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Synchronization limit of weakly forced nonlinear oscillators

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Journal of Physics A: Mathematical and Theoretical
J. Phys. A: Math. Theor. 47 (2014) 402002 (10pp)

doi:10.1088/1751-8113/47/40/402002

Fast Track Communication

Synchronization limit of weakly forced
nonlinear oscillators
Hisa-Aki Tanaka
Graduate School of Informatics and Engineering, The University ofElectro-Communications, Chofu, Tokyo 182–8585, Japan
E-mail: htanaka@uec.ac.jp
Received 13 May 2014, revised 22 July 2014
Accepted for publication 25 July 2014
Published 23 September 2014
Abstract

Nonlinear oscillators exhibit synchronization (injection-locking) to external
periodic forcings, which underlies the mutual synchronization in networks ofnonlinear oscillators. Despite its history of synchronization and the practicalimportance of injection-locking to date, there are many important open problems of an efﬁcient injection-locking for a given oscillator. In this work, I
elucidate a hidden mechanism governing the synchronization limit under weak
forcings, which is related to a widely known inequality; Hölderʼs inequality.
This mechanism enables us to understand how and why the efﬁcient injectionlocking is realized; a general theory of synchronization limit is constructed
where the maximization of the synchronization range or the stability of synchronization for general forcings including pulse trains, and a fundamental
limit of general m : n phase locking, are clariﬁed systematically. These synchronization limits and their utility are systematically veriﬁed in the Hodgkin–Huxley neuron model as an example.
S Online supplementary data available from stacks.iop.org/jpa/47/402002/
mmediaKeywords: synchronization, entrainment, injection-locking, fundamental limit,
Arnold tongue, optimization, Hölderʼs inequality
PACS numbers: 05.45.Xt, 05.45.-a, 84.35.+i(Some ﬁgures may appear in colour only in the online journal)

Content from this work may be used under the terms of the Creative Commons
Attribution 3.0 licence. Any further distribution of this work must maintain attribution to
the author(s) and the title of the work, journal citation and DOI.
1751-8113/14/402002+10$33.00 © 2014 IOP Publishing Ltd Printed in the UK

1

J. Phys. A: Math. Theor. 47 (2014) 402002

Figure 1. Best forcings revealed by the mechanism of synchronization limits whenmaximizing the range of entrainment: (a) power-reduced forcing in P1, (b) areareduced forcing in P2, and (c) magnitude-reduced forcing in P2, obtained for (d) agiven phase response function Z (taken from the Hodgkin–Huxley phase model [17] asan example). fopt, p ( p = 1, 2, and ∞) here is determined algorithmically in themain text.

1. Introduction
Entrainment, which adjusts the frequencies of oscillators to that of an external forcing (signal)
above a critical forcing amplitude, is a fundamental phenomenon of wide interest with a long
history and a large variety of applications [1–5]. Generally, the ratio of these two frequencies
is locked to m : n in the entrainment, which is called m : n frequency locking. Likewise, if theoscillation phase of the oscillator (= θ) and that of the external forcing (= θext ) satisfy nθ − mθext = const, it is called m : n phase locking. And, 1 : 1 phase locking is termed synchronization [4]. As opposed to conventional single-oscillator entrainment, entrainment has
signiﬁcance even in coupled oscillatory elements, thanks to recent studies on networks ofcoupled oscillators [6–9]. Simultaneously, in many branches of science and engineering,
engineering entrainment (injection-locking) with an efﬁcient forcing (injection signal) has
become more important, and methods for efﬁcient entrainment have been developed in recent
years [10–17], reﬂecting advances in experimental techniques for observing and manipulatingsuch oscillators (see [16–20] for instance).
Despite its history, and the advances made in and the wide utility of synchronization todate, it has been open problems of practical and theoretical signiﬁcance to realize the synchronization limit. Part of our results realizing the synchronization limit is schematicallyillustrated as in ﬁgure 1, which solve the three basic open problems, posed in P1, P2, and P3
below.
P1: What sort of mechanism determines the best (i.e., global optimal) power-reduced(average square of its waveform is bounded) forcing that produces the maximum entrainment
range or the most stable synchronization? And are the optimal power-reduced forcings
obtained in [16, 17] really the best ones if a more general class (function space) of forcings is
considered?
2

