Supplementary information for Letter “ Oscillation suppression and synchronization:Frequencies determine the role of control with time delays”

Wei Lin1,∗ Yang Pu1, Yao Guo1, and Jurgen Kurths2†1School of Mathematical Sciences, Centre for ComputationalSystems Biology, Fudan University, Shanghai 200433,

China, and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, China.2Potsdam Institute for Climate Impact Research,D-14412 Potsdam, and Department of Physics,

Humboldt University of Berlin, D-12489 Berlin, Germany.

In this supplementary information, we perform a rigorous and systematic analysis of a transcendental charac-teristic equation with complex coefficients, thereby establishing necessary and sufficient conditions for guaran-teeing the stability of the controlled supercritical-Hopf-bifurcation model with a time delay feedback. Besides,we expatiate, respectively, on the several transformations used for the oscillation suppression in the FitzHugh-Nagumo model and for the synchronization of a periodic orbitin a complex system ofN connected units withtime delay couplings.

CONTENTS

I. Preliminary formulations 2A. Non-real-valuedβ 2B. Real-valuedβ 6

II. Necessary and sufficient conditions for stability 8A. Along the transverse branches for non-real-valuedβ 9B. Along the concave branch for non-real-valuedβ 11C. Along the branches for real-valuedβ 18

III. Applications 20A. Oscillation suppression in the FitzHugh-Nagumo model 20B. Oscillation death and synchronization in complex networks 22

References 23

∗ wlin@fudan.edu.cn† kurths@pik-potsdam.de

2

I. PRELIMINARY FORMULATIONS

A linearization at the equilibriumz0 = 0 of the controlled supercritical-Hopf-bifurcation modelwith a timedelay feedback [refer to model (2) in Letter] yields:

z(t) = (a+ id)z(t) + keiψz(t − τ), (S1)

whereτ > 0 is a time delay of the feedback term andkeiψ is a complex-valued control gain. Actually, Eq. (S1)is the same as the linear model (3) which is in a matrix form in Letter. Thus, according to the theory in [1], theasymptotical stability of Eq. (S1) is determined by its characteristic equation

λ − (a+ id) − keiψe−λτ = 0, (S2)

which is the same as Eq. (4) in Letter. We import the following transformations of the argument and complexparameters:Z = λτ − aτ − idτ andβ = beiα, which have been defined in Letter. Here,b = −kτe−aτ andα = ψ − dτ (modπ), so thatb ∈ R andα ∈ [0, π). Hence, Eq. (S2) is transformed equivalently into an equationwith respect to the newly defined argumentZ:

h(Z) , Z + βe−Z= 0, (S3)

which is provided as Eq. (5) in Letter. LetA = aτ. Then, fromZ = λτ − A − idτ, it follows that all theeigenvalues of Eq. (S2) are on the left half-planeif and only ifevery root of Eq. (S3) in the complex plane ison the left side of the lineZ = −A, that is, for any rootZ = Z0 of Eq. (S3), Re{Z0} < −A.

Substitution ofZ = x+ iy into Eq. (S3) implies the equivalence between Re{Z} < −A andx < −A. Equatingthe real and imaginary parts separately yields:

{

x = −be−x cos(y− α),

y = be−x sin(y− α).(S4)

We are first to give an elementary description of the root distribution of Eq. (S4). For this purpose, we considertwo separate cases, viz.α , 0 (i.e. β is non-real)andα = 0 (i.e. β is real)in this section. It is noted that theparticular case ofα = 0 andA = 0 has been discussed in the Appendix of [1] and references therein. However,a more general case ofα = 0 andA ≥ 0 is considered here.

A. Non-real-valuedβ

Throughout this subsection,β is assumed to be non-real, so thatα ∈ (0, π). It thus is easy to validatey , 0sincey = 0 leads tob = 0 andβ = 0, which is not included in the present case. Hence, direct operation of thetwo equations in Eq. (S4) gives:

{

F(x, y) , x+ ycot(y− α) = 0,ycsc(y− α) = beycot(y−α).

(S5)

For consideration of a well-defined cot(y− α), it is required (C1):y , α + pπ for p ∈ Z. However, the straightlinesLp : y = α + pπ in thex-y plane have particular meanings for the curve represented byF(x, y) = 0, whichwill be illustrated later.

First, derivative of both sides ofF(x, y) with respect toy yields:

∂F∂y=−y+ sin(y− α) cos(y− α)

sin2(y− α),

This quantity is nonzero if (C2):y , sin(y−α) cos(y−α), i.e. 2y , sin 2(y−α), is satisfied. Under the conditions(C1)-(C2), the Implicit Function Theorem [2] could be implied toF(x, y) = 0. It thus is assured the existenceof a locally differentiable functiony = y(x) satisfyingy(x0) = y0 andF(x, y(x)) = 0 in the neighborhood ofx = x0. Hence,y(x) satisfies an initial value problem of the ordinary differential equation:

dydx=

sin2(y− α)y− sin(y− α) cos(y− α)

,

y(x0) = y0.

(S6)

3

The function on the right side of the first equation in problem(S6) fulfills a Lipschitz condition when 2y ,sin 2(y − α). It thus from the general theory of ordinary differential equations (ODEs) [3, 4] follows that theinitial value problem (S6) can be uniquely solved in the neighborhood ofx = x0 as the initial value satisfies2y0 , sin 2(y0 − α). In order to discuss the maximal interval of existence (MIE) for each solved solution, weneed the following result.

Lemma I.1 The root of the equation2y = sin 2(y− α) is uniquely existent forα ∈ (0, π).

proof. Consider the functionη(y) , 2y−sin 2(y−α). On the one hand, note thatη(y)→ ±∞ asy→ ±∞. Hence,by the continuity ofη(y), the zero ofη(y) is existent. On the other hand, the derivativeη′(y) = 2−2 cos 2(y−α) ≥0, where the equality is valid only whenys = sπ + α for s ∈ Z. However, none ofys is the zero ofη(y) asα ∈ (0, π), so that the zero ofη(y) is unique. Therefore, we complete the proof. �

With LemmaI.1, we get that, in problem (S6), dy/dx > 0 if y > y∗ and dy/dx < 0 if y < y∗, wherey = y∗ = y∗(α) is the unique root of 2y = sin 2(y− α) for a givenα ∈ (0, π). This means that the solutions ofproblem (S6) are strictly increasing when their initial values satisfyy0 > y∗ and strictly decreasing asy0 < y∗.Also note that the first equation in problem (S6) has constant solutions corresponding to the straight lines Lp.It is clear that these lines do not pass through the points (x0, y0) but bound all the trajectories of the solutionspassing through (x0, y0). Again, in light of the general theory of ODEs [3, 4], the MIE for every solutionof problem (S6) can be extended to±∞ as the initial value locates between the lineLp andLp−1 for p , 0;however, the MIE can be extended only to+∞ as the initial value locates betweenL0 andL−1 and on the rightside of the linex = x∗ wherex∗ = y∗ cot(y∗ − α).

Now, from the uniqueness of every solution starting from (x0, y0), it follows that each trajectory of thesolution on the entire MIE corresponds to a single curve branch, denoted byΓp, in between the straight linesLp andLp−1, and that these straight lines indeed become asymptotes of the branches. We thus obtain the curveof F(x, y) = 0, denoted byΓ =

⋃

p∈Z Γp, and arrive to the following specific description.

Lemma I.2 For p ∈ Z+ (p ∈ Z−, resp.), the branchΓp of the curveΓ is strictly increasing (decreasing, resp.)and transversely crosses the y-axis at(0, yp) with yp = α+ pπ−π/2. In addition, for p= 0, Γ0 is not monotonicbut concave to the right, thereby crossing the y-axis at both(0, 0) and(0, y0) with y0 = α− π/2. The coordinateof the vertex ofΓ0 is (− cos2(y∗ − α), y∗), which merges with(0, y0) into (0, 0) asα = π/2.

−10 −5 0 5 10−10

−5

0

5

10

x

y ηb=0

b>0

b<0

b>0

b<0

b>0

b<0

b>0

Γ−1

Γ−2

Γ0

Γ1

Γ2

Γ3

FIG. S-1. The curveΓ contains an infinite number of branchesΓp which alternately appear in between the asymptotesLp:y = α + pπ andLp−1 for non-realβ andp ∈ Z. Here, the curve is depicted particularly whenα = 1 in the first equation ofEq. (S5). All roots of Eq. (S4) are located on the branches. The arrow on each transverse branchΓp (p , 0) represents themovement direction of the root with the increase ofb from −∞ to 0 or from 0 to+∞. The arrow on the convex branchΓ0

shows the movement direction of the root from (+∞, α) to (+∞, α − π) with the increase ofb increases−∞ to +∞. Notethat x = y = 0 corresponding to the case ofb = 0 needs to be eliminated in the present situation.

The curve ofΓ is numerically depicted in Fig.S-1asα = 1. Furthermore, by the elementary trigonometric

4

formula, we have

[

− cos2(y∗ − α) +12

]2

+ (y∗)2=

14, (S7)

for all α ∈ (0, π). Then, with changingα, the vertex ofΓ0 varies on a circle with a center at (−1/2, 0) and aradius of 1/2. In particular, with the increase ofα from 0 toπ, the vertex moves clockwise on the circle from(−1, 0) to itself. According to LemmaI.2, the vertex passes by (0, 0) on the circle asα = π/2.

The analyses performed above manifest that all the roots of Eq. (S4) are located on the curveΓ. In whatfollows, we need to investigate how many roots are there and how the roots move along the branches ofΓ withchangingb.

This motivates us to consider the second equation in Eq. (S5). Actually, every root of the second equationcould be regarded as an intersection point of the two curves of functionsu1(y) = ycsc(y − α) andu2(y) =beycot(y−α), whereb , 0 could be viewed as a parameter determining the locus of the intersection point. Oncethe intersection is found for a givenb, the corresponding root of Eq. (S4) can be directly obtained throughsubstituting they coordinate of the intersection point into the first equationof Eq. (S5).

