple eigenvalue of W , pα ´ W q has a simple pole α “ µmax inside C while the other factor under the integral is regular. Thus, (3.48) yields

´Dp0; ξ, ηq 1 R W W ´ Φ 0 0 zrφspξ,η,xqdz “ 1 pα ´ q r s p , ,xqdα. (3.49) µmaxD p0q ¿c C¿

Noting that the integral on the right–hand side implements Riesz projection of W for µmax, we further have

1 ´Dp0; ξ, ηq 1 1 R W W ´ Φ 0 0 zrφspξ,η,xqdz “ 1 pα ´ q r s p , ,xqdα 2πi µmaxD p0q 2πi ¿c C¿ ´Dp0; ξ, ηq (3.50) WP Φ 0 0 “ 1 r s p , ,xq µmaxD p0q “ ρ1Dp0; ξ, ηqP rΦs p0, 0,xq,

´1 where ρ1 “ ´ pD1p0qq ą 0 by Lemma 3.9.

3.5 Asymptotic stability

We conclude this section with an application to linear asymptotic stability of mixing. For K P r0, Kcq, S has no eigenvalues, nonetheless mixing is asymptotically stable in the appropriate topology.

Theorem 3.12. Suppose Assumption 2.1 holds. Then for K P r0, Kcq the mixing state is linaerly asymptot- ` ´ ´ ically stable under the perturbations from Ha Ă Ha in the topology of Ha .

Proof. We estimate the semigroups of the linearized systems (2.13), (2.14). Each of the operators Lj, j ě 1, ` ´ ´ generates continuous semigroup on Ha Ă Ha in the topology of Ha : L e j trφspξ,η,xq“ φpξ ` t, η ` jt,xq, j ě 1, (3.51)

Lj t lim }e rφsptq}H´ “ 0. tÑ8 a

For L0, the same statement holds because of the constraint w0pt, 8,η,xq“ 0.

15 Since B is a bounded operator, S generates continuous semigroup expressed by the Laplace inversion ` formula for φ P Ha : bìc St 1 λt e rφspξ,η,xq“ lim e Rλrφspξ,η,xqdλ (3.52) 2πi cÑ8 żbíc for some positive b. Let K P r0, Kcq be arbitrary but fixed. By Theorem 3.6, there exists ǫ ą 0 such that there are no generalized eigenvalues in t´ǫ ď ℜzu and the generalized resolovent Rλ is holomorphic in this region. Hence, the integral pass can be moved to the left half plane and we obtain

´ǫìc St 1 λt e rφspξ,η,xq“ lim e Rλrφspξ,η,xqdλ 2πi cÑ8 ż´ǫíc (3.53) ´εt c e iλt “ lim ie Riλ´ǫrφspξ,η,xqdλ Ñ 0 2πi cÑ8 żć ´ in Ha as t Ñ8.

4 The pitchfork bifurcation

In this section, we study a pitchfork bifurcation of mixing underlying the onset of synchronization in the second order KM. This bifurcation is identiﬁed as the point in the parameter space at which a generalized eigenvalue crosses the imaginary axis. Its analysis requires generalized spectral theory [2].

4.1 Assumptions and the main result

Throughout this section, we assume the following

(A-1) µmax ą 0 is a simple eigenvalue of W with the corresponding eigenfunction Vmax; (A-2) g is an even unimodal probability density function subject to Assumption 2.1.

Assumption (A-1) holds for many common graph sequences encountered in applications including all–to– all, Erd˝os–R´enyi, small–world graphs, to name a few [3]. Assumption (A-2) covers the most representative case. It can be relaxed. We impose it for simplicity.

Remark 4.1. Suppose W has negative eigenvalues. Denote the smallest negative eigenvalue µmin and assume that it is simple with the corresponding eigenfunction Vmin. Then after setting W :“ ´W and K :“ ´K, we ﬁnd that the resultant system satisﬁes (A) and the analysis below shows that the original ´ ´1 system undergoes a pitchfork bifurcation at Kc “ pg0µminq ă 0.

