hadi susanto - professor, khalifa university (uae) & essex...

Nonlinear Dyn (2015) 81:353–371DOI 10.1007/s11071-015-1996-2

ORIGINAL PAPER

Dynamics of generalized PT-symmetric dimerswith time-periodic gain–loss

F. Battelli · J. Diblík · M. Feckan · J. Pickton ·M. Pospíšil · H. Susanto

Received: 12 May 2014 / Accepted: 19 February 2015 / Published online: 18 March 2015© Springer Science+Business Media Dordrecht 2015

Abstract A parity-time (PT)-symmetric system withperiodically varying-in-time gain and loss modeled bytwo coupled Schrödinger equations (dimer) is studied.It is shown that the problem can be reduced to a per-turbed pendulum-like equation. This is done by findingtwo constants of motion. Firstly, a generalized problemusing Melnikov-type analysis and topological degreearguments is studied for showing the existence of peri-odic (libration), shift- periodic (rotation), and chaoticsolutions. Then these general results are applied to thePT-symmetric dimer. It is interestingly shown that ifa sufficient condition is satisfied, then rotation modes,which do not exist in the dimer with constant gain–loss, will persist. An approximate threshold for PT-broken phase corresponding to the disappearance of

F. BattelliDepartment of Industrial Engineering and MathematicalSciences, Marche Polytecnic University, Via BrecceBianche 1, 60131 Ancona, Italye-mail: [email protected]

J. DiblíkDepartment of Mathematics, Faculty of ElectricalEngineering and Communication, Brno University ofTechnology, Technická 3058/10, 616 00 Brno, CzechRepublice-mail: [email protected]

M. FeckanDepartment of Mathematical Analysis and NumericalMathematics, Comenius University, Mlynská dolina,842 48 Bratislava, Slovakiae-mail: [email protected]

bounded solutions is also presented. Numerical studyis presented accompanying the analytical results.

Keywords PT-symmetry · PT-reversibility ·Schrödinger equation · Melnikov function ·Perturbation · Chaos

1 Introduction

We study the coupled Schrödinger equations

ı u1 = −u2 − |u1|2u1 − ıγ (t)u1

ı u2 = −u1 − |u2|2u2 + ıγ (t)u2, (1)

with a Kerr nonlinearity where γ (t) is a gain–loss para-meter. It is easy to see that (1) is invariant under theexchange

J. PicktonSchool of Mathematical Sciences, University of Nottingham,University Park, Nottingham, NG7 2RD, UK

M. PospíšilFaculty of Electrical Engineering and Communication,Centre for Research and Utilization of Renewable Energy,Brno University of Technology, Technická 3058/10,616 00 Brno, Czech Republice-mail: [email protected]

H. Susanto (B)Department of Mathematical Sciences, University of Essex,Wivenhoe Park, Colchester, CO4 3SQ, UKe-mail: [email protected]

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354 F. Battelli et al.

(u1, u2, u1, u2, t, ı) �→ (u2, u1,−u2,−u1,−t,−ı),

(2)

known as parity-time (PT) symmetry [1], provided thatγ (t) is even. However, for the sake of completeness, inthis paper, we consider a general, possibly not even,γ (t). In this case, we may consider a formal symmetryin the sense that under the transformation

(u1, u2, u1, u2, t, ı) �→ (u2, u1, u2, u1,−t,−ı),

the new equations have (u1(−t), u2(−t)) as a solutionwhenever (u1(t), u2(t)) solves (1). In this context, thePT-symmetry [1] can be referred to as PT-reversibility.

Equation (1) without nonlinearity with time-inde-pendent γ were derived in [2] as a finite dimensionalreduction of mode guiding optical systems with gainand loss regimes [3,4]. The following successful exper-iments [5,6] have stimulated extensive studies on thePT systems, including the interplay between the sym-metry and nonlinearity [7–12]. Later, equations similarto (1) were also derived [13] modeling Bose–Einsteincondensates with PT-symmetric double-well potentials[14–17].

The study of the above systems with equal and con-stant gain and loss was motivated by a relatively recentpostulate in quantum physics that Hamiltonian oper-ators do not have to be self-adjoint (Hermitian) forthe potential physical relevance; non-Hermitian oper-ators only need to respect Parity (P) and time-reversal(T ) symmetries and the spectrum of the operators canbe real if the strength of the anti-Hermitian part, i.e.,γ ≡ const in (1), does not exceed a critical value[1,18,19]. Above the critical value, the systems are saidto be in the PT-broken phase regime.

Note that system (1) with constant γ was firstly pro-posed and studied in [20]. The solution dynamics of (1)in that case has been studied in [21–24].

Recently, (1) with a gain and loss parameter that isperiodically a function of time was considered in [25].By deriving an averaged equation, the rapid modula-tion in time of the gain and loss profile was shownto provide a controllable expansion of the region ofexact PT-symmetry, depending on the strength and fre-quency of the imposed modulation. In the presenceof dispersion along the direction perpendicular to thepropagation axis, rapidly modulated gain and loss wasshown to stabilize bright solitons [26]. The same sys-tem featuring a “supersymmetry” when the gain andloss are equal to the inter-core coupling admits a vari-ety of exact solutions (we focus on solitons), which

are subject to a specific subexponential instability [27].However, the application of the “management” in theform of periodic simultaneous switch of the sign ofthe gain, loss, and inter-core coupling, effectively sta-bilizes solitons, without destroying the supersymmetry[27]. A significant broadening of the quasienergy spec-trum, leading to a hyperballistic transport regime, dueto a time-periodic modulation has also been reportedin [28]. When the damping and gain factor varies sto-chastically in time, it was shown in [29] that the sta-tistically averaged intensity of the field grows. A dif-ferent, but similar dimer to (1) involving a parametric(ac) forcing through periodic modulation of the cou-pling coefficient of the two degrees of freedom wasconsidered recently in [30]. It was reported that thecombination of the forcing and PT-symmetric potentialexhibits dynamical charts featuring tongues of para-metric instability, whose shape has some significantdifferences from that predicted by the classical theoryof the parametric resonance in conservative systems. Atime-periodic, piecewise-constant gain–loss parameterin the Schrödinger dimer has also been considered in[31], where it was shown that the dimer behaves likea single oscillator with time-periodic dissipation undercertain conditions.

In this work, we consider (1) with time-varying γ

whose modulation period is O(1). Using Melnikov-type analysis and topological degree arguments, weprove and derive conditions for the existence of peri-odic (libration), shift-periodic (rotation), and chaoticsolutions in the system. Note that the existence ofbounded rotation modes for the phase difference of thedimer wavefunctions is a novel and interesting featureas they cease to exist in the time-independent γ [24].We also derive an approximate threshold for PT-brokenphase corresponding to the disappearance of boundedsolutions.

The paper is organized as follows. In Sect. 2, wereduce the governing Eq. (1) into

ψ = 4(c2 sinψ + γ (t)z

),

z = γ (t)ψ, (3)

which becomes the main object that we study in thispaper. In Sect. 3, we consider a more general problemthan (3) of the form

ψ = f (ψ) + γ (t)h(z)

z = γ (t)ψ (4)

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Dynamics of generalized PT-symmetric dimers with time-periodic gain–loss 355

where γ (t) is 1-periodic and f (ψ), h(z) are specifiedlater. The motivation for such a generalization is toshow that our method is not restricted to the specificform of (3) but rather on its structure expressed in (4).Of course, we may derive back an equation like (1)by using transformations of variables given in Sect. 2for (4), but we do not make that here. First we con-sider small forcing (see (15)) and derive Melnikov-type conditions for bifurcation of subharmonics, shift-subharmonics and heteroclinics/homoclinics. As it iswell known [32], the Melnikov method is one of thefew analytical methods allowing us to detect and studychaotic motions for differential equations rigorously.Started in [33], many extensions and applications ofthis powerful approach have appeared since then, suchas [34,35]. Our present work also demonstrates itseffectiveness and applicability. For more details of themethod, we refer the interested reader to the books [36–40].

We also discuss the persistence of these solutionswhen (4) is reversible symmetric. Then an averaging-like result concerning the existence of periodic solu-tions is proved when all terms of (4) are small (see(26)). We return back to the general Eq. (4) in Sect. 4and use topological degree arguments for showingodd-periodic, shift-periodic and anti-periodic solutionsunder certain symmetric properties of (4). In Sect. 5,we apply results of Sects. 3 and 4 to (3). In Sect. 6 wepresent numerical integrations of (3) illustrating theanalytical results derived in the previous sections.

2 Model reduction

To simplify (1), by following [24], we use polar form

u j (t) = |u j (t)| eıφ j (t), j = 1, 2,

by assuming u1(t)u2(t) �= 0. Clearly

u j (t) =(d

dt|u j (t)| + ı |u j (t)|dφ j (t)

dt

)eıφ j , j = 1, 2.