Using a method similar to the discussion for the curveΓ, we know that both curves represented byu1(y) andu2(y) have an infinite number of branches separated by the vertical lines Vp : yp = α + pπ in they-u plane forp ∈ Z. More concretely, we get the following result.

Lemma I.3 If b > 0, roots of the equation u1(y) = u2(y) are only existent in the intervalsIp = (α + pπ, α +(p+ 1)π) for p = 2q− 2 and p= −2q− 1 (q ∈ N). Conversely, if b< 0, roots are only existent in the intervalsIp for p = 2q− 1 and p= −2q (q ∈ N). Each interval contains only one root for both cases.

proof. We first writeζ(y) = u1(y) − u2(y) and, without loss of generality, supposeb > 0, p = 2q − 2 andp = −2q− 1 (q ∈ N). Similar arguments can be performed for the case ofb < 0.

On the one hand,ζ(y) goes to+∞ asy → [α + (p + 1)π]−, and goes to−∞ asy → (α + pπ)+. Here, thedivergence ofu2 is exponentially faster than that ofu1 asy→ (α + pπ)+. Thus, from the continuity ofζ(y) inIp, it follows that the zero ofζ(y) is existent inIp.

On the other hand, derivative ofζ(y) yields:

ζ′(y) =

sin(y− α) − ycos(y− α)

sin2(y− α)− beycot(y−α)−y+ sin(y− α) cos(y− α)

sin2(y− α).

This derivative aty = y0 which is the zero ofζ(y) becomes:

ζ′(y0) =

1sin(y0 − α)

[

y0

sin(y0 − α)− cos(y0 − α)

]2

+ sin2(y0 − α)

.

Sincey0−α ∈ (pπ, (p+1)π), the sign of sin(y0−α) in eachIp keeps unchangeable. Moreover, sin2(y0−α) , 0in eachIp, so that the sign ofζ

′(y0), which is the same as that of sin(y0−α), is invariant inIp. This implies the

uniqueness of the zero in eachIp; otherwise, the sign ofζ′(y0) for differenty0 in the sameIp is changeable.

Therefore, we approach the conclusion of the lemma on the roots of the equationu1(y) = u2(y). �

Lemma I.4 If b , 0, the roots of the equation u1(y) = u2(y) are uniquely existent in the intervalI−1 = (α−π, α).

proof. It is clear that, for anyb , 0, ζ(y) = u1(y) − u2(y) goes to−∞ as y → α−, and goes to+∞ asy→ (α− π)+. This implies the existence of a zeroy0 ∈ I−1. Moreover, the uniqueness of the zero follows fromthe unchangeable sign ofζ

′(y0), which has been validated in the proof of LemmaI.3. �

From LemmasI.3 andI.4, we know that there are an infinite number of roots for the equation u1(y) = u2(y),which correspond to the intersection points of the curve branches in between the vertical linesVp andVp+1 forp ∈ Z. More importantly, in between every pair of the vertical lines, the intersection point is unique. Fig.S-2shows some of the branches and their intersection points forthe cases ofb = 1 andb = −1.

Once the intersection points are all determined for a givenb , 0, the roots of Eq. (S5) can be obtained.In particular, each intersection point corresponds to onlyone root which locates on one of the branches of thecurveΓ.

Next, we need to show how the root moves on each branch ofΓ for changingb. For this purpose, wesupposeZ to be a root of Eq. (S3), and so thatZ could be regarded as a function ofb. We thus are to check

5

−10 −5 0 5 10−10

−5

0

5

10

y

u

FIG. S-2. Some of the intersection points between the curve of u1(y) = ycsc(y − α) (blue solid line) and the curve ofu2(y) = beycot(y−α), respectively, forb = 1 (red solid line) andb = −1 (green dash line). Both curves have an infinite numberof branches separated by the vertical linesVp for p ∈ Z. Here,α = 1.

the signs of the real and the imaginary parts ofZ′(b), respectively. Note thatZ = −βe−Z= −beiα−Z. Then,

Z′(b) = Z/b− Z · Z′(b), which further implies

Z′(b) =Z

b(1+ Z)=

1b

(

1− 11+ Z

)

=1b

(

1− 1+ x− iy|1+ Z|2

)

,

whereZ(b) = x(b) + i · y(b) which we have set above. Equating the real and the imaginaryparts, respectively,yields:

x′(b) =1b·

(x+ 12)2+ y2 − 1

4

(1+ x)2 + y2, y′(b) =

1b· y

(1+ x)2 + y2.

It is clear thaty′(b) > 0 along the branchesΓp for b > 0, y > 0 andp = 2q− 1 (q ∈ N), and thaty′(b) < 0along the branchesΓp for b < 0, y > 0 andp = 2q (q ∈ N). Also it is easy to get the sign ofy′(b) asy > 0for differentb , 0. Because all the branches are uniquely obtained, the movement direction of the roots on thebranchesΓp (p , 0) can be determined consequently, as shown in Fig.S-1.

As for the concave branchΓ0, it crosses a circle with a center at (−1/2, 0) and a radius of 1/2 at, respectively,the origin and and the vertex ofΓ0. In particular, at the originb = 0, and at the vertex

b = b0 , cos(y∗ − α) · e− cos2(y∗−α), (S8)

where cos(y∗ − α) is positive for 0< α < π/2 and negative forπ/2 < α < π. In addition, the numerator onthe right side ofx′(b) is negative whenΓ0 is inside the circle but positive when it is outside. Thus, itis nothard to validate that, whether or notb0 is positive or negative,x′(b) < 0 for b < b0 andx′(b) < 0 for b > b0.Altogether, we summarize the above results into the following proposition.

Proposition I.5 Case I “b > 0”: For p = 2q− 1 (q ∈ N), the single root on each branchΓp that starts from(−∞, (p − 1)π + α) when b= 0 and goes to(+∞, pπ + α) when b= +∞, crossing the y-axis at the point(0, α+ pπ−π/2) when b= α+ pπ−π/2. For p = −2q (q ∈ N), the root on each branch starts from(−∞, pπ+α)when b= 0 and goes to(+∞, (p− 1)π + α) when b= +∞, crossing the y-axis at the point(0, α + pπ − π/2)when b= −α − pπ + π/2.

Case II “b < 0”: For p = 2q (q ∈ N), the single root on each branchΓp of Eq. (S4), denoted by(x0p, y

0p),

that starts from(+∞, pπ + α) when b= −∞ and goes to(−∞, (p− 1)π + α) when b= 0. It crosses the y-axisat the point(0, α + pπ − π/2) when b= −α − pπ + π/2. Moreover, for p= −2q+ 1 (q ∈ N), the root startsfrom (+∞, (p− 1)π + α) when b= −∞ and goes to(−∞, pπ + α) when b= 0, crossing the y-axis at the point(0, α − pπ − π/2) when b= −α + p+ π/2.

6

Case III “b , 0 and “p=0”: The single root on the branchΓ0 starts from(+∞, α) when b= −∞ and goesto (+∞,−π + α) when b= +∞, cro ssing the y-axis twice, viz.,(0, 0) when b= 0 and (0, α − π/2) whenb = −α + π/2. The zero approaches the leftmost position(− cos2(y∗ − α), y∗) when b= b0 as defined in (S8).

B. Real-valuedβ

In this subsection, we discuss the case of real-valuedβ. Then, lettingα = 0 leads Eq. (S4) to

{

x = −be−x cosy,

y = be−x siny.(S9)

Clearly, the second equation in Eq. (S9) has a zero root, i.e.y = 0. Therefore, any root ofx = −be−x is thereal root of Eq. (S4) and vise versa. The real root ofx = −be−x can be obtained from the intersection pointsbetween the curves represented byv1(x) = x andv2(x) = −be−x, as shown in Fig.S-3. Theoretically, we havethe following results.

−3 −2 −1 0 1 2−10

−5

0

5

10

x

y

FIG. S-3. The intersection points between the curves represented byv1(x) = x (blue) andv2(x) = −be−x where the parameterb = −1 (red),b = 1/e (black) andb = 1 (green).

Proposition I.6 For b < 0, Eq. (S4) has only one positive and real root, denoted by x+

b . With increasing b from−∞ to 0−, x+b decreases from+∞ to 0+. For b = 0, the root is zero. For0 < b < 1/e, Eq. (S4) has two negativeand real roots, denoted by x−1b and x−2b. These two roots merge into−1 as b= 1/e. For b > 1/e, Eq. (S4) hasno real roots.

proof. Write ρ(x) = v1(x)−v2(x). Then, it is clear that, forb ≤ 0,ρ(x) goes to+∞ asx→ +∞ andρ(0) = b ≤ 0.Thus, from the continuity ofρ(x), it has at least one zero on [0,+∞). It is easy to validate thatρ(x) has no zeroon (−∞, 0). Moreover,ρ′(x) = 1 − be−x > 0 for b ≤ 0. Therefore, the zero ofρ(x) is uniquely existent on(−∞,+∞) for b ≤ 0, and, in particular,x = 0 is the zero ofρ(x) asb = 0.

For the case ofb > 0, any root ofρ(x) is negative sincev2(x) < 0. ρ′x(x) = 1 − be−x= 0 givesρ′(x) < 0

asx ∈ J1 , (−∞, ln b) andρ′(x) > 0 asx ∈ J2 , (ln b,+∞). When 0< b < 1/e, we haveρ(ln b) < 0 andρ(±∞) = +∞. Therefore, from the continuity of the functionρ(x), it follows that there is a zero ofρ(x) in eachof J1,2 when 0< b < 1/e.

More accurately, writeρ(x, b) = v1(x) − v2(x). Thus, the derivativeρx(x, b) = 1− be−x= 0, which together

with ρ(x, b) = 0 impliesx = −1 andb = 1/e. Hence,ρ(−1, 1/e) = 0. Note thatρb(−1, 1/e) = e , 0. Then,according to the Implicit Function Theorem [2], there exists a functionb = b(x) in the neighborhood ofx = −1such thatρ(x, b(x)) = 0 andb(−1) = 1/e. Furthermore,

dρ(x, b(x))dx

= 1+ b′(x)e−x − b(x)e−x= 0,

7

which further yieldsb′(−1) = 0 andb′′(−1) = −b(−1) < 0. This actually implies that the graph ofb = b(x)describes the variation of the two zeros ofρ(x) with the change of 0< b < 1/e. Particularly, the two zerosmerge intox = −1 asb = 1/e and the real zero disappears asb > 1/e. �

Assumey , 0. We thus are able to investigate the non-real roots of Eq. (S9) as well as Eq. (S4). Eq. (S9)thus can be transformed equivalently into

{

x+ ycoty = 0,ycscy = beycoty.