J. Phys. A: Math. Theor. 47 (2014) 402002

P2: In conjunction with P1, what determines the best area-reduced (total absolute valueof its waveform is bounded) forcing or magnitude-reduced (its maximum amplitude isbounded) forcing?
P3: Do the answers to P1 and P2 regarding 1 : 1 phase locking remain valid for generalm : n phase locking? If so, how is the best forcing for m : n phase locking related to that for1 : 1 phase locking? Or, if not, what limits an ideal (most efﬁcient) forcing for m : n phaselocking?
Recent studies have approached these problems, from different formulations using calculus of variations ([16] for optimizing the locking range, [17] for optimizing the stability ofphase locking, and [21] for both of them in m : n phase locking, for a power-reduced case).
However, regarding P1, these conventional studies only suggest the existence of a synchronization limit by ﬁnding (possibly local optimal) forcings in each particular situation, andregarding P2 and P3, methods used in these conventional studies are not applicable, andhence a fundamental limit of entrainability has not been clariﬁed. Toward this end, in thispaper, we ﬁnd an underlying mechanism in the above three problems leads us to a uniﬁed,
global view of synchronization limits and their constructions.
2. Entrainment modeled by the phase equation
Here we introduce the well-known phase equation for the weakly forced nonlinear oscillators[1–5]. The entrainment process of a limit-cycle oscillator in the weak forcing limit can bemodeled by ψ˙ = ω + ϵZ (ψ ) f (Ωt ), where ψ is the phase variable of the oscillator(ψ ∈ [ − π , π ] ≡ S ), Z is the phase response (sensitivity) function, and ω and Ω are the natural
frequency of the oscillator and the frequency of the weak forcing ϵf (Ωt ), respectively,
following the notation in [2]. In general, m : n phase locking occurs when mω ≈ Ωn is satisﬁedfor positive relatively prime integers m and n. In this situation, the above equation is further
simpliﬁed by the method of averaging (after setting ϵ to 1) to the following phase equationdϕ= Δω + Γm n (ϕ),
dt

(1)
m

m

where ϕ and Δω satisfy ϕ = ψ − n Ωt and Δω = ω − n Ω , respectively, and the inter1

T

action function Γm n (ϕ) is determined by f and Z as Γm n (ϕ) = T ∫ Z
01
2π( Ωt + ϕ) f (Ωt )dt ≡mn

Z (mθ + ϕ) f (nθ ) , in which T = 2πn (n times the natural period of the oscillator), θ ∈Ωπ[ − π , π ] represents Ωt , and 〈 · 〉 denotes the integration over its period 2π : 〈 · 〉 ≡ ∫−π · dθ .
nWhen considering the case of m = n = 1, i.e., 1 : 1 entrainment, we will abbreviate Γm n as Γ,for simplicity.
Now, we deﬁne the synchronizability of the oscillator by equation (1), which is composed of the forcing f and the phase sensitivity Z, and we introduce some practical constraints
on f and Z.
We ﬁrst consider a general class of periodic functions f (θ ) as the weak forcing, namely,
those satisfying the following constraints

f p≡

1

f (θ ) p p = M and1f (θ ) = 0,2π(2)in which both p and M are positive constants; here we assume f ∈ L p (S ), namely, that f is an
Lp-function on S ≡ [ − π , π ]. Henceforth, we assume p ⩾ 1, due to the following physicalinterpretation of the constraints (2). First, for p = 2, || f || p = M becomes 〈f 2 〉 = M 2 , i.e., the3