Under these assumptions, as follows from the analysis in Section 3, at K “ Kc, S has a simple generalized eigenvalue λ “ 0. The corresponding eigenfunction is (cf. (3.19))

´ v0 “ Kc lim Dpλpµmaxq; ξ, ηqVmax P Ha . (4.1) λÑ0`

16 Finally, we postulate the existence of a one–dimensional center manifold for the vector ﬁeld in (2.12) for K “ Kc. As we already stated in the Introduction, the proof of this fact is a very technical problem, which is not addressed in this paper. The proof of existence of a center manifold in the classical KM can be found in [1].

In the remainder of this section, we perform a center manifold reduction and show that at K “ Kc, the system undergoes a pitchfork bifurcation. The branch of stable steady state solutions bifurcating from mixing is given in terms of the corresponding values of the order parameter:

´2 ´1{2 |h8pxq| “ Kc pµmaxρ2ρ3pxqq K ´ Kc ` OpK ´ Kcq, (4.2) a where ρ2 ą 0 and ρ3pxq are deﬁned below. In many applications, ρ3pxq ” 1 (see Remark 4.3 for more details).

4.2 The center manifold reduction

The existence of the one–dimensional center manifold at K “ Kc implies the following Ansatz for the 2 solutions of (2.12) with K “ Kc ` ǫ , 0 ă ǫ ! 1:

2 w1 “ Pw1 ` pI ´ Pqw1 “ ǫcptqv0 ` Opǫ q, (4.3)

wk “ qkpw1q, k P N1 :“ t0u t2, 3,... u, (4.4) ď where P is the Riesz projector (cf. (3.40)), cptq is the function determining the location on the slow manifold, N 1 and qk, k P 1, are smooth functions such that qkp0q“ qkp0q“ 0. 2 Using the Ansatz (4.3) and (4.4), below we derive (4.2). Let K “ Kc ` ǫ and rewrite equations for w0, w1, and w2 in (2.12) as follows

´ξ0 Btw0 “ Bξ0 w0 ´ Ke hpt,xqw´1 ´ hpt,xqw1 , (4.5)

2 ´ ´ξ1 ¯ Btw1 “ S0w1 ` ǫ 1 ´ e hpt,xqgˆpηq (4.6)

´ξ´1 ¯ ` K 1 ´ e hpt,xqw0 ´ hpt,xqw2 ,

´ ¯ ´ ´ξ2 ¯ Btw2 “ Bξ2 w2 ` 2Bηw2 ` Kp2 ´ e q hpt,xqw1 ´ hpt,xqw3 , (4.7) ´ ¯ where ξi, i P N, are as deﬁned in (2.9), and S0 is deﬁned by setting K “ Kc in the expression for S (cf. (2.13)). In particular, 2 ´ξ1 Sw1 “ S0w1 ` ǫ 1 ´ e hpt,xqgˆpηq. (4.8) ´ ¯

17 Lemma 4.2. Let e´ξ1 ´ 1 e´ξ1 ´ 1 e´ξ1 ´ 1 1 pe´ξ1 ´ 1q2 P0pλ; ξ1,ωq “ ´ ´ ´ (4.9) λ ´ iω λ ` iω λ ` 1 ´ iω 2 λ ` 1 ´ iω e´ξ1 ´ 1 1 pe´ξ1 ´ 1q2 ` ´ , λ ` 1 ` iω 2 λ ` 1 ` iω 1 1 pe´ξ1 ` 1q2 1 1 P2pλ; ξ1,ωq “ ` ` ´ (4.10) pλ ´ iωq2 2 pλ ` 1 ´ iωq2 λ ´ iω λ ` 1 ´ iω e´ξ1 ` 1 2pe´ξ1 ` 1q 3pe´ξ1 ` 1q ´ ´ ` λ ´ iω λ ` 1{2 ´ iω λ ` 1 ´ iω

´1 ´ξ1 iηω P1pλ; ξ1, ηq “ pλ ´ L1q 1 ´ e pP0 ´ P2q e gpωqdω. (4.11) R ż ”´ ¯ ı On the center manifold, we have 2 hpt,xq “ ǫcptqVmaxpxq` Opǫ q, (4.12) 2 2 2 2 iηω 3 w0 “ Kc ǫ |cptq| |Vmax| lim P0pλ; ξ1,ωqe gpωqdω ` Opǫ q (4.13) λÑ0` R ż 2 2 2 2 iηω 3 w2 “ Kc ǫ cptq Vmax lim P2pλ; ξ1,ωqe gpωqdω ` Opǫ q, (4.14) λÑ0` R ż In addition, ρ2 “´ lim P1pλ; 0, 0qą 0. (4.15) λÑ0`

The number ρ2 is determined solely by gpωq. Its expression will be given in the end of Sec.4.3. With Lemma 4.2 in hand, we now turn to the center manifold reduction. The proof of the lemma is given in the next subsection.