Hence setting θ(t) := φ2(t) − φ1(t) and separating(1) into the real and imaginary parts, we haved

dt|u1(t)| = −|u2(t)| sin θ(t) − γ (t)|u1(t)|,

d

dt|u2(t)| = |u1(t)| sin θ(t) + γ (t)|u2(t)|,d

dtθ(t) =

(|u2(t)|2 − |u1(t)|2

)

×(1 − cos θ(t)

|u1(t)||u2(t)|)

. (5)

Then it can be shown

d

dt

(|u1(t)|2|u2(t)|2 − 2|u1(t)||u2(t)| cos θ(t)

)= 0.

So we obtain

c4 = 1 + |u1(t)|2|u2(t)|2−2|u1(t)||u2(t)| cos θ(t). (6)

Note

1 + |u1(t)|2|u2(t)|2 − 2|u1(t)||u2(t)| cos θ(t)

= (1 − |u1(t)||u2(t)|)2+ 2(1 − cos θ(t))|u1(t)||u2(t)| ≥ 0.

Hence the constant c ≥ 0 in (6) is well defined.We study c > 0 in this paper. Note c �= 1 impliesu1(t)u2(t) �= 0. Next, using (6), we introduce a newvariable ψ as follows

c2 sinψ(t) = |u1(t)||u2(t)| sin θ(t),

c2 cosψ(t) = |u1(t)||u2(t)| cos θ(t) − 1. (7)

From (5) and (7), we obtain(|u1(t)|2 − |u2(t)|2

)|u1(t)||u2(t)| sin θ(t)

= d

dt(|u1(t)||u2(t)| cos θ(t)) = d

dt

(c2 cosψ(t)

)

= −c2ψ(t) sinψ(t) = −ψ(t)|u1(t)||u2(t)| sin θ(t),

which implies

ψ(t) = |u2(t)|2 − |u1(t)|2. (8)

Differentiating (8) and using (5), (7), we arrive at

ψ(t) = 4|u1(t)||u2(t)| sin θ(t)

+ 2γ (t)(|u1(t)|2 + |u2(t)|2

)

= 4c2 sinψ(t) + 2γ (t)(|u1(t)|2 + |u2(t)|2

).

(9)

On the other hand, using (5) and (8), we obtain

d

dt

(|u1(t)|2 + |u2(t)|2

)

= 2γ (t)(|u2(t)|2 − |u1(t)|2

)= 2γ (t)ψ(t). (10)

Finally setting

z(t) := |u1(t)|2 + |u2(t)|22

,

(9) and (10) give (3).

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Remark 1 Introducing the power P := |u1|2 +|u2|2 and the probability difference between the twowaveguides � := |u2|2 − |u1|2, and using (5), (8)–(10), instead of (3), we obtain from (1) [24]

P = 2γ (t)�

� = 2γ (t)P + 2√P2 − �2 sin θ

θ = �

(1 − 2 cos θ√

P2 − �2

), (11)

which reduces to the common pendulum-like systemfor wavefunctions in a double-well potential when γ ≡0 (see [24] and references therein). Note P2 − �2 =4|u1|2|u2|2, so P2 > �2 if and only |u1||u2| �= 0.Then

c4 = 1 + P2(t) − �2(t)

4−

√P2(t) − �2(t) cos θ(t).

Furthermore, one can compare the first equation of(11) with (8) to obtain

P = d

dt(2γψ) − 2γ ψ, (12)

which can be integrated to yield

P = 2γψ − 2∫

γ ψ dt. (13)

Using Eqs. (7), (8) and (13) in the second equationof (11) will yield the equation

1

4ψ = c2 sinψ + γ 2ψ − γ

∫γ ψ dt, (14)

which becomes (3) with z(t) = γψ − ∫γ ψ dt .

3 General bifurcation results and related topics

We study the coupled system

ψ = f (ψ) + εγ (t)h(z)

z = εγ (t)ψ (15)

where γ (t) is 1-periodic and C2-smooth, f (ψ), h(z)are C2 and ε ∈ R is small. Set:

F(ψ) =∫ ψ

0f (u)du, H(z) =

∫ z

0h(u)du.

From (15) it easy to derive the relationd

dt

[ψ2 − 2F(ψ) − 2H(z)

]= 0,

i.e.,

ψ2 − 2F(ψ) − 2H(z) = 2d (16)

for a constant d ∈ R. Of course, (16) holds even if γ (t)is not periodic.

Remark 2 The same first integral is such for the system

ψ = f (ψ) + εγ (t)h(z)h1(z)

z = εγ (t)h1(z)ψ (17)

where h1(z) is C2.

We assume that

(C1) H : I → J is a diffeomorphism from the openinterval I ⊆ R into the open interval J ⊆ R, i.e.,h(z) �= 0 for any z ∈ I .

Then we have:

z = H−1(

ψ2

2− F(ψ) − d

), (18)

provided ψ2

2 − F(ψ) − d ∈ J and (15) is reduced to

ψ = f (ψ)

+ εγ (t)h

(H−1

(ψ2

2− F(ψ) − d

)). (19)

This equation must be solved so that its solution sat-

isfies ψ2(t)2 −F(ψ(t))−d ∈ J for any t in some interval

(or t ∈ R if we look for globally defined solutions).Setting ε = 0 into (19) we obtain the unperturbed

equation:

ψ = f (ψ). (20)

We assume that

(H1) Equation (20) has a family of periodic solutions

ψλ(t) = ψλ(t + Tλ) (21)

with periods Tλ ∈ R and ψλ(t) is the unique, upto a multiplicative constant, Tλ-periodic solutionof the variational equation ψ = f ′(ψλ(t))ψ .

Remark 3 Assuming that dependence on λ in (21)is C1-smooth, from (20) we derive that ψλ(t) and∂∂λ

ψλ(t) solve ψ = f ′(ψλ(t))ψ . But from (21) wehave

ψλ(t) = ψλ(t + Tλ),

∂

∂λψλ(t) = ∂

∂λψλ(t + Tλ) + ψλ(t + Tλ)

∂

∂λTλ.

Hence ∂∂λ

ψλ(t) is Tλ-periodic if and only if ∂∂λTλ =

0. This observation is well known but we presented ithere for the reader’s convenience.

Next we assume that for some λ ∈ R the resonancecondition

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(R) Tλ = pq for p, q ∈ N

holds. Using (20) we see that cλ ∈ R exists so that

ψ2λ(t)

2− F(ψλ(t)) = cλ.

Then we assume:

(H2) cλ − d ∈ J .

Condition (H2) guarantees that

ψ2λ(t)

2− F(ψλ(t)) − d ∈ J

for any t ∈ R.Now,we follow the standard subharmonicMelnikov

method [32,36,41] to (19) based on the Lyapunov–Schmidt method, but for reader’s convenience, wepresent more details. First, we take Banach spaces

X :={ψ ∈ C2(R,R) | ψ(t + T ) = ψ(t)∀t ∈ R

}

Y := {ψ ∈ C(R,R) | ψ(t + T ) = ψ(t)∀t ∈ R}with the usual maximum norms ‖ψ‖2 and ‖ψ‖0,respectively, where T = p = qTλ, so that ψλ(t) isT -periodic. We want to find solutions in X of (19) suchthat

ψ2(t)

2− F(ψ(t)) − d ∈ J (22)

and ψ(t + T ) = ψ(t), for any t ∈ R. We consider theopen subset Ω ⊂ X of those ψ ∈ X such that (22)holds. For any ψ ∈ Ω we set

F(ψ, ε)(t) = ψ − f (ψ)

−εγ (t)h

(H−1

(ψ2

2− F(ψ) − d

)).

Note that, for any α ∈ R the function ψλ(· + α) :t �→ ψλ(t + α) satisfies (22) and

F(ψλ(· + α), 0)(t) = 0.

Then F(ψ, 0) vanishes on the one-dimensionalmanifold M = {ψλ(· + α) | α ∈ R}. The tangentspace toM at the point ψλ(· + α0) is

Tψλ(·+α0)M = span{ψλ(· + α0)}and of course the linear map

L : ψ �→ ψ − f ′(ψλ(t + α0))ψ

vanishes on Tψλ(·+α0)M. From (H1) it follows that

N L = Tψλ(·+α0)M.

Next, it is well known (see [32,36]) that b(t) ∈ Ybelongs toRL if and only if∫ T

0u(t)b(t)dt = 0

for any T -periodic solution u(t) of ψ = f ′(ψλ(t +α0))ψ . But then, since the space of periodic solutionsof this equation is spanned by ψλ(·+α0), we concludethat the equation

ψ − f ′(ψλ(t + α0))ψ = b(t)

has a periodic solution (i.e., b(t) ∈ RL) if and only if∫ T

0ψλ(t + α0)b(t)dt = 0.