(S10)

First, it is easy to see that all the roots appear symmetrically with respect to they-axis. Then, using thearguments as those performed in the preceding subsection, we know that all the roots are located on an infinitenumber of the branches represented by the first equation in Eq. (S10). In particular, in addition to thosebranches, denoted byΓp (p , 0) monotonically transversing they-axis, the branch, denoted byΓ0, is concaveto the right and occupies a strip with a wide of 2π. FigureS-4shows some of these branches and the movementdirection of the root on each branch with the change ofb. The concave branch here is slightly different from theconcave branch presented for the case of non-real-valuedβ and shown in Fig.S-1. Theoretically, we approachthe following results on the character of the non-real rootsof Eq. (S4) for any real-valuedβ.

−10 −5 0 5 10−10

−5

0

5

10

x

y

b>1/e

b<0

b>0

b<0

b>0

Γ0

b=0

b>1/e

b<0

Γ2

Γ1

Γ−2

Γ−1

0<b<1/e b=1/e

FIG. S-4. Some of the branchesΓp (p ∈ Z) of the curve represented by the first equation in Eq. (S10). All the real roots ofEq. (S4), obtained in PropositionI.6, are located on thex-axis. The arrows represent the movement direction of the rootson the branches and thex-axis with the increase ofb.

Proposition I.7 Case I “b< 0”: For any integer p= 2q+ 1 (q ∈ Z), there is one root of Eq. (S4) on eachΓp.As b changes from−∞ to 0, this root moves from(+∞, (2p+ 2)π) to (−∞, (2p+ 1)π), crossing the y-axis fromright to left at(0, 2pπ + 3π/2) when b= −2pπ − 3π/2.

Case II “b > 0”: For any integer p = 2q (q ∈ Z, q , 0), there is one root of Eq. (S4) on eachΓp. As bchanges from0 to+∞, this root moves from(−∞, 2pπ) to (+∞, (2p+ 1)π), crossing the y-axis from left to rightat (0, 2pπ + π/2) when b= 2pπ + π/2.

Case III “b > 1/e”: There are two roots symmetrically locating on the concavebranchΓ0. These two zerosmerge into(−1, 0) as b goes to1/e and tends towards, respectively,(+∞,±π) as b goes to+∞. They cross they-axis at(0,±π/2) when b= π/2.

Case IV “b< 1/e”: There is one real root of Eq. (S4) on the x-axis, moving from(+∞, 0) to (−1, 0) with theincrease of b from−∞ to 1/e. There is the other real root on the x-axis, moving from(−∞, 0) to (−1, 0) withthe increase of b from0+ to 1/e. When b> 0, two roots simultaneously appear, which have been described inPropositionI.6, and finally merge into(−1, 0) as b= 1/e.

Remark I.8 As a matter of fact, part of the arguments on the two cases discussed in the above section couldbe found briefly in [5]; however, for the sake of accuracy and integrity, we provide rigorous analyses and thenestablish systematic results here.

8

II. NECESSARY AND SUFFICIENT CONDITIONS FOR STABILITY

Based on the above-obtained results on the root distribution of Eq. (S4) as well as Eq. (S3), we, in thissection, are to establish necessary and sufficient (NS) conditions under which all the roots of Eq. (S4) satisfyRe{Z} < −A = −aτ, i.e., x < −A, and then the asymptotical stability of the controlled system (S1) is assured.For this purpose, we consider the two cases ofβ discussed in the preceding section, and finally summarize theestablished conditions into the NS conditions.

First, letx = −A in Eq. (S4). Then, we get

−A = −beA cos(y− α),

y = beA sin(y− α).(S11)

Clearly,b = −kτe−aτ= 0 impliesk = 0 andx = y = 0 in Eq. (S11). Fromx = 0 < −A = −aτ, it follows that

a < 0 becomes the stability condition whenk = 0. In the following discussion, we assume thatb , 0. Hence,Eq. (S11) can be transformed equivalently into

A2+ y2= b2e2A,

cos(y− α) =A

beA,

0 ≤ bysin(y− α).

Further calculation of the above equations yields:

y = ±√

b2e2A − A2,

y = α ± arccosA

beA+ 2pπ, p ∈ Z,

0 ≤ bysin(y− α).

(S12)

It is clear that the first equation in Eq. (S12) is solvable only if(C-I): b2e2A − A2 ≥ 0, i.e., |k| ≥ |a| in thecontrolled system (S1). This situation will be discussed in the following subsections; however, we first dealwith Eq. (S4) under the condition(C-II): b2e2A − A2 < 0, and then establish the following two propositions.

Proposition II.1 If A < 0 and b2e2A < A2 (i.e., a< 0, τ > 0, and|k| < −a), any root(x, y) of Eq. (S4) satisfiesRe{Z} = x < −A, so that the controlled system (S1) is asymptotically stable when a< 0, τ > 0, and|k| < −a.

proof. Suppose that there exists a solution of Eq. (S4), denoted by (x0, y0), such thatx0 ≥ −A > 0. Then wehave

(x0)2e2A ≤[

(x0)2+ (y0)2

]

e2A= b2e−2x0

e2A < A2e−2x0,

which implies

(x0)2 < A2e−2(x0+A) ≤ A2,

where the second inequality is due tox0+ A ≥ 0. However, (x0)2 < A2 contradicts the supposition, which

finally implies the completion of the proof. �

Proposition II.2 If A > 0 and b2e2A < A2 (i.e., a> 0, τ > 0, and |k| < a), then there exists at least one root(x0, y0) of Eq. (S4), such thatRe{Z0} = x0 > −A. Thus, the controlled system (S1) is unstable when a> 0,τ > 0, and|k| < a .

proof. For a givenA > 0, we consider a mapΨ : S ⊂ C→ C. Here, the setS , [−A,A] × [−A,A] is bounded,closed and convex, and the map

Ψ(x, y) ,(−be−x cos(y− α), be−x sin(y− α)

)

follows from Eq. (S11). Notice the conditionsA > 0 andb2e2A < A2 assumed in PropositionII.2. We thus get|be−x| ≤ |b|eA < A for any (x, y) ∈ S, so that

∣

∣

∣−be−x cos(y− α)∣

∣

∣ < A,∣

∣

∣be−x sin(y− α)∣

∣

∣ < A,

9

for any (x, y) ∈ S. This manifests thatΨ mapsS to itself, i.e.,Ψ : S → S. Also it is obvious that the mapΨ is continuous. Therefore, by virtue of the well-known Brouwer fixed-point theorem [6], we conclude thatthere exists a point (x0, y0) ∈ S such that (x0, y0) = Ψ(x0, y0). This means that Eq. (S4) at least possesses a root(x0, y0) with x0 ∈ [−A,A]. If x0

= −A, we get

A2 ≤ A2+ (y0)2

= (x0)2+ (y0)2

= b2e−2x0= b2e2A < A2,

for somey0 ∈ [−A,A]. However, this is a contradiction, which consequently implies x0 > −A as well as thecompletion of the proof. �

Now, under the condition(C-I): b2e2A − A2 ≥ 0, we divide the following discussions into three cases, viz.,the case of transverse branches for non-real-valuedβ, the case of a concave branch for non-real-valuedβ, andthe case of all branches for real-valuedβ.

A. Along the transverse branches for non-real-valuedβ

Here, we consider the case of all transverse branchesΓp+1 for non-real-valuedβ, i.e., p ∈ Z, p , −1 andα ∈ (0, π). According to PropositionI.5 in SectionI A, we need to discuss the following two sub-cases.

Sub-case A.I “b > 0”: each curveΓp+1 is located in between the straight linesLp+1 : y = α + (p + 1)π andLp : y = α + pπ for p = 0, 2, 4 · · · andp = −3,−5 · · · .

When A ≥ 0, we have 1≥ A/(beA) ≥ 0 and arccosAbeA ∈ [0, π/2]; conversely, whenA < 0, we have

−1 ≤ A/(beA) < 0 and arccosAbeA ∈ (π/2, π]. For y > 0, Eq. (S12) becomes

y =√

b2e2A − A2,

y = α + arccosA

beA+ 2(q− 1)π, q = 1, 2, 3 · · · ,

(S13)

where the second equation follows from the fact that notα−arccos AbeA +2(q−1)π butα+arccos A

beA +2(q−1)πlocates in the interval (α + pπ, α + (p+ 1)π) for p = 0, 2, 4 · · · .

In what follows, we regard Eq. (S13) as a group of equations with respect tob. Thereby, we denote the rootof Eq. (S13) as a function ofq by b11(q). Fig. S-5(a)numerically shows the root for particularly chosenq, αandA. With these settings, we approach the following conclusion.

−5 −A 0 51

1.5

2

2.5

3

3.5

4

4.5

x

y

b11

(1)

b > 0

Γ1

(a)

−6 −A 0 4−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5

b12

(−2)

b > 0Γ

−2

x

y

(b)

FIG. S-5. The transverse branchesΓ1 (a) andΓ−2 (b). Their intersection points with the vertical linex = −A producepositive values, denoted, respectively, byb11(1) andb12(−2). Any root of Eq. (S4) on branchΓ1 (Γ−2, resp.) satisfiesx < −A if b ∈ (0,b11(1)) [b ∈ (0,b12(−2)), resp.]. Here, the parameters areα = 1 andA = 1.5.

Lemma II.3 The function b11(q) is monotonically increasing with respect to q= 1, 2, 3, · · · .