J. Phys. A: Math. Theor. 47 (2014) 402002

average square of f (the power) is ﬁxed at M2, which is the case considered in [16, 17, 21],and this case corresponds to the power-reduced forcing in P11. (An example is shown inﬁgure 1(a).) For p = 1, || f || p = M corresponds to the area-reduced forcing in P2:| f (θ )| = M . (See ﬁgure 1(b).). On the other hand, the case of p = ∞ implies themagnitude-reduced forcing in P2 (ﬁgure 1(c)), because || f ||∞ = M : | f (θ )| ⩽ M for almost
every θ ∈ S , since || f ||∞ is the essential supremum of | f (θ )|. Thus, the constraints (2)continuously cover various situations in a natural way. Besides || f || p = M , another constraint
1〈f (θ ) 〉 = 0 in equation (2), i.e., a charge-balance constraint [22, 23], is introduced, because
2π
it is required in practical situations where total injection (injected charge) should be 0.
On the other hand, for the phase sensitivity Z (θ ), we assume a general class of Z being
twice differentiable for the case of 1 < p ⩽ ∞, and Z being locally Lipschitz continuous for
p = 1, which is required to prove existence of the optimal forcing; detailed information is
given in appendix A of the supplementary material (available at stacks.iop.org/jpa/47/402002/
mmedia). We note that these (mild) assumptions are normally satisﬁed for the oscillators inreal environments.
Phase locking occurs when the phase difference is locked, i.e., dϕ dt = Δω + Γ (ϕ) = 0.
The range of frequency difference Δω, where the solution for a stable steady state exists for ϕ,deﬁnes the locking range R [f ] for a certain ﬁxed forcing waveform [4]. Therefore, thelocking range is the difference between the maximum (at ϕ = ϕ+) and minimum (at ϕ = ϕ−)values of Γ (ϕ) where the phase locking is maintained. R [f ] is thus given by1
Γ (ϕ+) − Γ (ϕ−) = 2π 〈f (θ )[Z (θ + ϕ+) − Z (θ + ϕ−)] 〉 [16].
3. Fundamental limits of synchronization
Now, we formulate the optimal synchronization problem: the global optimal forcing waveform maximizes the locking range R [f ] under the constraints (2), which gives the maximum
width of the Arnold tongue for the weak forcing, as in ﬁgure 1. If we assume f to satisfy
1|| f || p = M , then for maximizing R [f ] under the constraint 2π 〈f (θ ) 〉 = 0, the functionalλJ [f ] ≡ R [f ] + 2π f (θ ) is introduced, where λ is a Lagrange multiplier. Moreover, J [f ] isrewritten as the following inner product of f and g

11f (θ ) ⎡⎣ Z¯ (θ ) + λ⎤⎦ ≡f (θ ) g (θ ) ,
(3)
2π
2π
where g (θ ) = Z¯ (θ ) + λ , Z¯ (θ ) ≡ Z (θ + Δϕ) − Z (θ ) and Δϕ ≡ ϕ+ − ϕ−, after moving to the
new coordinate: θ + ϕ− → θ . On the other hand, when maximizing the (linear) stability of the1phase locking, R [f ] is simply replaced with S [f ] ≡ − 2π 〈f (θ ) Z ′ (θ ) 〉 (≈ − Γ ′(ϕ ) where ϕ**denotes a stable ﬁxed point for equation (1). Note this ϕ is set to 0 on the new coordinate.) if*
Δω = 0 in (1) [17]. Thus, for the case of maximizing the stability S [f ], the same arguments
for maximizing the locking range R [f ] are possible simply by replacing g (θ ) = Z¯ (θ ) + λ tog (θ ) = − Z ′ (θ ) in equation (3). Henceforth, we restrict our argument to the case of
maximizing locking-range in the proof below, for the sake of simplicity. For the physicalsigniﬁcance of R [f ] and S [f ], see [16, 17] respectively.
The optimization of J [f ] is identical to the optimization of R [f ] under the constraints (2),and hereafter we denote the global optimal forcing for a given p as fopt, p . For this optimization problem, local optimal forcings can be captured by using the calculus of variations,
J [f ] =1