Finally,

∂

∂εF(ψλ(· + α0), 0)

= −γ (t)h

(H−1

(ψ2

λ(· + α0)

2− F(ψλ(· + α0)) − d

))

= −γ (t)h(H−1(cλ − d)).

Then, from thewell-known Poincaré–Melnikov the-ory we obtain the following result:

Theorem 1 Suppose that (H1), (R) and (H2) hold andthat the Melnikov function

Mλ(α) :=∫ T

0ψλ(t + α)γ (t)dt

has a simple zero at some α = α0. Then there existsρ > 0 such that for any ε sufficiently small there existsa unique α(ε) (C1 with respect to ε) such that |α(ε) −α0| → 0 as ε → 0 and such that Eq. (19) has a uniquep-periodic solution ψ(t, ε) satisfying:

‖ψ(t, ε) − ψλ(t + α(ε))‖2 ≤ ρ.

Moreover ‖ψ(t, ε) − ψλ(t + α(ε))‖2 → 0 as ε → 0.

Now suppose

(C2) f (ψ + 2π) = f (ψ), F(ψ + 2π) = F(ψ),(H3) Eq. (20) has a family of T l

λ-shift-periodic solu-tions

ψ lλ

(t + T l

λ

)= ψ l

λ(t) + 2π (23)

with shift-periods T lλ ∈ R and ψ l

λ(t) is theunique, up to a multiplicative constant, T l

λ-periodic solution of the variational equation ψ =f ′(ψ l

λ(t))ψ .

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Using again (20) we see that clλ ∈ R exists so that

ψ l2λ (t)

2− F

(ψ l

λ(t))

= clλ.

Then Theorem 1 can be directly extended as follows.

Theorem 2 Suppose that (C2), (H3) and

(R′) T lλ = p

q for p, q ∈ N,

(H ′2) clλ − d ∈ J

hold and that the Melnikov function

Mlλ(α) :=

∫ T

0ψ l

λ(t + α)γ (t)dt

has a simple zero at some α = α0. Then there existsρ > 0 such that for any ε sufficiently small there existsa unique αl(ε) (C1 with respect to ε) such that |αl(ε)−α0| → 0 as ε → 0 and such that Eq. (19) has a uniquep-shift-periodic solution ψ(t, ε) satisfying:

‖ψ(t, ε) − ψ lλ(t + αl(ε))‖2 ≤ ρ.

Moreover ‖ψ(t, ε) − ψ lλ(t + αl(ε))‖2 → 0 as ε → 0.

Similarly we can study heteroclinic Melnikov bifur-cation [32,36,42,43] to (15). In this case, we assume

(H4) the unperturbed equation (20) has a heteroclinicorbitψ∞(t) such that lim

t→±∞ ψ0(t) = ψ± andψ±are hyperbolic saddles of ψ = f (ψ).

Note that if ψ+ = ψ− then ψ∞(t) is a homoclinicorbit. Of course the quantity

ψ2

2− F(ψ)

is constant along ψ∞(t). Setting

c∞ = ψ∞(t)2

2− F(ψ∞(t))

we assume that

(H5) c∞ − d ∈ J .

Now we take functional spaces X = C2b (R,R) and

Y = Cb(R,R) and repeating the above arguments, wehave the following result.

Theorem 3 Under assumptions (H4) and (H5), thereexists ρ > 0 such that Eq. (19) has unique hyperbolicperiodic solutions u±(t) in a neighborhood of ψ± ofradius ρ. Moreover, if the Melnikov function

M∞(α) :=∫ ∞

−∞ψ∞(t + α)γ (t)dt

has a simple zero at some α = α0 then for any ε suffi-ciently small there exists a uniqueα(ε) (C1 with respectto ε) such that |α(ε) − α0| → 0 as ε → 0 and suchthat Eq. (19) has a unique heteroclinic (homoclinic)solution ψ(t, ε) satisfying:

‖ψ(t, ε) − ψ∞(t + α(ε))‖2 ≤ ρ.

Moreover ‖ψ(t, ε) − ψ∞(t + α(ε))‖2 → 0 as ε → 0and

limt→±∞ |ψ(t, ε) − u±(t)| = 0.

Remark 4 As a matter of fact, we have 1-parametricfamilies of subharmonic, shift-subharmonics andboun-ded solutions parameterized by d. Moreover, by [32,p. 197], these bounded solutions are accumulated withfamilies of subharmonics, shift-subharmonics fromTheorem 1. Furthermore, Theorem 3 implies the exis-tence of chaos near these bounded solutions creatingheteroclinic cycles (see [44] for more details). Now,γ (t) can be almost-periodic.

Remark 5 Note that∫ T

0Mλ(α)dα =

∫ T

0

∫ T

0ψλ(t + α)γ (t)dtdα

=∫ T

0γ (t)

(∫ T

0ψλ(t + α)dα

)dt

=∫ T

0γ (t) (ψλ(t + T ) − ψλ(t)) dt

=∫ qTλ

0γ (t) (ψλ(t + qTλ) − ψλ(t)) dt = 0,

∫ T

0Ml

λ(α)dα =∫ qT l

λ

0γ (t)

(ψ l

λ(t + qT lλ)−ψλ(t)

)dt

= 2πq∫ T

0γ (t)dt,

∫ 1

0M∞(α)dα =

∫ ∞

−∞γ (t) (ψ∞(t + 1) − ψ∞(t)) dt

=∫ ∞

−∞γ (t − 1)ψ∞(t)dt −

∫ ∞

−∞γ (t)ψ∞(t)dt = 0.

So if either the subharmonic Melnikov functionMλ(α) is not identically zero or

∫ T0 γ (t)dt = 0 and

the shift-subharmonic Melnikov function Mlλ(α) is not

identically zero, then they change signs over [0, T ], andhence we have existence (but not uniqueness) resultssimilar to Theorems 1 and 2. The same argument holdsfor the heteroclinic Melnikov function M∞(α) fromTheorem 3 with the corresponding chaotic behavior(see [44] for more details).

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Remark 6 Writing (19) as a system

ξ = f (ξ, t, ε), ξ = (ψ, θ), (24)

for

f (ξ, t, ε)

:=(

θ

f (ψ) + εγ (t)h(H−1

(θ2

2 − F(ψ) − d))

)

we see that, when γ (−t) = γ (t), i.e., when γ (t) iseven,

R1 f (ξ, t, ε) = − f (R1ξ,−t, ε) (25)

for the reflection R1(ψ, θ) = (ψ,−θ). Condition (25)means, that (24) is R1-reversible [45–48].Wemay sup-pose that the periodic solutions in (21) satisfy ψλ(0) =0. Then setting ξλ(t) = (ψλ(t), ψλ(t)), all ξλ(t) inter-sect transversally the fixed point set Fix R1 = {ξ ∈R2 | R1ξ = ξ} = {(ϕ, 0)} in 2 points.So each of these periodics is R1-symmetric, i.e., they

satisfy ξλ(t) = R1ξλ(−t) since ξλ(0) ∈ Fix R1. Sothey persist in (24) for ε small when γ (t) is even.Furthermore, if ψ∞(t) is homoclinic then we mayagain suppose that ψ∞(0) = 0. Hence ξ∞(t) :=(ψ∞(t), ψ∞(t)) intersects transversally the fixed pointset Fix R1 in one point. So, it is R1-symmetric and itpersists in (24) for ε small. Moreover there is an accu-mulation of these R1-symmetric periodics on the R1-symmetric homoclinic.

Next supposing that f (ψ) and γ (t) are odd, i.e.,f (ψ) = − f (−ψ) and γ (t) = −γ (−t), and takingthe reflection R2(ϕ, θ) = (−ϕ, θ), we see that (24)is R2-reversible. Note F(−ψ) = F(ψ), i.e., F(ψ) iseven. Then if the periodic solutions (21) and hetero-clinic cycle {ξ∞(t) | t ∈ R}∪ {R2ξ∞(t) | t ∈ R} inter-sect transversally the fixed point set Fix R2 = {(0, θ)},that is, if each of these orbits is R2-symmetric, theypersist in (24) for ε small when γ (t) is odd. Of course,here we suppose that ξλ(0), ξ∞(0) ∈ Fix R2. More-over, there is again an accumulation of R2-symmetricperiodic orbits on the R2-symmetric heteroclinic cycle.Finally, if in addition (C2) and (H3) hold then we havethe same persistence and accumulation results for shift-subharmonics.