10

proof. In order to show the monotonicity, we investigate the quantity√

b211(q+ 1)e2A − A2 −

√

b211(q)e2A − A2 = α + arccos

Ab11(q+ 1)eA

+ 2qπ −[

α + arccosA

b11(q)eA+ 2(q− 1)π

]

= 2π +

[

arccosA

b11(q+ 1)eA− arccos

Ab11(q)eA

]

,

where the last term is always positive because the range of arccos AbeA is [0, π] as mentioned above. Therefore,

b11(q) < b11(q+ 1) for all q = 1, 2, 3, · · · , which completes the proof. �

Now, according to PropositionI.5, all the roots of Eq. (S4) on the branchesΓp+1 for p = 0, 2, 4 · · · satisfyx < −A if

b ∈+∞⋂

q=1

(0, b11(q)) = (0, b11(1)).

Here, the last equality follows from LemmaII.3, andb11(1) satisfies the equation:√

b211(1)e2A − A2 = α + arccos

Ab11(1)eA

.

Moreover, supposeb12(q) to be the root of the following equation:

y = −√

b2e2A − A2,

y = α − arccosA

beA+ 2(q+ 1)π, q = −2,−3 · · · ,

which follows directly from Eq. (S12) for y < 0. Fig. S-5(b)shows this root for particularly selected pa-rameters. Analogously, we can verify thatthe function b12(q) is monotonically decreasing for q= −2,−3, · · · .Hence,

−∞⋂

q=−2

(0, b12(q)) = (0, b12(−2)),

which, together with PropositionI.5, implies that all the roots of Eq. (S4) on the branchesΓp+1 for p =−3,−5 · · · satisfyx <= A, provided withb ∈ (0, b12(−2)). Here,b12(−2) satisfies the equation

√

b212(−2)e2A − A2 = −α + arccos

Ab12(−2)eA

+ 2π.

Next, we compare the values ofb12(−2) andb11(1). To this end, we investigate the quantity√

b212(−2)e2A − A2 −

√

b211(1)e2A − A2 = −α + arccos

Ab12(−2)eA

+ 2π − α − arccosA

b11(1)eA

=

[

arccosA

b12(−2)eA− arccos

Ab11(1)eA

]

+ 2(π − α),

which is absolutely non-negative for anyα ∈ (0, π/2]. Hence,b12(−2) ≥ b11(1) if α ∈ (0, π/2].Altogether, on the branchesΓp+1 for p = 0, 2, 4 . . . and p = −3,−5 · · · , all the corresponding roots of Eq.

(S4) satisfyx < −A if and only if either one of the following conditions is satisfied:(I-1) α ∈ (0, π/2] andb ∈ (0, b11(1));(I-2) α ∈ (π/2, π) andb ∈ (0,min{b11(1), b12(−2)}).

Sub-case A.II “b < 0”: each curveΓp+1 is located in between the straight linesLp+1 andLp for p = 1, 3, 5 · · ·andp = −2,−4,−6 · · · .

Clearly, whenA ≤ 0, we have 0≤ A/(beA) ≤ 1 and arccosAbeA ∈ [0, π/2]; conversely, whenA > 0, we have

0 > A/(beA) ≥ −1 and arccosAbeA ∈ (π/2, π]. Supposeb21(q) andb22(q) to be, respectively, the roots of the

equations:

y =√

b2e2A − A2 > 0,

y = α − arccosA

beA+ 2(q− 1)π, q = 2, 3 · · ·

11

and

y = −√

b2e2A − A2 < 0,

y = α + arccosA

beA+ 2qπ, q = −1,−2,−3 · · · ,

where the above two equations follow from Eq. (S12) for y > 0 andy < 0, respectively. FiguresS-6(a)&S-6(b)show the rootsb21(q) andb22(q) when parameters are particularly selected.

−6 −A 0 44

4.5

5

5.5

6

6.5

7

7.5

8

b21

(2)

b < 0

Γ2

x

y

(a)

−6 −A 0 4−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

b22

(−1)

b < 0Γ

−1

x

y

(b)

FIG. S-6. The transverse branchesΓ2 (a) andΓ−1 (b). Their intersection points with the vertical linex = −A producenegative values, denoted, respectively, byb21(2) andb22(−1). Any root of Eq. (S4) on branchΓ2 (Γ−1, resp.) satisfiesx < −A if b ∈ (b21(2),0) [b ∈ (b22(−1),0), resp.]. Here, the parameters areα = 1 andA = 1.5.

Direct computations yield thatb21(q) is strictly decreasing for q= 2, 3, · · · , and that b22(q) is strictly in-creasing for q= −1,−2,−3, · · · . Therefore, we have

+∞⋂

q=2

(b21(q), 0)= (b21(2), 0),−∞⋂

q=−1

(b22(q), 0) = (b22(−1), 0),

whereb21(2) andb22(−1) satisfies, respectively, the equations:

√

b221(2)e2A − A2 = α − arccos

Ab21(2)eA

+ 2π,√

b222(−1)e2A − A2 = −α − arccos

Ab22(−1)eA

+ 2π.

In particular, we haveb21(2) ≤ b22(−1) forα ∈ [π/2, π) because ofb < 0 and because

√

b221(2)e2A − A2 −

√

b222(−1)e2A − A2 = α − arccos

Ab21(2)eA

+ 2π −[

−α − arccosA

b22(−1)eA+ 2π

]

= 2α −[

arccosA

b21(2)eA− arccos

Ab22(−1)eA

]

is always non-negative for anyα ∈ [π/2, π).Consequently, on the branchesΓp+1 for p = 1, 3, 5 · · · andp = −2,−4,−6 · · · , all the corresponding roots of

Eq. (S4) satisfyx < −A if and only if either one of the following conditions is satisfied:(II-1) α ∈ (0, π/2) andb ∈ (max{b21(2), b22(−1)}, 0);(II-2) α ∈ [π/2, π) andb ∈ (b22(−1), 0).

B. Along the concave branch for non-real-valuedβ

Secondly, we consider the branchΓ0, which is obtained in SectionI A. As shown in Fig.S-1, Γ0 is located inbetween the asymptotesL0 : y = α andL−1 : y = α − π. The upper half ofΓ0 corresponds to the case ofb < 0,while the lower half to the case ofb > 0. b = 0 corresponds to the origin (x, y) = (0, 0), which has been excludedfrom the following discussion. As illustrated in Eq. (S7), the leftmost vertex ofΓ0 is (− cos2(y∗ − α), y∗) where

12

y = y∗(α) is the unique solution of 2y = sin 2(y− α), and the locus of the vertex is on a circle of radius 1/2 andcenter (−1/2, 0), passing through the origin whenα = π/2.

First,− cos2(y∗ − α) < −A is required in the following discussion; otherwise no part of the concave branchΓ0 locates on the left side of the vertical linex = −A, which implies that the root of Eq. (S4) on Γ0 satisfiesx ≥ −A and then the asymptotical stability of the controlled system (S1) cannot be assured.

As mentioned above, they-axis of the root onΓ0 is allowed to be either positive or negative. Thus, from Eq.(S12), it follows that

y = ±√

b2e2A − A2,

y = α − arccosA

beA∈ (α − π, α).

(S14)

Moreover, according to LemmaI.2, we know that there are two interaction points betweenΓ0 and they-axis.One is the origin and the other is (0, y0) with y0 = α − π/2. Thus, forthe situation ofα ∈ (0, π/2), the vertexof Γ0 lies in the lower half plane (i.e.,y < 0). For this situation, we divide the following discussionsinto threesub-cases.

Sub-case B.I “0< A < cos2(y∗ − α)”: The concave branchΓ0 intersects with the linex = −A at the twopoints which are located on the left side of they-axis. These two points actually correspond to the two valuesof b, denoted byb01 andb02. From PropositionI.5 and Eq. (S14), it follows thatb01 andb02 are positive butdifferent-valued, satisfying the equation:

√b2e2A − A2 = −α + arccos

AbeA

(S15)

with respect tob for α ∈ (0, π/2). Assume thatb01 < b02, as shown in Fig.7(a). Then, according to PropositionI.5, the root of Eq. (S4) onΓ0 satisfiesx < −A if b ∈ (b01, b02).

Note that whenb = b01,02, we haveA/beA > 0, so that arccosAbeA ∈ (0, π/2), which is the same as the range ofα in the present situation. Thus, from Eq. (S15), we haveα ≤ arccos A

beA asb = b01,02. However, it is necessaryto analytically show the existence of two different roots of Eq. (S15) for the present sub-case.

−0.1 −A 0 0.2−1

−0.5

0

0.5

y2

y1

b=0 b < 0

b > 0

Γ0

x

y

b02

b01

(a)

−0.1 0 −A 0.2−1

−0.5

0

0.5

y4

y3

b=0b < 0

b > 0

Γ0

x

y

b03

b04

(b)

FIG. S-7. The intersecting points between the concave branch Γ0 and the vertical linex = −A determine the values ofb01,02

(a) andb03,04 (b), whenα ∈ (0, π/2). Here,b01,02,04 > 0 andb03 < 0. As 0< A < cos2(y∗ − α) (A < 0, resp.), the root of Eq.(S4) onΓ0 satisfiesx < −A if b ∈ (b01,b02) [b ∈ (b03, b04), resp.]. Here, the parameters areα = 1 ∈ (0, π/2), A = 0.025 (a),andA = −0.025 (b).

Lemma II.4 Assume thatα ∈ (0, π/2) is a given number and set A∗ = cos2(y∗ − α). Here, y∗ is the uniquesolution of2y = sin 2(y−α) as obtained in LemmaI.1. For A ∈ (0,A∗), Eq. (S15) with respect to b∈

[

A/eA,+∞)

has two positive but different roots, denoted by bA1 and bA

2 . In particular, with the increase of A from 0 to A∗,

bA1 increases from A/eA to

√A/eA; while bA

2 decreases from+∞ to√

A/eA. Both roots merge into√

A/eA asA = A∗. For A> A∗, Eq. (S15) has no root.