Besides the case for p = 2, the case for p > 2 can be useful for engineering nonlinear systems with generalizedenergy functions.
4J. Phys. A: Math. Theor. 47 (2014) 402002

such as the Euler–Lagrange equation for p > 1 (or the bang–bang principle [24] for p = ∞).
But, this approach has a limitation: it is intrinsically local and heuristic, and its result lacks
global information. Namely, we cannot understand ‘how’ and ‘when’ the global optimal
forcing is realized. Furthermore, it is impossible using this approach to show that a certain
arbitrarily tall pair of pulses realize the entrainment limit for p = 1 in P2, as explained later.
Hence, by using only the calculus of variations, it is hard or impossible to answer questions
P1, P2, and P3 regarding the physical limit of synchronizability.
However, if we realize that equation (3) with the constraint || f || p = M corresponds toHölderʼs inequality, || fg || 1 ⩽ || f || p|| g || q in which 1 ⩽ p, q ⩽ ∞ and p−1 + q−1 = 1, thenanswers to the basic questions P1, P2, and P3 are systematically obtained as theorems andtheir proofs are obtained, as follows.
3.1. 1 : 1 phase locking for 1 < p < ∞First, regarding P1 for 1 < p < ∞, the question is answered by direct use of the equality
condition of Hölderʼs inequality: ∥ fg ∥1 = ∥ f ∥p ∥ g ∥q holds if and only if there exist constants r and s, not both 0, such that r |f (θ )| p = s |g (θ )|q for almost all θ ∈ S [26]. Having thisequality condition in mind, a candidate of the optimal f* for J [f ] under the constraints (2) is
1
given as f (θ ) = Mσ (θ )(| g (θ )| ∥ g ∥q ) p − 1 with σ (θ ) being any function having either ±1*
values for θ ∈ S , by assuming the second equality holds in the following general relationship:
2πJ [f ] = 〈fg〉 ⩽ 〈 | fg | 〉 = ∥ fg ∥1 ⩽ ∥ f ∥p ∥ g ∥q = M ∥ g ∥q . Further, by assuming the ﬁrst
equality in the above relationship, we can narrow the above f* uniquely to

(

)

1

f (θ ) = M sgn [g (θ )] g (θ ) ∥ g ∥q p − 1 ,*(4)where sgn is the signum function deﬁned as sgn (x ) = − 1 (for x < 0), 1 (for x > 0),
0 (for x = 0).
Here we call this f* as an ideal forcing since it realizes a possible ideal entrainment of the
maximum locking range, and we now assume such an f* to exist (which is later veriﬁed fromequation (5)). Under this assumption, for any given Z, fg ( = 2πJ [f ]) is given as
1〈f g〉 = ∥ f ∥p ∥ g ∥q = M 〈 | Z¯ (θ ) + λ|q 〉 q ( = 2πJ [f ]). Then, the ideal locking range***
J [f ] is a function of Δϕ and λ, for a given Z and p. In order to maximize J [f ], the**function | Z¯ (θ ) + λ | q should be maximized by tuning the two parameters Δϕ and λunder the constraints (2). For this purpose, we deﬁne the following functions:1
F (Δϕ, λ ) ≡ |Z¯ (θ ) + λ|q
and G (Δϕ, λ ) ≡ 〈 sgn ⎡⎣ Z¯ (θ ) + λ ⎤⎦ | Z¯ (θ ) + λ | p − 1〉. Then
G (Δϕ, λ ) = 0 is obtained from the constraint 〈f (θ ) 〉 = 0 after plugging equation (4) into it.
Finally, for maximizing F (Δϕ, λ ) under the constraint G (Δϕ, λ ) = 0 , the function
H (Δϕ, λ ) ≡ F (Δϕ, λ ) + μG (Δϕ, λ ) is introduced, where μ is a Lagrange multiplier. Thus,
the optimal entrainment problem is reduced down to the ﬁnite-dimensional optimization of
H (Δϕ, λ ), and the above argument clariﬁes the mechanism of how optimal forcings are
realized.
Straightforward calculations show that the optimal solutions (Δϕ , λ* ) to H (Δϕ, λ ) are*determined from the following equations
1∂H ∂Δϕ = α sgn [g (θ )] g (θ ) p − 1 Z ′ (θ + Δϕ)