Finally, instead of (15), we consider the equation

ψ = ε f (ψ, ε) + εγ (t)h(z)

z = εγ (t)ψ (26)

where γ (t) is 1-periodic andC2-smooth, ε > 0 is smalland f (ψ, ε) is again C2-smooth. Arguing as before,(26) is reduced to (see (19))

ψ = ε f (ψ, ε) + εγ (t)

×h

(H−1

(ψ2

2− εF(ψ) − d

)). (27)

Note that, for ε = 0 Eq. (27) has the 1-periodicsolutionsψ(t) = α. Suppose that (27) has a 1-periodicsolution for any ε > 0 small uniformly bounded byM, so ‖ψ‖2 ≤ M . Since

∫ 10 ψ(t)dt = 0, there is an

t0 ∈ [0, 1) such that ψ(t0) = 0. Then we derive

ψ(t) = ε

∫ t

t0f (ψ(s), ε) + γ (s)

×h

(H−1

(ψ(s)2

2− εF(ψ(s)) − d

))ds

(28)

and hence taking absolute values and using the bound-edness ofψ(t), and the smoothness of f, h and H−1 wesee that

‖ψ‖0 = O(ε)

but then −d ∈ J . We suppose (see (H2))

− d ∈ J. (29)

Next we note that the equation

ψ = b(t) = b(t + 1)

has a 1-periodic solution if and only if∫ 10 b(t)dt =

0. So, repeating the above method for (28) with thesplitting

Y = Y1 ⊕ Y2, X = X1 ⊕ X2,

Y1 := span{1}, Y2 :={ψ ∈ Y

∣∣∣∫ 1

0ψ(t)dt = 0

}

X1 := Y1, X2 := X ∩ Y2,

and the projections P : Y → Y1 and Q = I − P :Y → Y2 defined as Pψ := ∫ 1

0 ψ(t)dt , we get the nextresult related to averaging theory [32,49].

Theorem 4 Suppose (29) and the Melnikov-like func-tion

Maver(α) := f (α, 0) + h(H−1(−d))

∫ 1

0γ (t)dt (30)

has a simple zero at some α0 ∈ R. Then Eq. (27) has aunique 1-periodic solution ψ(t) = α0 + O(ε) for anyε sufficiently small.

123



4 Forced anti-symmetric periodic solutions

In this section, we consider the problem of existenceof periodic solutions of the differential Eq. (4). Aswe apply topological degree arguments [36,46,50], weintroduce a new parameter λ ∈ [0, 1], changing f (ψ)

with λ f (ψ), and γ (t) with λγ (t):

ψ = λ f (ψ)+λγ (t)h

(H−1

(ψ2

2− λF(ψ) − d

)).

(31)

Note that, for λ = 1 we have equation (19) withε = 1 and with λ = 0 we obtain the equation ψ = 0.

For any w ∈ J we have:

d

dwH−1(w) = 1

H ′(H−1(w))= 1

h(H−1(w))

hence from Eq. (31) we obtain:

d

dtH−1

(ψ(t)2

2− λF(ψ(t)) − d

)= λγ (t)ψ(t),

(32)

which implies

H−1(

ψ(t)2

2− λF(ψ(t)) − d

)

= H−1(

ψ(t0)2

2− λF(ψ(t0)) − d

)

+ λ

∫ t

t0γ (s)ψ(s)ds

that is

ψ(t)2

2− λF(ψ(t))

= d + H

(H−1

(ψ(t0)2

2− λF(ψ(t0)) − d

)

+ λ

∫ t

t0γ (s)ψ(s)ds

). (33)

To proceed,we recall the followingBihari’s inequal-ity [51].

Theorem 5 If w(t) is a nonnegative continuous func-tion such that:

w(t) ≤ α +∫ t

0f (s)g(w(s))ds

with a constant α > 0, g : [0,∞[→ [0,∞[ non-decreasing continuous with g(u) > 0 if u > 0, and

f (t) is continuous with f (t) ≥ 0 for any t ≥ 0 then,with G(x) = ∫ x

11

g(u)du,

w(t) ≤ G−1(G(α) +

∫ t

0f (s) ds

)

for all t ≥ 0 for which G(α) + ∫ t0 f (s) ds belongs to

the domain of G−1.

Suppose that (32), or (33), has a 1-periodic or ahomoclinic, solution ψ(t). Then at some point t0 itmust result ψ(t0) = 0. Motivated by this, we provethe following.

Lemma 1 Let Γ := ∫ 10 |γ (s)|ds and assume the fol-

lowing conditions hold:

(i) |F(ψ)| ≤ M for any ψ ∈ R,(ii) |H(z)| ≤ g(|z|) for any z ∈ R,(iii) [−d − λ supψ∈R F(ψ),−d − λ infψ∈R F(ψ)] ⊂

J for any λ ∈ [0, 1],(iv) Γ <

∫ ∞

N+Γ√2(M+|d|)

du√2g(u)

,

where

N := sup{|H−1(−λF(ψ) − d)| | ψ ∈ R, λ ∈ [0, 1]},and g : [0,∞[→ [0,∞[ is a non-decreasing contin-uous function satisfying g(r) > 0 for r > 0. Thenthere exists a positive constant M1 such that for anyλ ∈ [0, 1], any 1-periodic solution ψ(t) of Eq. (33)whose derivative vanishes at t0 ∈ [0, 1[ satisfies:‖ψ‖0 ≤ M1.

Proof From Eq. (33) we get:

1

2ψ(t)2 ≤ |d| + M + g

(∣∣∣H−1 (−λF(ψ(t0)) − d)

+λ

∫ t

t0γ (s)ψ(s)ds

∣∣∣∣)

≤|d|+M+g(∣∣∣H−1

(−λF(ψ(t0))−d

)∣∣∣

+∫ t

t0|γ (s)ψ(s)|ds

)

≤ |d| + M + g

(N +

∫ t

t0|γ (s)ψ(s)|ds

)

for t ≥ t0. Note that here we need that[−d − λ sup

ψ∈RF(ψ),−d − λ inf

ψ∈RF(ψ)

]⊂ J

123



for any λ ∈ [0, 1]. Then

|ψ(t)| ≤√2(|d| + M)+2g

(N +

∫ t

t0|γ (s)ψ(s)|ds

)

≤ √2(|d| + M) +

√2g

(N +

∫ t

t0|γ (s)ψ(s)|ds

).

Setting w(t) := N +∫ t

t0|γ (s)ψ(s)|ds:

w′(t) ≤ |γ (t)|√2(|d| + M) + |γ (t)|√2g(w(t))

and then

w(t)≤N + Γ√2(|d| + M)+

∫ t

t0|γ (s)|√2g(w(s)) ds.

Bihari’s inequality implies

w(t) ≤ G−1(G

(N + Γ

√2(|d| + M)

)+

∫ t

t0|γ (s)| ds

)

where G−1 is the inverse function of

G(x) :=∫ x

1

du√2g(u)

and for all t ∈ [t0, t0 + 1] for whichG

(N + Γ

√2(|d| + M)

)+

∫ t

t0|γ (s)| ds

belongs to the domain of G−1. Since g(r) > 0 for allr > 0 we get:

G : [0,∞[→[

−∫ 1

0

du√2g(u)

,

∫ ∞

1

du√2g(u)

[.

So G(N + Γ

√2(|d| + M)

)+ ∫ tt0

|γ (s)| ds belongs tothe domain of G−1 if and only if

G(N + Γ

√2(|d| + M)

)+

∫ t

t0|γ (s)| ds

<

∫ ∞

1

du√2g(u)

,

and this holds if:∫ N+Γ

√2(|d|+M)

1

du√2g(u)

+ Γ <

∫ ∞

1

du√2g(u)

⇔ Γ <

∫ ∞

N+Γ√2(|d|+M)

du√2g(u)

.

Then the thesis follows with

M1

=√2

(|d| + M + g

(G−1

(G

(N+Γ

√2(|d| + M)

)+Γ

))).

The proof is finished. ��

Now we assume the following hold:

(A1) f (ψ) is odd and continuous,(A2) γ (t) is odd and continuous,

(A3)

[−d − M,−d + M + M2

12

]⊂ J .

Note (A3) implies (iii) of Lemma 1. Suppose thatψ(t) solves equation (31). Then−ψ(−t) is also a solu-tion of (31). Let X,Y, Z be, resp., the space of C2, C0,and C1, 1-periodic odd functions. By (A3), we maytake M1 < M1 so that[−d − M,−d + M + M2

1

2

]⊂ J.