13

proof. As verified in LemmaI.1, the rooty = y∗ of 2y = sin 2(y− α) is uniquely existent. Use the notationη(y)defined in LemmaI.1 and notice that

η(0) = − sin(−2α) = sin 2α > 0,η(−π/2) = −π − sin[2(−π/2− α)] = −π − sin 2α < 0,η′(y) = 2− 2 cos 2(y− α) ≥ 0,

for any givenα ∈ (0, π/2). Therefore, the rooty = y∗ ∈ (−π/2, 0), which givesy∗−α ∈ (−π, 0) and sin(y∗−α) <0. Notice that

0 > 2y∗ = sin 2(y∗ − α) = 2 sin(y∗ − α) cos(y∗ − α),

Then, we obtain

cos(y∗ − α) > 0, y∗ − α ∈ (−π/2, 0). (S16)

As set in the lemma,A∗ = cos2(y∗ − α). This with (S16) implies 0< A∗ < 1. Furthermore, cos(y∗ − α) =√

A∗,which yieldsy∗ = α − arccos

√A∗.

Denote byu(b,A) , u1(b,A) − u2(b,A) where

u1(b,A) =√

b2e2A − A2, u2(b,A) = −α + arccosA

beA, (S17)

andb ≥ A/eA > 0 because of the condition (C-I) as assumed above. Thus, the derivative ofu(b,A) with respectto b is

ub(b,A) =∂u(b,A)∂b

=b2e2A − A

b√

b2e2A − A2, (S18)

so thatub(b,A) = 0 gives a critical valueb = b(A) ,√

A/eA. Moreover,ub(b,A) = 0 coupled withu(b,A) = 0yields an equation with respect toA:

√

A(1− A) = −α + arccos√

A. (S19)

By using the results obtained in (S16), one can verify directly thatA = A∗ = cos2(y∗ −α) is a root of Eq. (S19).Hence,u(b(A∗),A∗) = 0.

Now, we consider the case of 0< A < A∗ < 1. Then,A/eA < b(A), and the derivativeub(b,A) is negativeasb ∈

(

A/eA, b(A))

but positive asb ∈(

b(A),+∞)

. In particular,u(A/eA,A) = α > 0 andu(b,A) → +∞ asb→ +∞; while

u(b(A),A) =√

A(1− A) + α − arccos√

A < 0,

for 0 < A < A∗, becauseu(b(A∗),A∗) = 0 as verified above, and because

du(b(A),A)dA

=1− A√

A− A2> 0, (S20)

for 0 < A < A∗ < 1. Therefore, from the continuity of the functionu(b,A) with respect to the argumentb,it follows that there are two positive zerosbA

1 andbA2 of u(b,A), respectively, in the intervals

(

A/eA, b(A))

and(

b(A),+∞)

if 0 < A < A∗.As a matter of fact, notice the derivative

uA(b(A∗),A∗) =∂u(b,A)∂A

∣

∣

∣

∣

∣

b=b(A∗), A=A∗=

1− A∗√

A∗ − (A∗)2, 0.

Then, according to the Implicit Function Theorem [2], there exists a smooth functionA = A(b) in the neigh-borhood ofb = b(A∗) such thatu(A(b), b) = 0 andA(b(A∗)) = A∗. Moreover,

du(b,A(b))db

=

[

b2e2A(b) − 2A(b) + 1]

A′(b) + be2A(b) − A(b)/b√

b2e2A(b) − A(b)2,

14

which yieldsA′(b(A∗)) = 0 andA′′(b(A∗)) = −2e2A∗/(1 − A∗) < 0. This implies that the graph ofA = A(b)describes the variation of the two zerosbA

1,2 with the change ofA ∈ (0,A∗). Indeed, these two zeros merge into

b = b(A∗) asA = A∗, sou(b,A) has no zero asA > A∗.In fact, we can directly verifyu(b,A) has no zero asA > A∗. On the one hand, forA ≥ 1 > A∗, we have

b(A) ≤ A/eA andub(b,A) ≥ 0 asb ∈[

A/eA,+∞)

. Hence,u(b,A) ≥ u(A/eA,A) = α > 0 asb ∈[

A/eA,+∞)

. Onthe other hand, consider 1> A > A∗. From the monotonicity ofu(b,A) with respect tob as verified above, itfollows thatu(b,A) approaches the minimal value asb = b(A). Note that the minimal valueu(b(A),A) equalsto zero asA = A∗ and thatu(b(A),A) is strictly increasing due to the validity of the inequality in (S20) for1 > A > A∗. Consequently,u(b,A) > 0 for 1 > A > A∗ andb ∈

[

A/eA,+∞)

. Finally, we complete the entireproof. �

FiguresS-8(a)-(c)show the variation of the roots of Eq. (S15) with changingA for particularly selectedparameters. As shown in Fig.S-8(c), there are two positive but different-valued roots of Eq. (S15) for someA ∈ (0,A∗).

0 0.5 1−0.5

0

0.5

1

u

0 0.5 1−0.5

0

0.5

1

0 0.5 1−0.5

0

0.5

1

b

u2u2 u2

u1(a) u1u1 (c)(b)

FIG. S-8. The variation of the roots of Eq. (S15) with the change ofA. Here, the parameters are taken as:α = 1 ∈ (0, π/2),A = 0.1 (a),A = 0.0838 (b), andA = 0.05 (c), so thatA∗ = cos2(y∗ − α) = 0.0838.

Sub-case B.II “A = 0”: The concave branchΓ0 intersects directly with they-axis at the two points whichcorrespond to the two values ofb, viz., 0 andπ/2− α > 0. From PropositionI.5, it follows that the root of Eq.(S4) onΓ0 satisfiesx < −A if b ∈ (0, π/2− α).

Sub-case B.III “A < 0”: The concave branchΓ0 intersects with the linex = −A at the two points which arelocated on the right side of they-axis, as shown in Fig.7(b). These two points correspond to the two values ofb, denoted byb03 andb04. From PropositionI.5 and Eq. (S14), it follows thatb03 < 0 satisfies the equation:

√b2e2A − A2 = α − arccos

AbeA

, (S21)

with respect tob for α ∈ (0, π/2). Meanwhile,b04 > 0 satisfies Eq. (S15) with respect tob for α ∈ (0, π/2).Then, according to PropositionI.5, the root of Eq. (S4) on the branchΓ0 satisfiesx < −A if b ∈ (b03, b04).

Also note that arccosAb03eA ∈ (0, π/2) asA < 0 andb03 < 0; while arccos A

b04eA ∈ (π/2, π) as A < 0 andb04 > 0. With these,A < 0, andα ∈ (0, π/2), the existence of a unique root for each of Eqs. (S21) and (S15)can be verified as follows.

Lemma II.5 For any givenα ∈ (0, π/2) and A< 0, Eq. (S21) with respect to|b| ≥ −A/eA has a unique root,which is negative and denoted by bA

3 ; while, Eq. (S15) with respect to|b| ≥ −A/eA has a unique root, which ispositive and denoted by bA

4 .

proof. Consider the functionu(b,A) as defined in (S17). Clearly, from the derivative (S18), it follows that, forA < 0, u(b,A) as a function ofb is decreasing in

(

−∞,A/eA)

but increasing in(

−A/eA,+∞)

. Moreover,u(b,A)

15

goes to+∞ asb→ ±∞. This, together withu(A/eA,A) = α > 0 andu(−A/eA,A) = α − π < 0, implies that Eq.(S15) has only one positive rootbA

4 ∈(

−A/eA,+∞)

.For investigating the roof of Eq. (S21), we writeu(b,A) = u1(b,A) + u2(b,A), where|b| ≥ −A/eA, A < 0,

andu1,2(b,A) are defined in (S17). Thus, the derivative ofu(b,A) with respect tob can be computed as:

ub(b,A) =∂u(b,A)∂b

=b2e2A

+ A

b√

b2e2A − A2. (S22)

For the case ofA ≤ −1, the derivative in (S22) implies thatu(b,A) as a functionb is decreasing in(

−∞,A/eA)

but increasing in(

−A/eA,+∞)

. Note thatu0(b,A)→ +∞ asb→ ±∞, u(A/eA,A) = −α < 0, andu(−A/eA,A) =−α + π > 0. Therefore, from the continuity and monotonicity ofu(b,A), it follows that Eq. (S15) has only onenegative rootbA

3 ∈(

−∞,A/eA)

.However, for the case of−1 < A < 0, the derivative in (S22) implies thatu(b,A) as a functionb is decreasing,

respectively, in(

−∞,−√−A/eA

)

and(

−A/eA,√−A/eA

)

but increasing, respectively, in(

−√−A/eA,A/eA

)

and(√−A/eA,+∞

)

. Also note thatu(b,A) → +∞ asb → −∞ andu(−√−A/eA,A) ≤ u(A/eA,A) = −α < 0.

Therefore, from the continuity and monotonicity ofu(b,A), it follows that Eq. (S15) has only one negativeroot bA

3 ∈(

−∞,−√−A/eA

)

but no root in(

−√−A/eA,A/eA

)

. In addition,u(√−A/eA,A) =

√−A(1+ A) − α +

arccos(−√−A) , v(A). Note that the derivativev′(A) = (−1 − A)/

√−A− A2 < 0 for −1 < A < 0, and that

v(0) = −α + π/2 > 0. Hence,u(√−A/eA,A) > 0 for A ∈ (−1, 0). Consequently, Eq. (S15) has no root in the

intervals(

−A/eA,√−A/eA

)

and(√−A/eA,+∞

)

. �

Next, we deal withthe situation ofα ∈ (π/2, π). Correspondingly, the vertex of the concave branchΓ0 liesin the upper half plane (i.e.,y > 0). Still, we have three sub-cases pending for discussion.

−0.1 −A 0 0.2−0.4

−0.2

0

0.2

0.4

0.6

0.8

y6

y5

b=0

b < 0

b > 0

Γ0

x

y

b06

b05

(a)

−0.1 0 −A 0.2−0.5

0

0.5

1

y8

y7

b=0

b > 0

b < 0Γ

0

x

y

b08

b07

(b)

FIG. S-9. The intersecting points between the concave branch Γ0 and the linex = −A determine the values ofb05,06 (a) andb07,08 (b), whenα ∈ (π/2, π). Here,b05,06,07 < 0 andb08 > 0. As 0< A < cos2(y∗ − α) (A < 0, resp.), the root of Eq. (S4)on Γ0 satisfiesx < −A if b ∈ (b05,b06) [b ∈ (b07,b08), resp.]. All the parameters areα = 2 ∈ (π/2, π), A = 0.025 (a), andA = −0.025 (b).