We want to project both sides of (4.6) onto the eigenspace spanned by v0. To this end, we note

PrBtw1s“ ǫc9ptqv0, and PrS0w1s“ 0, (4.16) as follows from (4.3). Further, from Lemma 3.11,

´ξ1 ´1 ´ξ1 P 1 ´ e hpt,xqgˆpηq “ ρ1 lim Dpλ; ξ1, ηq P pλ ´ L1q 1 ´ e gˆpηqhpt,xq . λÑ0` ξ1“η“0 ”´ ¯ ı " ” ´ ¯ ı ˇ * ˇ (4.17) By the ﬁrst line in (3.15), ˇ ´1 ´ξ1 pλ ´ L1q 1 ´ e gˆpηq“ Dpλ; ξ1, ηq. (4.18) After plugging (4.18) and (4.12) into (4.17),´ we have ¯

´ξ1 P 1 D 1 P D 1 1 ´ e hpt,xqgˆpηq “ ρ lim pλ; ξ , ηq r pλ; ξ , ηqhpt,xqsξ1“η“0 λÑ0` ”´ ¯ ı ! ) 2 “ ρ1 lim tDpλ; ξ1, ηqDpλqu P rǫcptqVmaxs p0, 0,xq` Opǫ q λÑ0` 2 (4.19) “ ǫρ1cptqVmaxpxq lim tDpλ; ξ1, ηqDpλqu ` Opǫ q λÑ0` ǫρ1cptqv0 2 “ 2 ` Opǫ q, Kc µmax

18 where we used (3.47) and (4.1) to obtain the last equality.

Finally, we turn to the last group of terms of the right–hand side of (4.6). By Lemma 4.2,

3 2 2 2 hpt,xqw0 ´ hpt,xqw2 “ ǫ Kc cptq|cptq| Vmax|Vmax| ˆ iηω 4 (4.20) ˆ lim rP0 ´ P2s pλ; ξ1,ωqe gpωqdω ` Opǫ q. λÑ0` R ż Further,

L ´1 ´ξ1 3 2 2 2 pλ ´ 1q p1 ´ e q hw0 ´ hw2 “ ǫ Kc cptq|cptq| Vmax|Vmax| lim P1pλ; ξ1, ηq λÑ0` (4.21) 4 ” ` ˘ı ` Opǫ q, where P1 is deﬁned in (4.11). This yields

By projecting both sides of (4.6) onto the subspace spanned by v0 and using (4.16), (4.19), and (4.22), we obtain 3 2 ǫ ρ1 3 2 2 P rVmax|Vmax| s 4 ǫcv9 0 “ cptqv0 ` ǫ K ρ1cptq|cptq| v0 lim P1pλ; 0, 0q` Opǫ q. 2 c 0 Kc µmax Vmax λÑ ` This yields 2 P 2 9 ǫ ρ1 4 rVmax|Vmax| s 2 3 c “ 2 cptq 1 ` Kc µmax lim P1pλ; 0, 0q |cptq| ` Opǫ q. (4.23) K µ λÑ0` V c max ˆ max ˙ 2 Equation (4.23) describes the dynamics on the center manifold. Recalling that hpt,xq“ ǫcptqVmax ` Opǫ q from Lemma 4.2, we rewrite this equation to the equation of the local order parameter h “ hpt,xq as 2 dh ρ1 2 4 P rVmax|Vmax| s 2 4 “ 2 h ǫ ` Kc µmax lim P1pλ; 0, 0q 2 |h| ` Opǫ q dt K µ λÑ0` V |V | c max ˆ max max ˙ ρ1 2 4 2 4 “ 2 h ǫ ´ Kc µmaxρ2ρ3|h| ` Opǫ q (4.24) Kc µmax ` ˘ where ρ2 ą 0 is deﬁned in (4.15) and ρ3 is deﬁned by 2 P rVmaxpxq|Vmaxpxq| s ρ3pxq :“ 2 . (4.25) Vmaxpxq|Vmaxpxq|

Equation (4.24) has a ﬁxed point given by (4.2). It undergoes a pitchfork bifurcation at K “ Kc. Since ρ1 and ρ2 are positive, the ﬁxed point is stable when ρ3 ą 0 (see below).