Sinceψ(t) = ∫ t0 ψ(s)ds implies ‖ψ‖0 ≤ ‖ψ‖0, we

can take the norm ‖ψ‖1 := ‖ψ‖0 on Z . Let BM1(0) be

the ball in Z of radius M1 and center 0. Then the map:

Fλ : ψ �→ λ f (ψ)

+ λγ (t)h

(H−1

(ψ2

2− λF(ψ) − d

))

from BM1(0) to Y is well defined. Moreover, for any

function v(t) ∈ Y the equation

ψ(t) = v(t)

has the unique odd 1-periodic solution

ψ(t) = (L pv)(t) :=∫ t

0

∫ s

0v(τ)dτds

−t∫ 1

0

∫ s

0v(τ)dτds

=∫ t

0(t − s)v(s)ds + t

∫ 1

0sv(s)ds ∈ X

with‖ψ‖2 ≤ 2‖v‖0, since‖ψ‖0 ≤ ‖ψ‖0 ≤ 2‖v‖0 and‖ψ‖2 = max{‖ψ‖0, ‖ψ‖0, ‖ψ‖0} ≤ 2‖v‖0. HenceL p : Y → X is continuous and so L pFλ : BM1

(0) →X is well defined and continuous. Arzela-Ascoli theo-rem gives the compactness of the embedding X ⊂ Z .Then L pFλ : BM1

(0) → Z is well defined, continuousand compact. Next from Lemma 1 it follows that equa-tion ψ = L pFλψ has no solutions in the boundary ofball BM1

(0). Hence we get

deg(I − L pF1, BM1

(0), 0) = 1.

So, we proved the following result.

Theorem 6 Assume (i), (ii) and (iv) of Lemma 1, (A1),(A2), (A3) and h is continuous satisfying (C1). Thenequation

ψ = f (ψ) + γ (t)h

(H−1

(ψ2

2− F(ψ) − d

))(34)

has a 1-periodic odd solution.

123



In a similar way, we may also consider the problemof existence of solutions of (34) satisfying the condition

ψ(t + q) = ψ(t) + 2π (35)

for some q ∈ N by supposing (C2) in addition. Sowe are looking for q-shift-periodic solutions. This isaccomplished by looking for solutions ψ(t) of (31)satisfying ψ(t) = ϕ(t) + 2π

q t for all t ∈ R with ϕ(t +q) = ϕ(t). Plugging the equality into (31) we get theequation for ϕ(t)

ϕ = λ f

(ϕ + 2π

qt

)+ λγ (t)

×h

(H−1

(1

2

(ϕ + 2π

q

)2− λF

(ϕ + 2π

qt

)−d

)).

(36)

As we observed previously, if f (ψ) and γ (t) areodd and ϕ(t) is a solution of (36), then −ϕ(−t) is alsoa solution of the same Eq. (36). Then we see, as in theproof of Lemma 1, that the following inequality holdsfor any q-periodic function ϕ(t) such that ϕ(t0) = 0:1

2

(ϕ(t)+ 2π

q

)2

≤ M+|d|

+H

(H−1

(2π2

q2−λF

(ϕ(t0)+ 2π t0

q

)−d

)

+λ

∫ t

t0γ (s)

[ϕ(s) + 2π

q

]ds

)

≤ M + |d|+g

(∣∣∣∣H−1(2π2

q2−λF

(ϕ(t0)+ 2π t0

q

)−d

)∣∣∣∣

+∫ t

t0|γ (s)|

∣∣∣∣ϕ(s) + 2π

q

∣∣∣∣ ds)

≤ M+|d|+g

(Nq+

∫ t

t0|γ (s)|

∣∣∣∣ϕ(s)+ 2π

q

∣∣∣∣ ds)

where

Nq = sup{|H−1(x)|

∣∣∣∣2π2

q2−M ≤ x+d ≤ 2π2

q2+ M

}.

(37)

Note that we need that 2π2

q2− λF

(ϕ(t0) + 2π t0

q

)−

d ∈ J (the domain of H−1) in order that the inequalitymakes sense. Next arguing as before with

w(t) = Nq +∫ t

t0|γ (s)|

∣∣∣∣ϕ(s) + 2π

q

∣∣∣∣ ds,we get:

w(t) ≤ |γ (t)|√2[M + |d| + g(w(t))]≤ |γ (t)|√2(M + |d|) + |γ (t)|√2g(w(t))

and then

w(t) ≤ Nq + √2(M + |d|)Γ

+∫ t

t0|γ (s)|√2g(w(s))ds

for any t ∈ [t0, t0 + q], with Γ := ∫ q0 |γ (s)|ds. As

before Bihari’s inequality implies:

w(t) ≤ G−1(G

(Nq + Γ

√2(|d| + M)

)+

∫ t

t0|γ (s)| ds

),

∀t ∈ [t0, t0 + q]and then, since ϕ(t) is q-periodic:

|ϕ(t)| ≤ M2 := 2π

q+

(2(M + |d|

+ g(G−1

(G

(Nq + Γ

√2(|d| + M)

)+ Γ

))))1/2

,

for all t ∈ R. Thus, arguing as before we conclude withthe following.

Theorem 7 Assume (i), (ii) of Lemma 1, (C2), (A1),(A2), h is continuous satisfying (C1), and

(iv′) Γ <

∫ ∞

Nq+Γ√2(M+|d|)

du√2g(u)

,

(A′3)

[2π2

q2− d − M, 2π2

q2− d + M + M2

22

]⊂ J .

Then Eq. (34) has a q-shift-periodic solution, i.e.,satisfying (35).

Remark 7 Since 2π2

q2> 0 it is enough that J ⊃ [a,∞[

for some a ∈ R and −d > M + a for getting (A3) and(A′

3).

We conclude this section with the following.

Theorem 8 Assume (i), (ii) and (iv) of Lemma 1, (A1),(A3) and h is continuous satisfying (C1). Moreover,suppose it holds

(A4) γ (t + 12 ) = −γ (t) for any t ∈ [0, 1] and γ is

continuous.

Then Eq. (34) has a 1-periodic solution satisfying

ψ

(t + 1

2

)= −ψ(t) ∀t ∈ [0, 1]. (38)

Proof Let X,Y, Z be, resp., the space of C2, C0, andC1, 1-periodic functions satisfying (38). By (A3), wemay take M1 < M1 so that[−d − M,−d + M + M2

1

2

]⊂ J.

123



Furthermore, for any function v ∈ Y the equation

ψ(t) = v(t)

has the unique 1-periodic solution

ψ(t) = (Lav)(t) :=∫ t

0v(s)ds − 1

2

∫ 1/2

0v(s)ds

satisfying (38). Clearly ‖ψ‖0 ≤ 54‖v‖0, so La : Y →

Z is continuous. This gives also ‖ψ‖0 ≤ 54‖ψ‖0. Sowe

can take the norm ‖ψ‖1 := ‖ψ‖0 on Z . Like above, letBM1

(0) be the ball in Z of radius M1 and center 0. ThenL2aFλ : BM1

(0) → Z is well defined, continuous andcompact. Next from Lemma 1 it follows that equationψ = L2

aFλψ has no solutions in the boundary of ballBM1

(0). Hence we get

deg(I − L2

aF1, BM1(0), 0

)= 1.

The proof is finished.

Remark 8 Condition (38) means that ψ(t) is 12 -anti-

periodic [52,53] and of course such ψ is 1-periodic.

5 Applications to PT-symmetric dimer withvarying gain–loss

First, we apply results of Sect. 3. Following the Intro-duction, we study the coupled system

ψ = 4(c2 sinψ + εγ (t)z

)

z = εγ (t)ψ (39)

where γ (t) is 1-periodic andC2-smooth, ε is small andc > 0. It has a form of (15) with f (ψ) = 4c2 sinψ

and h(z) = 4z. So now F(ψ) = 4c2(1 − cosψ) andH(z) = 2z2 with either I =] − ∞, 0[ or I =]0,∞[,J =]0,∞[, and (C1) holds. Then (18) has the form

z = ±√

ψ2

4+ 2c2(cosψ − 1) − d

2, (40)

and (19) is now

ψ = 4

⎛⎝c2 sinψ ± εγ (t)

√ψ2

4+ 2c2(cosψ − 1) − d

2

⎞⎠ .

(41)

The unperturbed equation of (41) is given by (see(20))

ψ = 4c2 sinψ, (42)

which possesses a family of periodic solutions [54,pp. 114–115]

ψk,c(t) = π + 2 arcsin (k sn (2ct, k))

with periods

T (k, c) = 2

cK (k),

where K (k) is the complete elliptic integral of the firstkind, sn is a Jacobi elliptic function and k ∈ (0, 1).Note T (k, c) → ∞ as k → 1 and T (k, c) → π

c ask → 0. Moreover it holds

ψk,c(t) = 4ck cn (2ct, k) ,

where cn is a Jacobi elliptic function. Finally, we note

ψ2k,c

2+ 4c2

(cosψk,c − 1

) = −8c2(1 − k2).

So we suppose

d < −8c2(1 − k2) (43)

to get (H2). Resonance assumption (R) has the form

(Rp) K (k) = cp2q for p, q ∈ N.

Applying Theorem 1, we obtain the following result.

Theorem 9 Suppose (Rp), (43) and that there is anα0 ∈ R such that∫ T

0γ (t + α0) cn (2ct, k) dt = 0

and∫ T

0γ (t + α0) cn (2ct, k) dt �= 0.