Sub-case B.IV “0< A < cos2(y∗ − α)”: The concave branchΓ0 intersects with the linex = −A at the twopoints which are located on the left side of they-axis, which is analogous to Sub-case B.I. As shown in Fig.9(a), these two points actually correspond to the two values ofb, denoted byb05 andb06. Using the methodsimilar to the proof performed for LemmaII.4, we can verify thatb05 andb06 are negative but different-valued,satisfying Eq. (S21) with respect tob for α ∈ (π/2, π). Whenb = b05,06, we haveA/beA < 0, such thatarccos A

beA ∈ (π/2, π). Assumeb05 < b06. Then, according to PropositionI.5, the root of Eq. (S4) onΓ0 satisfiesx < −A if b ∈ (b05, b06).

Sub-case B.V “A = 0”: Analogous to Sub-case B.II,Γ0 intersects with they-axis at the two points whichcorrespond to the two values ofb, viz., 0 andπ/2− α < 0. From PropositionI.5, it follows that the root of Eq.(S4) onΓ0 satisfiesx < −A if b ∈ (π/2− α, 0).

16

Sub-case B.VI “A < 0”: The concave branchΓ0 intersects with the linex = −A at the two points which arelocated on the right side of they-axis, which is analogous to Sub-case B. III. As shown in Fig.9(b), these twopoints correspond to the two values ofb, denoted byb07 andb08. From PropositionI.5 and Eq. (S14), it followsthatb07 < 0 andb08 > 0 satisfy, respectively, Eqs. (S21) & (S15) for α ∈ (π/2, π). The unique root existenceof each of these equations can be verified through a method analogous to the proof of LemmaII.5. Also it isnoted that arccosA

b07eA ∈ (0, π/2) asA < 0 andb07 < 0; while arccos Ab08eA ∈ (π/2, π) asA < 0 andb08 > 0.

Finally, according to PropositionI.5, the root of Eq. (S4) on the branchΓ0 satisfiesx < −A if b ∈ (b07, b08).

Thirdly, we cope withthe critical situation ofα = π/2. Thus, the vertex of the branchΓ0 is located at theorigin (0, 0), so we only need to consider the case ofA < 0. As shown in Fig.S-10, the concave branchΓ0

intersects with the linex = −A at the two points which correspond to the two values ofb, denoted byb09 andb10. From PropositionI.5 and Eq. (S14), it follows thatb09 < 0 andb10 > 0 satisfy, respectively, Eqs. (S21) and(S15) for α = π/2. Consequently, according to PropositionI.5, the root of Eq. (S4) on the branchΓ0 satisfiesx < −A if b ∈ (b09, b10). This conclusion actually could be included in the Sub-case B.VI.

−5 0 −A 10−5

0

5

x

y

Γ0

Γ1

Γ−1

b09

b10

y10

b<0

b>0

b<0y9

b>0

FIG. S-10. The intersecting points between the branchΓ0 and the linex = −A determine the values ofb09,10 for the criticalsituationα = π/2, whereb09 < 0 andb10 > 0. ForA < 0, the root of Eq. (S4) onΓ0 satisfiesx < −A if b ∈ (b09, b10). TheparameterA = −2.5 is used in simulation.

In order to summarize all the conclusions obtained in the aboveSubsections II-A & II-Band provide compactconditions for stability, we first give results on the relations among the ending points of the intervals forb.

Lemma II.6 For any givenα ∈ (0, π/2), max{

b03, b21(2), b22(−1)}

= b03.

proof. First, the functionu1(b,A) as defined in (S17) is decreasing with respect tob in L− , (−∞,−|A|/eA) andincreasing inL+ , (|A|/eA,+∞) due to the unchangeable sign of the derivative

∂u1(b,A)∂b

=be2A

√b2e2A − A2

asb belongs to either one of the intervalsL±. With this monotonicity, we are able to validate the conclusion ofthe lemma.

To this end, we investigate the following two quantities:

u1(b21(2),A) − u1(b03,A) = α − arccosA

b21(2)eA+ 2π −

[

α − arccosA

b03eA

]

= 2π −[

arccosA

b21(2)eA− arccos

Ab03eA

]

> 0

and

u1(b22(−1),A) − u1(b03,A) = −α − arccosA

b22(−1)eA+ 2π −

[

α − arccosA

b03eA

]

= 2(π − α) −[

arccosA

b22(−1)eA− arccos

Ab03eA

]

> 0,

17

where the validity of the two inequalities is due to arccosAbeA ∈ [0, π] andα ∈ (0, π/2). Note thatb03, b21(2),

and b22(−1) are all negative, which have been verified in the precedingargument. Then, from the mono-tonicity of the functionu1(b,A), it follows that b03 > b21(2) andb03 > b22(−1). This therefore validatesmax

{

b03, b21(2), b22(−1)}

= b03. �

Lemma II.7 For any givenα ∈ (π/2, π), min{

b08, b11(1), b12(−2)}

= b08.

proof. We investigate the following two quantities:

u1(b11(1),A) − u1(b08,A) = α + arccosA

b11(1)eA−

[

−α + arccosA

b08eA

]

= 2α +

[

arccosA

b11(1)eA− arccos

Ab08eA

]

< 0

and

u1(b12(−2),A) − u1(b08,A) = −α + arccosA

b12(−2)eA+ 2π −

[

−α + arccosA

b08eA

]

= 2π +

[

arccosA

b12(−2)eA− arccos

Ab08eA

]

< 0,

where the validity of the two inequalities follows from arccos AbeA ∈ [0, π] and the assumed conditionα ∈

(π/2, π). Thus, the monotonicity ofu1(b,A), together with the fact thatb11(1), b12(−2), andb08 are all positive,impliesb11(1) > b08 andb12(−2) > b08. This therefore completes the proof. �

Lemma II.8 For any givenα ∈ (0, π/2), max{

b11(1), b02}

= b11(1) if 0 < A < 1; while max{

b11(1), b04}

=

b11(1) if A < 0.

proof. For the case of 0< A < 1, we consider the function

u3(b) =√

b2e2A − A2 − arccosA

beA,

with |b| ≥ |A|/eA > 0. From the derivativeu′3(b) which is the same as the derivative in (S18), it follows that

u3(b) is decreasing in(

A/eA,√

A/eA)

but increasing in(√

A/eA,+∞)

.

Moreover,u3(A/eA) = 0 andu3(√

A/eA) =√

A(1− A) − arccos√

A. The derivative

du3(√

A/eA)dA

=1− A√

A− A2> 0,

asA ∈ (0, 1). Sinceu3(A/eA)∣

∣

∣

A=1= 0,u3(

√A/eA) is negative asA ∈ (0, 1) . This, together with the monotonicity

of u3(b) and

0 < α = u3(b11(1)) > u3(b02) = −α < 0, (S23)

implies thatb11(1) > b02.As for the case ofA < 0, u3(b) is increasing in

(

−A/eA,+∞)

and the inequalities in (S23) are still valid.Therefore, we concludeb11(1) > b04. �

Lemma II.9 For any givenα ∈ (π/2, π), min{

b22(−1), b05}

= b22(−1) if 0 < A < 1; while min{

b22(−1), b07}

=

b22(−1) if A < 0.

proof. For the case of 0< A < 1, we consider the function

u4(b) =√

b2e2A − A2 + arccosA

beA,

with |b| ≥ |A|/eA > 0. From the derivativeu′4(b) which is the same as the derivative in (S22), it follows that

u4(b) is decreasing in(

−∞,−A/eA)

. Note that bothb22(−1) andb05 are less than−A/eA, and thatu4(b22(−1)) =

18

2π − α > α = u4(b05) because ofα ∈ (π/2, π). Therefore, from the monotonicity ofu4(b), it follows thatb22(−1) < b05.

As for the case ofA < 0, we are to consider two situations. On the one hand, ifA ≤ −1, u4(b) is decreasingin

(

−∞,A/eA)

. This, withu4(b22(−1)) = 2π − α > α = u4(b07), implies thatb22(−1) < b07.

On the other hand, if−1 < A < 0,u4(b) is decreasing in(

−∞,−√−A/eA

)

but increasing in(

−√−A/eA,A/eA

)

.

Moreover,u4(A/eA) = 0 andu4(−√−A/eA) =

√−A(1+ A) − arccos

√−A is negative asA ∈ (−1, 0) because

the derivative

du4(−√−A/eA)

dA=−1− A√−A− A2

< 0,

for A ∈ (−1, 0) andu4(−√−A/eA)

∣

∣

∣

A=−1= 0. Hence, bothb22(−1) andb07 ∈

(

−∞,−√−A/eA

)

. Consequently,from u4(b22(−1)) = 2π − α > α = u4(b07) > 0, it follows thatb22(−1) < b07. �

Altogether, we are able to summarize all the conclusions above, and finally approach a conclusion that, forany non-real-valuedβ, every root of Eq. (S4) on the curveΓ satisfiesx < −A if and only if one of the followingconditions is satisfied:

(1) α ∈ (0, π/2), 0< A < cos2(y∗ − α), andb ∈ (b01, b02);(2) α ∈ (0, π/2), A = 0, andb ∈ (0, π/2− α);(3) α ∈ (0, π/2), A < 0, andb ∈ (b03, b04);(4) α ∈ (π/2, π), 0 < A < cos2(y∗ − α), andb ∈ (b05, b06);(5) α ∈ (π/2, π), A = 0, andb ∈ (π/2− α, 0);(6) α ∈ [π/2, π), A < 0, andb ∈ (b07, b08).

C. Along the branches for real-valuedβ

Finally, we consider the case of real-valuedβ (i.e.,α = 0), in which the elementary root distribution of Eq.(S4) has been discussed in SectionI B.