19 Remark 4.3. In many applications, a graphon is the form W px,yq“ Gpx ´ yq such that Gpxq“ Gp´xq. It admits Fourier series expansion

2πikpx´yq R 2 W px,yq“ cke , ck “ c´k P , ck ă8. (4.26) kPZ kPZ ÿ ÿ Then, eigenvalues of the operator W are the coefﬁcients of the expansion. The largest eigenvalue is given by µmax “ cm :“ suptck : k P Zu, (4.27) 2πimx and the corresponding eigenfunction is Vmax “ e [4]. In this case, ρ3 “ 1. For many important graphs such as Erdos–R˝ enyi´ and small-world graphs, µmax “ c0 and Vmax “ ρ3 “ 1. In this case, the stable branch of equilibria is given through the values of the order parameter

´2 ´1{2 |h8pxq| “ Kc pµmaxρ2q K ´ Kc ` OpK ´ Kcq, (4.28) a 4.3 Proof of Lemma 4.2

By plugging (4.3) and (4.4) into (4.6) and recalling S0v0 “ 0, we immediately get

2 Btw1 “ Opǫ q. (4.29)

From this, (4.3) and (4.4), we further observe that

1 3 N Btwk “ qkpw1qBtw1 “ Opǫ q, k P 1. (4.30)

From (4.3) and the equation for h in (2.12), we have

2 hpt,xq“ W px,yqpǫcptqv0p0, 0,yq` Opǫ qqdy żI 2 “ ǫcptqKc lim Dpλq W px,yqVmaxpyqdy ` Opǫ q 0 λÑ ` I ż 2 “ ǫcptqKcµmax lim DpλqVmaxpxq` Opǫ q λÑ0` 2 “ ǫcptqVmaxpxq` Opǫ q.

This shows (4.12).

Next we turn to the equation for w2:

´ξ2 Btw2 “ Bξ2 w2 ` 2Bηw2 ` Kp2 ´ e qphw1 ´ hw3q.

3 Since Btw2 and hw3 are of order Opǫ q, we have

2 2 ´ξ2 3 pBξ2 ` 2Bηq w2 “´Kcǫ cptq p2 ´ e qVmaxv0 ` Opǫ q. (4.31)

20 Lemma 4.4. On the center manifold, w2 is expressed as 2 2 2 2 w2 “ Kc ǫ cptq Vmax 1 1 e´2ξ2 1 1 ˆ lim 2 ` 2 ` ´ λÑ0` R pλ ´ iωq 2 pλ ` 1 ´ iωq λ ´ iω λ ` 1 ´ iω ż ´ξ2 ´ ´ξ2 ´ξ2 e 2e 3e 3 ´ ´ ` eiηωgpωqdω ` Opǫ q. (4.32) λ ´ iω λ ` 1{2 ´ iω λ ` 1 ´ iω ¯ Proof. A straightforward substitution shows that (4.32) satisﬁes (4.31) up to Opǫq3.

By the relation e´ξ2 “ e´ξ1 ` 1 derived from (2.9), we obtain (4.14) with (4.10).

Next, the equation for w0 is given by

´ξ0 Btw0 “ Bξ0 w0 ´ Ke hw´1 ´ hw1 .

Lemma 4.5. On the center manifold, w0 is expressed as ` ˘ 2 2 2 2 w0 “ Kc ǫ |cptq| ¨ |Vmax| e´ξ0 e´ξ0 e´ξ0 ` e´2ξ0 {2 e´ξ0 ´ e´2ξ0 {2 ˆ lim ´ ´ ` eiηωgpωqdω λÑ0` R λ ´ iω λ ` iω λ ` 1 ´ iω λ ` 1 ` iω 3ż ´ ¯ `Opǫ q. (4.33)