Then there is a δ > 0 such that for any 0 �= ε ∈(−δ, δ) there are precisely two p-periodic solutions(ψ±, z±) of (39) with

(ψ±(t), z±(t))

=(

ψk,c(t − α0),±√

−4c2(1 − k2) − d

2

)+ O(ε).

Next, (C2) clearly holds. (H3) is satisfied for [54,p. 117]

ψ lk,c(t) = π + 2 am

(2ct

k, k

)

with periods

T l(k, c) = K (k)k

c,

where am is a Jacobi elliptic function and k ∈ (0, 1).Moreover it holds

ψ lk,c(t) = 4c

kdn

(2ct

k, k

),

123



where dn is a Jacobi elliptic function. Finally, we note

ψ l2k,c

2+ 4c2

(cosψ l

k,c − 1)

= 8c2

k2(1 − k2).

So we suppose

d <8c2

k2(1 − k2) (44)

to get (H ′2). Resonance assumption (R′) has the form

(R′p) K (k)k = cp

q for p, q ∈ N.

Applying Theorem 2, we obtain the following result.

Theorem 10 Suppose (R′p), (44) and that there is an

α0 ∈ R such that∫ T

0γ (t + α0)dn

(2ct

k, k

)dt = 0

and∫ T

0γ (t + α0)dn

(2ct

k, k

)dt �= 0.

Then there is a δ > 0 such that for any 0 �= ε ∈(−δ, δ) there are precisely two p-shift-periodic solu-tions (ψ l±, zl±) of (39) with

(ψ l±(t), zl±(t))

=⎛⎝ψk,c(t − α0),±

√4c2

k2(1 − k2) − d

2

⎞⎠ + O(ε).

Similarly, we can study heteroclinicMelnikov bifur-cation to (39). The unperturbed Eq. (42) has a hetero-clinic cycle

ψc,±(t) = π ± 2 arctan (sinh(2ct))

with

ψc,±(t) = ±4c sech(2ct),

and

ψ2c,±2

+ 4c2(cosψc,± − 1

) = 0.

So we consider

d < 0, (45)

which is a limit of (43) and (44) as k → 1−. ApplyingTheorem 3, we have the following result.

Theorem 11 Suppose (45) and that there is an α0 ∈ R

such that∫ ∞

−∞γ (t + α0) sech(2ct)dt = 0

and∫ ∞

−∞γ (t + α0) sech(2ct)dt �= 0.

Then there is a δ > 0 such that for any 0 �=ε ∈ (−δ, δ) there are precisely four bounded solutions(ψ±± , z±±) of (39) with

(ψ±± (t), z±±(t))

=(π ± 2 arctan (sinh(2c(t − α0))) ,±

√−d

2

)+ O(ε).

Clearly, Remarks 4 and 5 can be applied to (39). Toapply Remark 6, we shift (41) by ψ(t) = π + ϕ(t) toget

ϕ = θ,

θ = 4

⎛⎝−c2 sin ϕ ± εγ (t)

√θ2

4− 2c2(cosϕ + 1) − d

2

⎞⎠ .

(46)

The unperturbed equation of (46) when ε = 0 is justthe pendulum equation

ϕ = θ,

θ = −4c2 sin ϕ,

which possesses well-known families of solutions [54,pp. 115–117] mentioned above, and so Remark 6 canbe used to (46).

Remark 9 The Melnikov functions in Theorems 9, 10and 11 have concrete forms for a concrete function γ .For instance, taking γ (t) = a + b cos 2π t , a, b ∈ R

and following the standardway as in, e.g., [32, pp. 191–193, 198–204], [54, pp. 509, 867], we can derive thatthey have the form Υ cos 2πα0 for suitable constant Υfor each case, where of course, Υ has non-trivial ana-lytical form in general, but independent of a. For thisreason, we do not go into details but rather we under-take numerical computations in the next Sect. 6. Cer-tainly, we obtainmore complicatedMelnikov functionsof the form Υ1 cos 2πnα0 +Υ2 cos 2πmα0 for suitableconstants Υ1,2 by considering γ (t) = a cos 2πnt +b cos 2πmt , a, b ∈ R, n,m ∈ N. We recall Remark 5for a general γ .

Now we suppose in (39) that c → 0 as ε → 0, sowe consider

ψ = 4 (εc(ε) sinψ + εγ (t)z)z = εγ (t)ψ

(47)

where ε > 0 is small and c(ε) > 0 isC2-smooth aroundε = 0. Then (47) has a form of (26). Assumption (29)reads now

d < 0. (48)

123



The Melnikov function (30) has now the form

Maver (α) = c(0) sin α ±√−d

2

∫ 1

0γ (t)dt. (49)

By applying Theorem 4 we get the next.

Theorem 12 Assume that

c(0) > 0,

√−d

2

∣∣∣∣∫ 1

0γ (t)dt

∣∣∣∣ < c(0), (50)

then there are exactly four 1-periodic solutions of (39)with c2 = εc(ε) and ε > 0 small.

Proof Condition (50) gives that (49) has 2 simple rootsover [0, 2π). The proof is finished.

On the other hand, if√−d

2

∣∣∣∣∫ 1

0γ (t)dt

∣∣∣∣ > c(0),

such (39) has no 1-periodic solutions for ε > 0 small.In the last part of this section, we apply results of

Sect. 4 to (3) when γ (t) is 1-periodic and continuous,and c > 0. First, assumptions (i), (ii) of Lemma 1hold with M = 8c2 and g(z) = 2z2. Since

∫ ∞ dz2z =

∞, assumptions (iii) and (iv) of Lemma 1 are satisfiedprovided

d < −8c2. (51)

We already verified (C1), (C2), and (A1) holds aswell. Since now J =]0,∞[, it follows from Remark 7that (A3) and (A′

3) are satisfied when (51) holds.Summarizing, we see that results of Sect. 4 can be

used to (3) under assumption (51) and γ (t) satisfieseither (A2) or (A4), respectively. On the other hand,we need

ψ2

4≥ 2c2(1 − cosψ) + d

2. (52)

Since

ψ2

4≥ 2c2(1 − cosψ) + d

2≥ d

2,

we see that if

d > 0

then ψ(t) �= 0, so ψ(t) can be neither periodic norhomoclinic nor heteroclinic. If d < 0, then by (43)and (44), (3) may have oscillating solutions for suitableγ (t). If d = 0, then condition ψ(t0) = 0 and (52)give cosψ(t0) = 1, and this situation occurs whenγ (t) = 0, for instance.

6 Numerical results

To illustrate the analytical results reported in the previ-ous sections, we integrate the governing Eq. (3) numer-ically using a multistep Adam–Bashforth–Moultonmethod. The initial conditions ψ(0) and ψ(0) are cho-sen independently, while z(0) satisfies (40) for a fixedd ∈ R, such that the solution curves of (3) lie on thethree-dimensional surface (16). The solution trajecto-ries can therefore be conveniently plotted on the, e.g.,(ψ, ψ)-plane (cf. (19), (41)). Throughout the section,we set d = −8.5. This value is taken following theresult of Sects. 3 and 4 (see also (51)) as a necessarycondition for the existence of the various periodic solu-tions.

First, we consider c = 1. In the absence of gain–loss parameter, i.e., γ = 0, the governing equationreduces to the well-known pendulum equation that hastwo families of solutions, namely the periodic (libra-tion) and the p-shift-periodic (rotation) solutions. Thesolution families are separated by heteroclinic man-ifolds. In Fig. 1, the phase portrait of (3) is plottedfor a small value of constant γ . In the presence ofa non-vanishing gain–loss parameter, while periodicsolutions persist, there are no longer p-shift-periodicsolutions. Instead of heteroclinic trajectories, one hashomoclinic connections with unbounded solutions out-side the invariant manifolds. By increasing the gain–loss parameter γ , there will be a critical value γc abovewhich no periodic solutions exist and all solutionsbecome unbounded. This region is referred to as the

0 1 2 3 4 5 6−5

0

5

ψ

ψ

Fig. 1 Phase portrait of Eq. (3) with γ = 0.01

123



0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(a)

0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(b)

Fig. 2 Stroboscopic plots of the system (3) with (53) for twovalues of ε. a ε = 0.2. b ε = 2.0

PT-broken phase regime and the transition has beencompletely discussed in [24]. For the parameters above,γc ≈ 0.79.

When γ (t) is a periodic function of time, rather thanshowing the phase portraits with a continuous time, itbecomes convenient to represent the solution trajec-tories in Poincaré maps (stroboscopic plots at everyperiod T = 1). In recurrence maps, a stable periodicorbit will correspond to an elliptical fixed point encir-cled by closed regions (islands).