5

= 0,

(5a)

J. Phys. A: Math. Theor. 47 (2014) 402002

Figure 2. Overview of all optimal forcings for various p ∈ [1, ∞] obtained for the
Hodgkin–Huxley (HH) neuron phase model [17]. Black, red, and blue curves represent the
global optimal, the second optimal, and Z, respectively. Panel (a), (b), respectively, show
1 : 1 and 1 : 2 phase locking optimal forcings for the HH neuron model [17], having theassociated Z (θ ) = 0.176116 + 0.371736 cos θ − 0.740283 sin θ − 0.819478 cos 2θ
+ 0.00225226 sin 2θ + 0.181875 cos 3θ + 0.403816 sin 3θ + 0.111446 cos 4θ
− 0.0892503 sin 4θ −0.0127103 cos 5θ − 0.0165083 sin 5θ as shown in ﬁgure 2(c).

1∂H ∂λ = α sgn [g (θ )] g (θ ) p − 1

(

)

= 0,

2
 (H ) = 13 α12− 13 22 > 0,

(5b)

(5c)

p
where α = p − 1 , g (θ ) = Z¯ (θ ) + λ , 1 < p < ∞, and  (H ) represents the bordered Hessian
matrix of H; detailed information on  (H ) is given in appendix B of the supplementarymaterial (available at stacks.iop.org/jpa/47/402002/mmedia). Note, for every (Δϕ , λ* ), the*associated f* indeed maximizes the associated Γ at ϕ+ and minimizes Γ at ϕ− bystraightforward calculation, although its detail is omitted here.
To determine (Δϕ , λ* ) of the ideal forcing f* in equation (4), we have numerically*solved equations (5a) and (5b), and checked whether the obtained (Δϕ, λ ) satisﬁes
equation (5c); the results for the example in ﬁgure 2 are listed in appendix C of the supplementary material (available at stacks.iop.org/jpa/47/402002/mmedia). Since all possible(Δϕ , λ* ) can be obtained numerically, and from the above argument concerning the equality*
condition and the associated equation (4), all global or local optimal forcings are captured in
L p (s ) 2. Among them, the global optimal (i.e., the best) forcing fopt, p is identiﬁed, simply bycomparing the associated locking ranges R [f ], as shown in ﬁgures 2(a) and (b) for
p = 1, 1.01, 2, 5, ∞ for Z (θ ) shown in ﬁgure 2(c), respectively. The above steps constitute the algorithm for realizing the global optimal forcing, i.e., the fundamental limit of
injection-locking3. Thus, the answer to P1 has been obtained.

2
These optimal forcings are obtained in L p (S ). This implies that these optimals belong to a broader class offunctions, compared with the one considered in the calculus of variations. More precisely, in the calculus ofvariations, we usually assume the function space to consist of piecewise-smooth functions (or absolutely continuousfunctions, at best), which is a subset of Lp. Namely, the result here is stronger than the one obtained by theconventional calculus of variations.
3Note, for the special case of p = 2, equation (5b) gives λ = 0 , equation (5a) becomes 〈Z¯ (θ ) Z ′ (θ + Δϕ) 〉 = 0 ,equation (5c) becomes 〈Z¯ (θ ) Z ″ (θ + Δϕ) 〉 < 0 , and hence the result for the power-reduced forcing in [16] isnaturally recovered. This implies the optimal forcing obtained in [16] (as well as in [17]) is nothing other than the
global optimal forcing in the more general function space of forcings, i.e., L 2 (S ) (cf footnote 2).