3 Proof. By using w´1pt,ξ´1,η,xq“ w1pt,ξ´1, ´η,xq and Bw0{Bt “ Opǫ q, we get

Bw0 ´ξ0 2 2 3 “ Kce ¨ ǫ |cptq| Vmaxv0pξ´1, ´η,xq´ Vmaxv0pξ1,η,xq ` Opǫ q Bξ0 2 2 2 ´ 2 ´ξ0 ¯ “ Kc ǫ |cptq| ¨ |Vmax| e ´ξ0 ´ξ0 1 1 1 ´ e 1 ` e 3 ˆ lim ´ ´ ` eiηωgpωqdω ` Opǫ q, λÑ0` R λ ` iω λ ´ iω λ ` 1 ` iω λ ` 1 ´ iω ż ´ ¯ ξ 1 ξ0 where the relations e´ ˘ “ 1 ˘ e´ from (2.9) is used to write the right hand side as a function of ξ0. Integrating both sides in ξ0 with the boundary condition w0pt, 8,η,xq“ 0 veriﬁes the lemma.

The relation e´ξ1 “ 1 ` e´ξ0 gives (4.13) with (4.9).

Finally, we prove (4.15). Recall

´1 ´ξ1 iηω P1pλ; ξ1, ηq :“ pλ ´ L1q 1 ´ e pP0 ´ P2q e gpωqdω R ż ”´ ¯ ı By (3.6),

´1 ´ξ1 iηω pλ ´ L1q 1 ´ e pP0 ´ P2q e pξ1,η,xq

ipξ1ζ`η”´ωq ¯ ı e ´ξ1 iηω “ Frp1 ´ e q pP0 ´ P2q e spζ, ωqdζdω. R2 λ ´ iζ ´ iω ż r r r r 21 ξ1 iηω Here, Frp1 ´ e´ q pP0 ´ P2q e s is the Fourier transform with respect to ξ1 and η with parameters λ, ω. Thus,

´1 ´ξ1 iηω pλ ´ L1q 1 ´ e pP0 ´ P2q e pξ1,η,xq

ipξ1ζ`η”´ωq ¯ ı e ´ξ1 “ Frp1 ´ e q pP0 ´ P2qspζq ¨ δpω ´ ωqdζdω R2 λ ´ iζ ´ iω ż r ipξ1ζ`ηωq e ´ξ1 “ Frp1 ´ e q pP0 ´ P2qspζqdζ. r r R λ ´ iζ ´ iωr ż r Hence, we obtain

ipξ1ζ`ηω e q ´ξ1 P1pλ; ξ1, ηq “ Frp1 ´ e q pP0 ´ P2qspζqgpωqdζdω R2 λ ´ iζ ´ iω ż 1 ´ξ1 P1pλ; 0, 0q “ Frp1 ´ e q pP0 ´ P2qspζqgpωqdζdω, R2 λ ´ iζ ´ iω ż ξ1 ξ1 where Frp1 ´ e´ q pP0 ´ P2qs is the Fourier transform with respect to ξ1. Since p1 ´ e´ q pP0 ´ P2q is a ξ1 2ξ1 3ξ1 ξ1 linear combination of 1, e´ , e´ , e´ , Frp1 ´ e´ q pP0 ´ P2qs is a linear combination of δpζq, δpζ ´ iq, δpζ ´ 2iq, δpζ ´ 3iq, and the integration with respect to ζ is easy. A straightforward albeit lengthy calculation shows that 1 1 P1pλ; 0, 0q“ ´ 3 gpωqdω R pλ ´ iωqpλ ` iωq pλ ´ iωq ż ˆ ˙ 3 1 1 ` ´ gpωqdω 2 R λ ´ iω λ ` iω ż ˆ ˙ 12iω7 ` 4ω6 ` 43iω5 ` 13ω4 ` 58iω3 ` 25ω2 ` 21iω ` 10 ´ 2 3 2 gpωqdω. R p1 ` ω q p1 ` 4ω q ż ˆ ˙ Since g is an even function,

1 gpωq 1 1 2 P1pλ; 0, 0q“ dω ` g pωqdω R λ ´ iω iω 2 R λ ´ iω ż ż 4ω6 ` 13ω4 ` 25ω2 ` 10 ´ 2 3 2 gpωqdω. R p1 ` ω q p1 ` 4ω q ż ˆ ˙ Further, Sokhotski–Plemelj formula [12] shows

8 1 1 2 g pωq lim P1pλ; 0, 0q“ πg p0q` 2 lim dω λÑ0` 2 εÑ0 w żε 4ω6 ` 13ω4 ` 25ω2 ` 10 ´ 2 3 2 gpωqdω, R p1 ` ω q p1 ` 4ω q ż ˆ ˙ which is a negative number and ρ2 is positive.