By an argument similar to those performed in SectionII A and by using PropositionI.7 and Eq. (S12),we can show that on every branchΓp+1 for p ∈ Z with p , −1, the corresponding root of Eq. (S4) satisfiesx < −A if b ∈ (b0(1), b0(2)). Here,b0(1) is negative andb0(2) is positive, as shown in Fig.S-11. They satisfy,respectively, the equations:

√

b20( j)e2A − A2 = arccos

Ab0( j)eA

+ jπ, (S24)

for j = 1, 2. Secondly, still according to PropositionI.7 and Eq. (S12), the concave branchΓ0 surely passesthrough the point (−1, 0). In order to establish the stability condition, we have tolocate the point (−1, 0) on theleft side of the linex = −A, which implies a prerequisiteA < 1. Then, we easily approach a conclusion that onthe branchΓ0 andx-axis, all the corresponding roots of Eq. (S4) satisfyx < −A if A < 1 andb ∈

(

A/eA, b0(0))

.Here,b0(0) is a positive root satisfying Eq. (S24) for j = 0.

Finally, in order to summarize the stability condition for the present case, we perform the computation asfollows:

u1(b0(2),A) − u1(b0(0),A) = arccosA

beA+ 2π − arccos

AbeA= 2π +

[

arccosA

beA− arccos

AbeA

]

> 0,

whereu1(b,A) is defined in (S17). Hence,b0(2) > b0(0). Moreover,b0(1) ≤ A/eA due to the domain of thefunctionu1(b,A). Consequently, for the case of real-valuedβ (i.e.,α = 0), all the corresponding roots of Eq.(S4) satisfyx < −A if and only if A< 1 andb ∈

(

A/eA, b0(0))

.Now, we can summarize all the condition established in the preceding subsections into the NS condition

for the asymptotical stability of the controlled system (S1). In the statement of the following theorems, wesubstitute the original variables of the controlled system(S1) into the conditions obtained above.

Theorem II.10 The asymptotical stability of the controlled system (S1) is assured if and only if one of thefollowing conditions is satisfied:

(1) α = 0, aτ < 1, and−kτe−aτ ∈ (

aτe−aτ, b0(0))

;

19

−10 −5 −1 −A 10

−6

−4

−2

0

2

4

6

x

y

b>1/e

b>1/e

Γ1

Γ−1

Γ0

b (0)0

b (0)0

b (1)0

b (1)0

b=0|

b<00<b<1/e

b<0

b<0

b=1/e

FIG. S-11. The intersecting points betweenΓ0 and the vertical linex = −A determine the positive value ofb0(0) and thenegative value ofb0(1). Every pair of symmetric intersecting points correspond to the same value ofb. Particularly, Forreal-valuedβ (i.e.,α = 0), every root of Eq. (S4) satisfiesx < −A if and only if A < 1 andb ∈

(

A/eA, b0(0))

.

TABLE I. The NS conditions for the asymptotical stability ofthe controlled model (S1).

Three Cases I, II, III α = ψ − dτ (modπ) Nonzero and finite-valuedτ τ = 0 τ→ +∞I: a > 0 andτ < 1/a α ∈ [0, α0) k ∈

(

− B02τ,− B01

τ

)

k ∈(

−∞,− acosψ

)

∅

α ∈ (α1, π) k ∈(

− B06τ,− B05

τ

)

k ∈(

− acosψ ,+∞

)

∅

II: a = 0 α ∈ [0, π/2) k ∈(

α−π/2τ,0

)

k ∈ (−∞,0) ∅

α ∈ (π/2, π) k ∈(

0, α−π/2τ

)

k ∈ (0,+∞) ∅III: a < 0 α ∈ [0, π/2) k ∈

(

− B04τ,− B03

τ

)

k ∈(

−∞,− acosψ

)

k ∈ (a,−a)

α ∈ [π/2, π) k ∈(

− B08τ,− B07

τ

)

k ∈(

− acosψ ,+∞

)

k ∈ (a,−a)

(2) α ∈ (0, π/2), 0 < aτ < cos2(y∗ − α), and−kτe−aτ ∈ (b01, b02);(3) α ∈ (0, π/2), a = 0, and−kτe−aτ ∈ (0, π/2− α);(4) α ∈ (0, π/2], a < 0, and−kτe−aτ ∈ (b03, b04);(5) α ∈ (π/2, π), 0 < aτ < cos2(y∗ − α), and−kτe−aτ ∈ (b05, b06);(6) α ∈ (π/2, π), a = 0, and−kτe−aτ ∈ (π/2− α, 0);(7) α ∈ [π/2, π), a < 0, and−kτe−aτ ∈ (b07, b08).

Note that the conditions (2) and (5) in TheoremII.10 need a determination ofa andτ for a givenα. Some-times in real applications, one needs to a determination ofα for properly givena andτ, since the originalsystem is given in advance butα = ψ − dτ (modπ) containsψ, the phase of the feedback control gain, whichcan be adjusted for a given system. Thus, we rewrite TheoremII.10 equivalently into a more compact butfeasible form in Tab.I. Here, TableI is the same as the table listed in Letter.

More concretely, the asymptotical stability of the controlled system (S1) is assuredif and only if thoseconditions in one of the three cases listed in Tab.I are satisfied. In Tab.I, the conditions for the limitsituations ofτ = 0 andτ → +∞ can be directly obtained from the established conditions byletting τ goto the corresponding situations. For integrity and practical usefulness, we thus include these results also inTab. I. Those parameters used in Tab.I are illustrated as follows:α∗ = α0,1 in Case I are the two solutionssatisfying the equation cos2[y∗(α∗) − α∗] = A, whereA = aτ < 1 andy = y∗(α∗) is the unique solution of2y = sin 2(y− α∗). For Case I andα ∈ [0, α0), bothB = B01,02 > 0 are the only two solutions of the equationH1(B) = −α + H2(B) which follows from Eq. (S15) and settingH1(B) = (B2 − A2)1/2, H2(B) = arccos(A/B),andB = beA. Correspondingly, for Case I andα ∈ (α1, π), B = B05,06 < 0 satisfyH1(B) = α − H2(B) whichfollows from Eq. (S21). In addition, for Case III,B = B03,07 < 0 are the solutions ofH1(B) = α − H2(B),respectively, forα ∈ [0, π/2) andα ∈ [π/2, π), andB = B04,08 > 0 satisfyH1(B) = −α + H2(B), respectively,for α ∈ [0, π/2) andα ∈ [π/2, π). The notation∅ represents an empty set for the choice ofk.

20

III. APPLICATIONS

As an application of the NS conditions established above, wenow consider the following examples: one isthe oscillation suppression in the FitzHugh-Nagumo model,and the other is the oscillation death and synchro-nization in complex networks.

A. Oscillation suppression in the FitzHugh-Nagumo model

First, we consider the FitzHugh-Nagumo model (FHNM) which usually is regarded as a simplification ofthe Hodgkin-Huxley model of spike generation in squid giantaxons [7, 8]. It thus is used for describing nervepulses and sometimes for modeling spiral waves in a two-dimensional excitable medium. The typical FHNMreads:

v = v− v3

3− w+ I ,

w = ǫ(v+ ζ − δw),(S25)

wherev represents the membrane potential,w is a recovery variable,I standards for the magnitude of the inputstimulus current, andǫ, ζ andδ are three nonnegative constant parameters withǫ > 0. With these settings andafter a few computations, we directly approach the following conclusion.

Proposition III.1 The equilibrium, denoted by E(v0,w0), of model (S25) becomes:

(v0,w0) =(

−ζ, ζ3/3− ζ + I)

, (v0,w0) =(

(3I − 3ζ)13 , (3I − 3ζ)

13 + ζ

)

when, respectively,δ = 0 andδ = 1. Generally, whenδ , 0, model (S25) has a unique equilibrium if and onlyif D , 9(I − ζ/δ)2 − 4(1− 1/δ)3 > 0.

Once the equilibrium is obtained, it is natural to have the following results on the oscillations generated bythe model (S25) through a route of Hopf bifurcation [9].

Proposition III.2 Assume thatδ2ǫ < 1 andδǫ < 1, and set s= (1−δǫ)1/2. Then model (S25) undergoes a Hopfbifurcation at the equilibrium E when I is around either I1 , (s+ ζ)/δ+ s3/3− s or I2 , (−s+ ζ)/δ− s3/3+ s.Correspondingly, the first Lyapunov coefficients for both I1,2 are the same, i.e.

l1 =δ2[−(δǫ − 1)2 + 1− ǫ]

8(1− δ2ǫ)2.

Consequently, l1 < 0 asδ2ǫ > 2δ− 1, which implies that a stable oscillation is generated around I1,2 through aroute of supercritical Hopf bifurcation.

Actually, according to the bifurcation theory [9] and through a tedious computation, we around the equilib-rium E obtain the FHNM’s normal form which reads

z= (a+ id)z+ c1|z|2z+O(|z|4), (S26)

wherez= x+ iy,

a =−v2

0 − δǫ + 1

2, d =

√

−(v20 − δǫ − 1)2 + 4ǫ

2,

Re{c1} = −ρ2

8d4+

v20ρ

2(−aρ2+ ad2

+ 4d2ρ)

16(a2 + d2)d4+

av20ρ

2(ρ2+ d2)

(a2 + 9d2)d4,

and

Im{c1} =ρ3

8d3+

2v20ρ

2(d2 − ρ2 − aρ)

(a2 + d2)d3−

av20ρ

2(ρ2+ d2)

(a2 + 9d2)d3.

21

Particularly, Re{c1} = l1 whenI = I1 or I = I2, as concluded in PropositionIII.2 above. Here,ρ = (−v20 + δǫ +

1)/2 and the frequency of the oscillation can be represented by

f ,12π

[

d− a · Im{c1}Re{c1}

]

.

Moreover, since we only focus on the local asymptotical stability of the controlled FHNM, it is sufficient forus to consider a linearization of the normal form (S26). In fact, the linearization of the normal form can beobtained directly by using the linear transformation (x, y)⊤ = T (v − v0,w − w0)⊤ to model (S25), where thematrix

T =

1 − 1ρ

d2ρ 0

.