6

J. Phys. A: Math. Theor. 47 (2014) 402002

3.2. 1 : 1 phase locking for p = 1 and p ¼ ∞Next, regarding P2, the question is answered by utilizing the cases of p = 1 and p = ∞ in
Hölderʼs inequality. This can be done in a rigorous way [25], but due to space limitations, wehere present a more intuitive explanation; taking the limits p → ∞ and p → 1 in equation (4),results respectively inf (θ ) → M sgn [g (θ )] ,*pointwise for any θ ∈ S ,⎧ 0,pointwise for θ ≠ θ *,⎪f (θ ) → ⎨ ∞ , for θ = θ * with g ( θ * ) > 0,*⎪
⎩ −∞ , for θ = θ * with g ( θ * ) < 0,

(6a)

(6b)

where θ* represents a maximal point of | g (θ )| in S. A derivation of equations (6) is given inappendix D of the supplementary material (available at stacks.iop.org/jpa/47/402002/
mmedia). In fact, these two limits are, respectively, consistent with the optimal forcings
fopt, ∞ = M sgn [g (θ )] which satisﬁes the bang–bang principle [24] and fopt, 1 which is a pairconsisting of one arbitrarily tall negative pulse and one arbitrarily tall positive pulse (i.e.,
bipolar pulses) separated by Δϕmax 4, as proved in appendix A.2 of the supplementary material(available at stacks.iop.org/jpa/47/402002/mmedia). Note this Δϕmax is determined algorithmically from Z, as shown in appendix A.2 of the supplementary material (available atstacks.iop.org/jpa/47/402002/mmedia). Thus, the answer for the case p = 1, ∞ in P2 isobtained, and the resulting best forcings are shown in ﬁgure 2(a), respectively, for p = 1, ∞.
For the practical signiﬁcance of such pulse forcings (p = 1) as well as square forcings( p = ∞), see [14, 15, 18, 23] for instance.
Now, we can design efﬁcient injection-locking toward the synchronization limits. Such adesign example is given in appendix E of the supplementary material (available at stacks.iop.
org/jpa/47/402002/mmedia) for the HH neuron model [17].
3.3. General m : n phase locking

Finally, regarding P3, for simplicity, here we assume that Z, f, and the associateda
Zn and fm are given as Fourier series: Z (θ ) = 20 + ∑α ∈ 1 (aα cos αθ + bα sin αθ ),acf (θ ) = 20 + ∑α ∈ 1 (cα cos αθ + dα sin αθ ), Z n (θ ) ≡ 20 + ∑α ∈ n1 (aα cos αθ + bα sin αθ ),cand fm (θ ) ≡ 20 + ∑α ∈ m 1 (cα cos αθ + dα sin αθ ), in which 1 is the natural numbers={1, 2, ...}, n 1 denotes the multiples of n = {n , 2n , ⋯}, and m 1 denotes the multiples ofm. Namely, Zn and fm are respectively the partial sum of the Fourier series Z and f. Also, we
1assume the resulting Γm n (ϕ) = 2π 〈Z (mθ + ϕ) f (nθ ) 〉 to be obtained by integrating term byterm from the Fourier series of Z and f 5. Then, from a trigonometric identity, we obtain
1
1
Γm n (ϕ) = 2π 〈Z n (mθ + ϕ) f (nθ ) 〉 = 2π 〈Z n (mθ + ϕ) fm (nθ ) 〉. This implies that for generalm : n phase locking, the global optimal forcing fopt, p is obtained in the form of the abovefm (nθ ) simply by replacing Z by Zn in the optimization algorithms related to P1 and P2. Note,4

The reason we represent fopt, 1 in such an asymptotic form (rather than using a formal delta function) is that fopt, 1belongs to L1 (S ) from the context of Hölderʼs inequality, and that what counts here for optimization is the resultingΓ0 (ϕ) (rather than the form of fopt, 1 itself).
5Since we have assumed Z is twice differentiable for 1 < p ⩽ ∞, here we further assume f [∈L p (S )] is piecewise
continuous, which implies that 〈Z f 〉 is piecewise smooth and it is obtained by integrating term by term.
7