As a particular choice, we consider

γ (t) = ε cos(2π t). (53)

0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(a)

0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(b)

Fig. 3 The same as Fig. 2, but for γ (t) defined in (54). a ε = 0.2.b ε = 0.5

The cosine function is taken to respect the PT-reversibility of the governing equation. Plotted in Fig. 2is the Poincaré map of (3) with (53), obtained fromvarious sets of initial conditions using direct numericalintegrations of the governing equations. If Fig. 1 is anordered dynamical picture, Fig. 2 have visible chaotic(stochastic) regions.

Figure 2a corresponds to a relatively small value ofε. The chaotic layer in the figure is caused by the pres-ence of heteroclinic manifolds connecting (ψ, ψ) =(0, 0) and (2π, 0), i.e., separatrix chaos, as proven inTheorem 11 (see also Remark 4).

Within the region bounded by the chaotic layer, thepresence of closed islands is observable. The centers

123



of the islands correspond to stable p-periodic solutionsshown to persist in Theorem 9. Beyond the chaoticlayer, there are also islands with centers correspond-ing to stable p-shift-periodic solutions, i.e., boundedrunning modes with period p, discussed in Theorem10, which do not exist in the system with constantγ .

Wehave increased the strength of the gain–loss coef-ficient ε. Depicted in Fig. 2b is the Poincaré map forε = 2. Despite the significant expansion of the chaoticregion, one can observe similar features as in Fig. 2a.The presence of periodic solutions and p-shift-periodicsolutions are guaranteed respectively by Theorems 6(and 8) and 7 as the gain–loss function (53) satis-fies the condition (A2) and (A4) under the translationt → (t −1/4). Note as well that the gain–loss strengthε is beyond the critical value of PT-broken phase forconstant γ . Hence, similarly to the result reported in[25], we prove and observe a controllable expansion ofthe exact PT-symmetry region.

On the interesting persistence of p-shift-periodicsolutions observed in Fig. 2, it may be unsurprisingas the parametric drive (53) has zero average. Never-theless, the sufficient condition in Theorem 10 doesnot require a zero-averaged γ . To illustrate it, next weconsider the gain–loss function

γ (t) = 10−2 + ε cos(2π t). (54)

When ε = 0, one will obtain the phase portrait inFig. 1.

Shown in Fig. 3 are the phase portraits of the systemwith the gain–loss function (54) for two values of ε.In both cases, periodic solutions exist. For ε = 0.2 inpanel (a), we have a similar plot as in Fig. 1, wherethere is no bounded p-shift-periodic solution. Interest-ingly, when ε is large enough such that the conditionsin Theorem 10 can be satisfied, we observe islandsin the region of unbounded solutions, see panel (b).Hence, we obtain (stable) bounded rotating solutionsas expected from the theorem (see also Remark 9).

Next, we consider the case when both the gain–lossstrength and the parameter c2 are small. We present inFig. 4 the Poincaré maps of the system for two smallvalues of c and the gain–loss function (54) with ε =0.2.

Note that using Theorem 12 (cf. Theorem 4), peri-odic solutions exist if and only if the parameter c2

exceeds the threshold value

0 1 2 3 4 5 6−4

−3

−2

−1

0

1

2

3

4

ψ

ψ

(a)

0 1 2 3 4 5 6−4

−3

−2

−1

0

1

2

3

4

ψ

ψ

(b)

Fig. 4 The same as Fig. 3, but for a c2 = 0.05 and b c2 = 0.02

√−d

2

∣∣∣∣∫ 1

0γ (t) dt

∣∣∣∣ ≈ 0.0206. (55)

In panel (a) of Fig. 4, c2 is above the value and thepresence of periodic solutions can be clearly seen. Inpanel (b), we decrease the value of c2 below (55) and ingood agreement with the theorem there is nomore peri-odic solution. All solutions become unbounded. There-fore, Theorem 12 provides an approximate thresholdvalue for PT-broken phase.

Finally, note that for the analysis in Sect. 3 the gen-eral nonlinearity f (ψ) and the driving function γ (t)are required to be C2-smooth, while in Sect. 4, theyneed to be continuous and odd only. In the following,we consider the situation when the driving function or

123



Fig. 5 The same as Fig. 2,but for γ (t) defined in (56).In a, b, we set c=1 and taketwo different values of ε, aε = 0.15, b ε = 0.20. In c,d, we take ε = 0.2 andc2 = 0.05, d c2 = 0.02

0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(a)

0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(b)

0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(c)

0 1 2 3 4 5 6−10

−5

0

5

10

ψψ

(d)

the nonlinearity does not satisfy the assumption and iseven discontinuous. First, we consider the same Eq. (3)with the driving amplitude given by

γ (t) = 10−2 + ε sign (cos(2π t)) . (56)

It can be noted that the drive is even, i.e., odd witha phase-shift in time, but not continuous. Shown inFig. 5a, b are the stroboscopic plots of the system forc=1 and two different values of ε.

Comparing the figure with Fig. 3, it is interesting tonote that despite the drive beingdiscontinuous, the plotsare still similar, i.e., we observe islands in the region ofunbounded solutions when the conditions in Theorem10 are satisfied for large enough driving amplitude.

We also plot the Poincaré maps of the system withthe same drive for two different c in Fig. 5c, d. Compar-ing the panels with Fig. 4, one can conclude that eventhough the drive does not satisfy the continuity con-dition, Theorem 12 may still provide an approximatethreshold value for PT-broken phase.

Next, we also consider the general Eq. (4) with

f (ψ) = 4c2 sign

(cos

(ψ

2

))sin

(ψ

2

), h(z) = 4z.

(57)

We can also easily note that the nonlinearity f isdiscontinuous. Nevertheless, the phase portrait of thesystem especially when γ (t) ≡ 0 is simply a compos-ite of those of (4) with f (ψ) = ±4c2 sin (ψ/2) (thereader is referred to, e.g., [55] and references thereinfor the ‘composite’ method).

Using the driving function (54), our numerical sim-ulations show that the system (4) with the discontinu-ous nonlinearity (57) also admits the same qualitativebehavior as those with the continuous nonlinearity. Asillustrations, we depict in Fig. 6 the corresponding plotsof Figs. 3 and 4 (see also Fig. 5), where one can notethe similarity between them in the existence of p-shift-periodic solutions and that of PT-broken phase.

Our numerical simulations indicate that the theoret-ical analysis above may still be applicable and relevantto cases that do not satisfy the required assumptions.

123



Fig. 6 The same as Fig. 5,but for the nonlinearity (57)and γ (t) defined in (54). Ina, b, we set c = 1 and taketwo different values of ε aε = 0.2, b ε = 0.5. In c, d,we take ε = 0.2 and cc2 = 0.05, d c2 = 0.02

0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(a)

0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(b)

0 1 2 3 4 5 6−10

−5

0

5

10

ψ

ψ

(c)

0 1 2 3 4 5 6−10

−5

0

5

10

ψψ

(d)

However, careful analysis is required as in [46,56–58]and will be addressed elsewhere.

Acknowledgments FB is partially supportedbyPRIN-MURSTEquazioni Differenziali Ordinarie e Applicazioni. JD is par-tially supported by Grant GACR P201/11/0768. MF is par-tially supported by Grant VEGA-MS 1/0071/14. MP is sup-ported by Project No. CZ.1.07/2.3.00/30.0005 funded by Euro-pean Regional Development Fund.

References

1. Bender, C.M.: Making sense of non-Hermitian Hamiltoni-ans. Rep. Prog. Phys. 70, 947–1018 (2007)

2. El-Ganainy, R., Makris, K.G., Christodoulides, D.N., Mus-slimani, Z.H.: Theory of coupled optical PT-symmetricstructures. Opt. Lett. 32, 2632–2634 (2007)

3. Klaiman, S., Guenther, U., Moiseyev, N.: Visualization ofbranch points in PT-symmetric waveguides. Phys. Rev. Lett.101, 080402 (2008)

4. Ruschhaupt, A., Delgado, F., Muga, J.G.: Physical realiza-tion of PT-symmetric potential scattering in a planar slabwaveguide. J. Phys. A 38, L171–L176 (2005)

5. Guo, A., Salamo, G.J., Duchesne, D., Morandotti,R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A.,

Christodoulides, D.N.: Observation of PT-symmetry break-ing in complex optical potentials. Phys. Rev. Lett. 103,093902 (2009)

6. Rueter, C.E.,Makris,K.G., El-Ganainy,R., Christodoulides,D.N., Segev,M., Kip, D.: Observation of paritytime symme-try in optics. Nat. Phys. 6, 192–195 (2010)

7. Alexeeva, N.V., Barashenkov, I.V., Rayanov, K., Flach, S.:Actively coupled optical waveguides. Phys. Rev. A 89,013848 (2014)

8. Duanmu, M., Li, K., Horne, R.L., Kevrekidis, P.G.,Whitaker, N.: Linear and nonlinear parity-time-symmetricoligomers: a dynamical systems analysis. Philos. Trans. R.Soc. A 371, 20120171 (2013)