Hence, we obtain the controlled FHNM in a matrix form:

x(t)

y(t)

=

a −d

d a

x(t)

y(t)

+

m −n

n m

x(t − τ)y(t − τ)

. (S27)

Clearly, this equation is a particular case of system (S1). Moreover,f , a andd obtained above can be regardedas functions of the input stimulus currentI . Figures12(a)& 12(b)shows these functions when the parametersare set asζ = 0.02, δ = 0.004, ǫ = 200. In particular,a = a(I ) is slightly above zero forI ∈ (90, I1) withI1 = 116.386 and then model (S25) can produce stable oscillations. As shown in Fig.12(a), despite of thedifferent values,f andd have the same variation tendency asI ∈ (90, I1). This implies that in such an intervalof the input current,d almost determines the frequency variation of the produced stable oscillations. Moreover,in model (S27), the control gain matrix in front of the feedback control term with time delays has the followingcharacters: (1)n , 0 corresponds to an asymmetric control, butn = 0 to a symmetric (diagonal) control. (2)m< 0 corresponds to a stable control; conversely,m> 0 invites an unstable control.

0 100 200 300 400 5000

5.0

10.0

15.014.1

2.2

I

f(I)

d(I)

(a)

FIG. S-12. The plots of the functionsf = f (I ), a = a(I ) andd = d(I ) as the input stimulus currentI ∈ (0,500) (a) andI ∈ (90, I1) (b) with I1 = 116.386. (c) The FHNM (S25) cannot be stabilized by a symmetric control with time delayforanym. The colors in theI -m plane represent the positive exponential rates of the divergence of the trajectories generatedby the controlled model (S28). Here,τ = 0.1 andI ∈ (90,120).

Using the inverse transformation ofT to model (S27) yields the controlled FHNM in a form of the originalvariables as follows:

v

w

=

v− v3

3 − w+ I

ǫ(v+ ζ − δw)

+

m+ ρ

dn − 1dn

d2+ρ2

d n m− ρ

dn

v(t − τ) − v0

w(t − τ) − w0

. (S28)

Clearly, the control gain matrix here still preserves the corresponding characters (1) and (2) as illustrated abovefor the gain matrix of model (S27).

Finally, according to the NS condition of Case I listed in Tab. I, we can conclude that, for anym, thesymmetric control is useless to suppress the oscillation when τ = 0.1, ζ = 0.02, δ = 0.004, ǫ = 200, andI ∈ (90, I1). This is becaused(I ) ∈ Jp , ( pπ−α1

τ,

pπ−α0

τ) for I ∈ (90, I1) and for the parameters and time delay

set above. The corresponding numerical validation is shownin Figs. 12(c)& 13(b). However, if we choosea suitable asymmetric control and time delay, it is possibleto achieve a successful OS for the FHNM. This isverified by numerical simulations in Figs.13(a)& 13(c).

22

−5

0

5

v

0 5 100.420

0.422

0.425

t

v

(b)

(c)

FIG. S-13. (a) The FHNM (S25) can be stabilized by the asymmetric control with time delay. The colors in them-n planerepresent the exponential rates of the divergence or convergence of the trajectories generated by the controlled FHNM(S28). Here,τ = 0.1 andI = 110. (b) An unsuccessful OS in the controlled FHNM (S28) as a symmetric and stable gainmatrix (m= −4) is used. (c) A successful OS as an unstable and asymmetric gain matrix (m= 4, n = −5) is used.

B. Oscillation death and synchronization in complex networks

We consider a complex system ofN identical nonlinear dynamical units connected through time delayedcouplings:

xi(t) = f (xi(t)) +N

∑

j=1

ci j x j(t − τ), i = 1, · · · ,N, (S29)

wherexi ∈ Rq represents the state variable of each unit andx0 = 0 is assumed as the equilibrium of each unit.In addition,{ci j }N×N is a diagonalizable coupling matrix whose row sum is supposed to bem for eachi. Here,m could be either zero or non-zero. In particular,m = 0 corresponds to the conventional couplings which arenon-invasive to the dynamics of uncoupled unit within the synchronization manifold (SM).

Oscillation death.By using the standard technique developed in [10], we conclude directly that a study ofthe oscillation death (OD) in system (S29) is equivalent to an investigation of the asymptotical stability of agroup ofN variational equations (VEs):

ξσ(t) = D f (0)ξσ(t) + λσξσ(t − τ), σ = 1, · · · ,N, (S30)

where eachξσ is aq-dimensional variable along different direction. It is easy to see thatλ1 = m is the eigenvalueof {ci j }N×N associated to perturbations atx0 within the SM andλ2,··· ,N are the transversal eigenvalues of{ci j }N×N.

More concretely, we suppose that each uncoupled unit satisfies the normal form of the supercritical-Hopf-bifurcation, so thatq = 2,D f (0) = [a, − d; d, a], and the coupling matrix in front of the delay term can bewritten as:

kcosψ −ksinψ

ksinψ kcosψ

=

λσ 0

0 λσ

,

whered determines the oscillation frequency,f = (d+γa)/(2π), of the uncoupled unit, and the coupling matrixcould be regarded as a symmetric gain matrix for Eq. (S30) due that it is in the diagonal form. To investigatethe stability, we only need to analyze the characteristic equations of Eq. (S30) for σ = 1, · · · ,N.

For the first VE [i.e., Eq. (S30) asσ = 1] , the characteristic equation of is the same as Eq. (S2) ask = λ1 = mandψ = 0. Thus, according to the NS conditions established in Tab.I for the symmetric control (ψ = 0) withtime delay, despiteA = aτ < 1, for anym the first VE is unstable and so ODcannotbe observed physicallywhen the oscillation frequencyf of the uncoupled unit is in the intervals

Jp =

( pπ − α1 + γA2πτ

,pπ − α0 + γA

2πτ

)

.

However, the OD can be observed only when the frequency of theuncoupled unit satisfiesf < Jp. Moreover,the stability of the transversal VEs [i.e., Eq. (S30) asσ = 2, · · · ,N], should be guaranteed through adjusting

23

the eigenvaluesλσ of {ci j } in light of the NS conditions. For some particular frequencies, the transversaleigenvalues might satisfy Re{λσ} > 0, which invites unstable gain matrix.

Oscillation synchronization.We also can discuss periodic orbit synchronization in system (S29) by usingthe NS condition established above. For a clear illustration, we suppose the uncoupled unit to be a three-dimensional system having a stable periodic orbitS(t) with periodT. The Floquet exponents ofS(t) thus areµ1 = 0 andµ2,3 = a± id (a < 0) [4]. We further setλ1 = m= 0. It is easy to see thatS(t) is within the SM, andthat the other transversal VEs become:

ξσ(t) = D f (S(t))ξσ(t) + λσξσ(t − τ), (S31)

which is different from Eq. (S30) since the synchronization of the periodic orbitS(t) is considered here. Now,according to the Floquet theory [4], for eachσ and corresponding linear system

ξσ(t) = D f (S(t))ξσ(t), (S32)

there exists a time-varying transformationξσ = P(t)ησ such that system (S32) can be equivalently transformedinto a linear system

ησ(t) = Rησ(t).

Here,P(t) = Φ(t)e−Rt is periodic with a periodT, Φ(t) is a fundamental solution matrix of the linear system(S32), andR is a nonsingular constant matrix, having eigenvaluesµ1,2,3 and satisfyingΦ(t + T) = Φ(t)eRT.Then,P−1(t) exists and also is periodic with a periodT. Now, we still use the transformationξσ = P(t)ησ toevery transversal VE (S31), and thus obtain

ησ(t) = Rησ(t) + λσP−1(t)P(t − τ)ησ(t − τ),

which further becomes:

ησ(t) = Rησ(t) + λσησ(t − τ), (S33)

if τ = pT with p ∈ N, because of the periodicity ofP(t). Also we can find a nonsingular matrix to diagonalizethe matrixR. Hence, the above transformed VE (S33) reads:

ησ(t) = diag{µ1, µ2, µ3}ησ(t) + λσησ(t − τ), (S34)

where, for simplicity, the notationησ for the state variable is preserved although an additional transformationobtained from diagonalizing the matrixR is used on VE (S33).

Now, a study of the synchronization of the periodic orbit in system (S29) becomes an investigation of theasymptotical stability of Eq. (S34) for σ = 2, · · · ,N andτ = pT. For a givenσ, there actually exist threecharacteristic equations of Eq. (S34): one is

λ − λσe−λτ = 0,

and the other two are

λ − µ j − λσe−λτ = 0, j = 2, 3.

We thus can study the root distribution of the above three characteristic equations for everyσ, and derive someconditions for stability and synchronization according tothe NS conditions for Case II (a = 0) and Case III(a < 0) established in Tab.I. For example, if allλ2,··· ,N take real values,−dτ (modπ) ∈ [0, π2), andτ = pT forp ∈ N, one possible condition for synchronization of periodic orbit becomesλ2,··· ,N ∈ (− π

2τ , 0)∩ (− B04τ,− B03

τ).

[1] J. Hale,Theory of Functional Differential Equations(Springer-Verlag, London, 2003).[2] C. Robinson,Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (CRC Press, London, 1998).[3] W. Walter,Ordinary Differential Equations(Springer-Verlag, New York, 1998).[4] G. Teschl,Ordinary Differential Equations and Dynamical Systems(Amer. Math. Soc., Providence, 2012).[5] M.V. DeFazio and M.E. Muldoon, On the zeros of a transcendental function, preprint (2005).[6] K. Yoshida,Functional Analysis(Springer-Verlag, New York, 1980).

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[7] C. Rocsoreanu, A. Georgescu, and N. Giurgiteanu,The FitzHugh-Nagumo model: Bifurcation and Dynamics(KluwerAcademic Publishers, Dordrecht, 2000).

[8] E.M. Izhikevich,Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting(MIT Press, Cam-bridge, 2007).

[9] Y.A. Kuznetsov,Elements of Applied Bifurcation Theory(Spring-Verlag, New York, 1998).[10] L.M. Pecora and T.L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett.80, 2109

(1998).