J. Phys. A: Math. Theor. 47 (2014) 402002

there is only one exceptional case: m : 1 phase locking, where an ‘asymptotically’ best forcing
is constructed6. Thus, the answer to P3 has been obtained.
Figure 2(b) shows the global optimal 1 : 2 phase locking forcings of the Hodgkin–Huxley(HH) neuron phase model [17] for various p. We observe that these forcings have simpler
waveforms (compared with the 1 : 1 cases in ﬁgure 2(a)), uniformly spaced bipolar pulses(p = 1), nearly sinusoidal (p = 2), and almost uniformly spaced rectangles ( p = ∞), which is atypical feature for 1 : n global optimal forcing. The reason is as follows. In 1 : n phase locking,the above Z n (θ ) determines the optimal forcings (through the algorithms related to P1 andP2); only the multiples of n-th harmonics in the original Z (θ ) affect the optimal forcings. In
addition, as we observed the Fourier coefﬁcients of Z in ﬁgure 2 caption, the second (n = 2)
harmonic dominates other higher harmonics, resulting in a virtually sinusoidal oscillation(analogous to the one near the Hopf bifurcation point). This situation becomes more typical if
we consider a larger n, since the magnitude of higher multiples of n-th harmonics in Z (θ )decays sufﬁciently fast as the multiples of n becomes large for generic limit-cycle oscillators.
In fact, this insight explains the reason for the more sinusoidal-like best forcings systematically obtained for a larger n in the general m : n case (under a different but similar setting) [21].
4. Conclusion and discussion
In conclusion, we have proved a mechanism governing synchronization (injection-locking)limits, and clariﬁed how and why the best forcing realizes the synchronization limit, by
unveiling a hidden aspect, i.e., Hölderʼs inequality, behind it, for a general class of externallyforced limit-cycle oscillators. Namely, synchronization limit is now characterized as theequality condition of Hölderʼs inequality. To the best of the authorʼs knowledge, no previous
study has addressed this mechanism or the existence of the synchronization limit. Since thephase equation (1) appears in many areas of science and engineering, the obtained results here
have direct, broad impacts, as follows.
(i) Designing the best forcing (beyond power-reduced forcings, including pulse trains) and(simultaneously) designing a better phase sensitivity (i.e., oscillators with better
entrainability) is one of the direct consequences, since optimization of the forcing f
and that of Z are now equivalent by permuting Z and f in equation (3). Such an
application is feasible for a practical system in real environments, for instance as in[27, 28]. Also, it is noted that our results for the area-reduced forcings here enables us todesign the optimal pulse trains for injection-locking for the ﬁrst time.
(ii) In addition, designing an efﬁcient ‘coupling’ between oscillators for better mutualsynchronization [29] is also promising, since the results of a single oscillator with one
external forcing here is modiﬁed to the case of mutual synchronization straightforwardly.
(iii) Furthermore, the algorithm for realizing the optimality here should provide a uniﬁed,
systematic method for optimally entraining a given oscillation pattern in an ensemble ofoscillators (or excitable elements) with global coupling [6], as well as with local coupling

6
The construction is as follow. Starting from m copies with the optimal forcing with prime period T0 for 1 : 1 phaselocking, add a certain small perturbation such that the m copies of the forcing become a single forcing with prime
period mT0 while still satisfying the constraints (2). The resulting locking range becomes arbitrarily close to the ideal
one (which is realized only in 1 : 1 phase locking) as the perturbation becomes smaller, since the associated Γm/1 inequation (1) becomes arbitrarily close to the Γ1 1 of the best forcing for 1 : 1 phase locking.

8

J. Phys. A: Math. Theor. 47 (2014) 402002[8], since this oscillation pattern is regarded as a limit-cycle oscillation in higher phasespace, which is described by equation (1).
(iv) Though our present framework focuses on the noiseless case, noisy oscillators can betreated in the same way as here, by virtue of the recent progress in this direction [30, 31].
These applications and extensions (i)–(iv) of our ﬁndings are of high importance forfurther theoretical as well as experimental work, which will be reported in the near future.

Acknowledgments
The author (HT) is indebted to Dr Y Tsubo and Dr T Shimada for their enlightening suggestions and critical reading of the manuscript. H T is also grateful to the anonymous referees
for their critical comments. This work was supported by MEXT (No. 23360047) and by the
Support Center for Advanced Telecommunications Technology Research (SCAT).

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