9. Li, K., Kevrekidis, P.G.: PT-symmetric oligomers: analyti-cal solutions, linear stability, and nonlinear dynamics. Phys.Rev. E 83, 066608 (2011)

10. Miroshnichenko, A.E., Malomed, B.A., Kivshar, Yu.S.:Nonlinearly PT-symmetric systems: spontaneous symme-try breaking and transmission resonances. Phys. Rev. A 84,012123 (2011)

11. Ramezani, H., Kottos, T., El-Ganainy, R., Christodoulides,D.N.: Unidirectional nonlinear PT-symmetric optical struc-tures. Phys. Rev. A 82, 043803 (2010)

12. Sukhorukov, A.A., Xu, Z., Kivshar, Yu.S.: Nonlinear sup-pression of time reversals in PT-symmetric optical couplers.Phys. Rev. A 82, 043818 (2010)

123



13. Rodrigues, A.S., Li, K., Achilleos, V., Kevrekidis, P.G.,Frantzeskakis, D.J., Bender, C.M.: PT-symmetric double-well potentials revisited: bifurcations, stability and dynam-ics. Rom. Rep. Phys. 65, 5–26 (2013)

14. Cartarius, H., Wunner, G.: Model of a PT-symmetric Bose–Einstein condensate in a δ-function double-well potential.Phys. Rev. A 86, 013612 (2012)

15. Dast, D., Haag, D., Cartarius, H., Wunner, G., Eichler, R.,Main, J.: A Bose–Einstein condensate in a PT symmetricdouble well. Fortschr. Phys. 61, 124–139 (2013)

16. Graefe, E.M.: Stationary states of a PT symmetric two-mode Bose–Einstein condensate. J. Phys. A: Math. Theor.45, 444015 (2012)

17. Heiss, W.D., Cartarius, H., Wunner, G., Main, J.: Spectralsingularities in PT-symmetric Bose–Einstein condensates.J. Phys. A: Math. Theor. 46, 275307 (2013)

18. Bender, C.M., Boettcher, S.: Real spectra in non-HermitianHamiltonians having PT symmetry. Phys. Rev. Lett. 80,5243 (1998)

19. Bender, C.M., Boettcher, S., Meisinger, P.N.: PT-symmetricquantum mechanics. J. Math. Phys. 40, 2201–2209 (1999)

20. Chen, Y.J., Snyder, A.W., Payne, D.N.: Twin core nonlinearcouplers with gain and loss. IEEE J. Quantum Electron. 28,239–245 (1992)

21. Barashenkov, I.V.: Hamiltonian formulation of the standardPT-symmetric nonlinear Schrödinger dimer. Phys. Rev. A90, 045802 (2014)

22. Barashenkov, I.V., Jackson, G.S., Flach, S.: Blow-upregimes in the PT-symmetric coupler and the actively cou-pled dimer. Phys. Rev. A 88, 053817 (2013)

23. Kevrekidis, P.G., Pelinovsky, D.E., Tyugin, D.Y.: Nonlineardynamics in PT-symmetric lattices. J. Phys. A:Math. Theor.46, 365201 (2013)

24. Pickton, J., Susanto, H.: Integrability of PT-symmetricdimers. Phys. Rev. A 88, 063840 (2013)

25. Horne, R.L., Cuevas, J., Kevrekidis, P.G., Whitaker, N.,Abdullaev, F.Kh., Frantzeskakis, D.J.: PT-symmetry man-agement in oligomer systems. J. Phys. A: Math. Theor. 46,485101 (2013)

26. Driben, R., Malomed, B.A.: Stability of solitons in parity-time-symmetric couplers. Opt. Lett. 36, 4323–4325 (2011)

27. Driben, R., Malomed, B.A.: Stabilization of solitons in PTmodels with supersymmetry by periodic management. EPL96, 51001 (2011)

28. Valle, G.D., Longhi, S.: Spectral and transport propertiesof time-periodic PT-symmetric tight-binding lattices. Phys.Rev. A 87, 022119 (2013)

29. Konotop, V.V., Zezyulin, D.A.: Stochastic parity-time-symmetric coupler. Opt. Lett. 39, 1223–1226 (2014)

30. D’Ambroise, J., Malomed, B.A., Kevrekidis, P.G.: Quasi-energies, parametric resonances, and stability limits in ac-driven PT-symmetric systems. Chaos 24, 023136 (2014)

31. Psiachos, D., Lazarides, N., Tsironis, G.P.: PT-symmetricdimers with time-periodic gain/loss function. Appl. Phys. A117, 663–672 (2014)

32. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations,Dynamical Systems, and Bifurcations of Vector Fields.Springer, New York (1983)

33. Melnikov, V.K.: On the stability of the center for time peri-odic perturbations. Trans.Mosc.Math. Soc. 12, 1–57 (1963)

34. Holmes, P.J., Marsden, J.E.: A partial differential equationwith infinitely many periodic orbits: chaotic oscillations of aforced beam.Arch. Ration.Mech. Anal. 76, 135–165 (1981)

35. Hu, W.P., Deng, Z.C., Wang, B., Ouyang, H.J.: Chaos in anembedded single-walled carbon nanotube. Nonlinear Dyn.72, 389–398 (2013)

36. Chicone, C.: Ordinary Differential Equations with Applica-tions. Texts in Applied Mathematics, vol. 34. Springer, NewYork (2006)

37. Awrejcewicz, J., Holicke, M.M.: Smooth and NonsmoothHigh Dimensional Chaos and the Melnikov-Type Methods.World Scientific Publishing Co, Singapore (2007)

38. Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory.Springer, New York (1982)

39. Palmer, K.J.: Shadowing in Dynamical Systems, Theoryand Applications. Kluwer Academic Publishers, Dordrecht(2000)

40. Wiggins, S.: Global Bifurcations and Chaos, AnalyticalMethods. AppliedMathematical Sciences, vol. 73. Springer,New York (1988)

41. Berger, M.S.: Nonlinearity and Functional Analysis. Acad-emic Press, New York (1977)

42. Battelli, F., Lazzari, C.: Exponential dichotomies, hetero-clinic orbits, and Melnikov functions. J. Differ. Equ. 86,342–366 (1990)

43. Gruendler, J.: Homoclinic solutions for autonomous ordi-nary differential equations with nonautonomous perturba-tions. J. Differ. Equ. 122, 1–26 (1995)

44. Battelli, F., Feckan, M.: Chaos arising near a topologicallytransversal homoclinic set. Topol.Meth.NonlinearAnal. 20,195–215 (2002)

45. Devaney, R.: Reversible diffeomorphisms and flows. Trans.Am. Math. Soc. 218, 89–113 (1976)

46. Feckan, M.: Topological Degree Approach to BifurcationProblems. Springer, Berlin (2008)

47. Feckan,M.: Topologically transversal reversible homoclinicsets. Proc. Am. Math. Soc. 130, 3369–3377 (2002)

48. Vanderbauwhede, A., Fiedler, B.: Homoclinic period blow-up in reversible and conservative systems. Z. Angew. Math.Phys. 43, 292–318 (1992)

49. Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Meth-ods in Nonlinear Dynamical Systems. Applied Mathemati-cal Sciences, vol. 59, 2nd edn. Springer, New York (2007)

50. Mawhin, J.: Equivalence theorems for nonlinear operatorequations and coincidence degree theory for somemappingsin locally convex topological vector spaces. J. Differ. Equ.12, 610–636 (1972)

51. Bihari, I.: A generalization of a lemma of Bellman and itsapplication to uniqueness problems of differential equations.Acta Math. Hung. 7, 81–94 (1956)

52. Aizicovici, S., Feckan, M.: Anti-periodic forced oscillationsof damped beams on elastic bearings. Dyn. Partial Differ.Equ. 1, 339–357 (2004)

53. Haraux, A.: Anti-periodic solutions of some nonlinear evo-lution equations. Manuscr. Math. 63, 479–505 (1989)

54. Lawden,D.F.: Elliptic Functions andApplications. Springer,New York (1989)

55. Knight, C.J.K., Derks, G., Doelman, A., Susanto, H.: Sta-bility of stationary fronts in a non-linear wave equation withspatial inhomogeneity. J. Differ. Equ. 254, 408–468 (2013)

123



56. Battelli, F., Feckan,M.: Nonsmooth homoclinic orbits, Mel-nikov functions and chaos in discontinuous systems. Phys.D 241, 1962–1975 (2012)

57. Feckan, M.: Bifurcation and Chaos in Discontinuous andContinuous Systems. HEP-Springer, Berlin (2011)

58. Feckan, M., Pospíšil, M.: On the bifurcation of periodicorbits in discontinuous systems. Commun. Math. Anal. 8,87–108 (2010)

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