22 1 1

SYNC

0.5 0.5 MIXING TWISTED MIXING STATE SYNC

0 0 0 1.5 3 -15 -5 5 a b

Figure 2: The bifurcation diagrams for the second order KM on ER (a) and SW (b) random graphs. In these experiments, the intrinsic frequences are taken from the normal distributrion with mean 0 and standard deviation equal to 0.3. The parameters used for generating ER and SW graphs are p “ 0.5 for the former and p “ 0.1 and r “ 0.3 for the latter. The plot in (a) shows a pitchfork bifurcation at Kc ą 0. The ` ´ bifurcation diagram for the small-world family shows two bifurcations at Kc ą 0 and Kc ă 0. The latter features a spatially heterogeneous pattern bifurcating from mixing, because the eigenfunction of W corresponding to the smallest negative eigenvalue µmin is not constant (see text for details). The red dashed lines indicate pitchfork bifurcations identiﬁed analytically. These values are in good agreement with the results of numerical simulations.

5 Examples

In this section, we apply the theory developed in the previous sections to identify the pitchfork bifurcations in the KM with inertia on Erd˝os-R´enyi (ER) and small-world (SW) graphs. The eigenvalues and eigenfunctions of the integral operator W corresponding to these graphs were computed in our previous work [3]. Below, we will use this information without any further explanations. An interested reader is referred to [3] for more details.

5.1 ER graphs

An ER graph Gpn,pq is a graph on n nodes, for which the probability of two given nodes being connected by an edge is equal to p P p0, 1q. The edges between distinct pairs of nodes are assigned independently. Gpn,pq is a convergent sequence of graphs, whose limit is given by a constant function W ” p. The largest eigenvalue of W is µmax “ p, and the corresponding eigenfunction is constant Vmax ” 1 (cf. [3]). Thus, there is a pitchfork bifurcation at 1 Kc “ , pg0 where g0 is defined in (3.23). The bifurcation diagram in Fig. 2 shows that in the vicinity of Kc the order parameter undergoes a sharp transition from very small positive values to the values close to 0.5. This corresponds to the loss of stability of mixing and the onset of synchronization. Since Vmax is constant, the unstable mode is spatially homogeneous. With minor modifications the analysis of the pitchfork bifurcation can be extended to sparse Erd˝os–Rényi graphs [10].

23 5.2 SW graphs

There are two types of random connections in a SW graph: the short-range and the long-range. The former are assigned with probability 1´p and the latter are assigned with probability p P p0, 1{2q. The connectivity is defined by another parameter r P p0, 1{2q (see [9] for the exact definition the SW graph sequence used here). SW graphs in [9] are defined as W-random graphs with the limiting graphon

1 ´ p, dpx,yqă r, W px,yq“ , p, otherwise. " where dpx,yq “ mint|x ´ y|, 1 ´ |x ´ y|u. The eigenvalues and the corresponding eigenfunctions of W for the SW graphs can be found in [3]. In particular, the largest positive eigenvalue µmax “ 2r ` p ´ 4rp ` is simple and the corresponding eigenfunction Vmax ” 1. This yields a pitchfork bifurcation at Kc “ ´1 pg0p2r ` p ´ 4rpqq .

In addition, W has negative eigenvalues. As was explained in Remark 4.1, if the smallest negative ´ ´1 eigenvalue µmin is simple, then there is another bifurcation at Kc “ pg0µminq . The value of µmin has the following form ˚ ´1 ˚ µmin “ pπk q p1 ´ 2pq sin p2πk q for some integer k˚ ‰ 0, which is not available analytically in general. The corresponding eigenfunction 2πik˚x ` Vmin “ e is not constant anymore. This implies that in contrast to the bifurcation at Kc , the steady ´ state bifurcating from mixing at Kc ă 0 is heterogeneous (Fig. 2b). In [4], a bifurcation for a non-constant eigenfunctions and the corresponding twisted state is studied for the classical KM in detail.

Acknowledgements. The authors thank Matthew Mizuhara for providing numerical bifurcation diagrams. This work was supported in part by NSF grant DMS 2009233 (to GSM).

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