Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials
Two non-harmonic canonical-dissipative limit cycle oscillators are considered that oscillate in one-dimensional Smorodinsky-Winternitz potentials. It is shown that the standard approach of the canonical-dissipative framework to introduce dissipative forces leads naturally to a coupling force betwee...
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irk-123456789-1570352019-06-20T01:27:47Z Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials Mongkolsakulvong, S. Frank, T.D. Two non-harmonic canonical-dissipative limit cycle oscillators are considered that oscillate in one-dimensional Smorodinsky-Winternitz potentials. It is shown that the standard approach of the canonical-dissipative framework to introduce dissipative forces leads naturally to a coupling force between the oscillators that establishes synchronization. The non-harmonic character of the limit cycles in the context of anchoring, the phase difference between the synchronized oscillators, and the degree of synchronization are studied in detail. Розглянуто канонiчно-дисипативнi граничнi цикли для двох негармонiчних осциляторiв, що коливаються в одномiрних потенцiалах Смородинського-Вiнтернiца. Показано, що стандартний канонiчнодисипативний пiдхiд iз введенням дисипативних сил природно приводить до появи взаємодiї мiж осциляторами, яка синхронiзує їх рух. Детально дослiджено негармонiчний характер граничних циклiв у контекстi анкерування, рiзницю фаз мiж синхронiзованими осциляторами та ступiнь їх синхронiзацiї. 2017 Article Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials / S. Mongkolsakulvong, T.D. Frank // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 44001: 1–7. — Бібліогр.: 30 назв. — англ. 1607-324X PACS: 05.45.-a, 05.40.Jc, 87.19.rs DOI:10.5488/CMP.20.44001 arXiv:1712.05378 http://dspace.nbuv.gov.ua/handle/123456789/157035 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Two non-harmonic canonical-dissipative limit cycle oscillators are considered that oscillate in one-dimensional
Smorodinsky-Winternitz potentials. It is shown that the standard approach of the canonical-dissipative framework to introduce dissipative forces leads naturally to a coupling force between the oscillators that establishes
synchronization. The non-harmonic character of the limit cycles in the context of anchoring, the phase difference between the synchronized oscillators, and the degree of synchronization are studied in detail. |
| format |
Article |
| author |
Mongkolsakulvong, S. Frank, T.D. |
| spellingShingle |
Mongkolsakulvong, S. Frank, T.D. Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials Condensed Matter Physics |
| author_facet |
Mongkolsakulvong, S. Frank, T.D. |
| author_sort |
Mongkolsakulvong, S. |
| title |
Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials |
| title_short |
Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials |
| title_full |
Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials |
| title_fullStr |
Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials |
| title_full_unstemmed |
Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials |
| title_sort |
synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via smorodinsky-winternitz potentials |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157035 |
| citation_txt |
Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials / S. Mongkolsakulvong, T.D. Frank // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 44001: 1–7. — Бібліогр.: 30 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT mongkolsakulvongs synchronizationandanchoringoftwononharmoniccanonicaldissipativeoscillatorsviasmorodinskywinternitzpotentials AT franktd synchronizationandanchoringoftwononharmoniccanonicaldissipativeoscillatorsviasmorodinskywinternitzpotentials |
| first_indexed |
2025-07-14T09:22:35Z |
| last_indexed |
2025-07-14T09:22:35Z |
| _version_ |
1837613665059078144 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 4, 44001: 1–7
DOI: 10.5488/CMP.20.44001
http://www.icmp.lviv.ua/journal
Rapid communication
Synchronization and anchoring of two non-harmonic
canonical-dissipative oscillators via
Smorodinsky-Winternitz potentials
S. Mongkolsakulvong1, T.D. Frank2,3
1 Faculty of Science, Department of Physics, Kasetsart University, Bangkok 10900, Thailand
2 CESPA, Department of Psychology, University of Connecticut, 406 Babbidge Road, Storrs, CT 06269, USA
3 Department of Physics, University of Connecticut, 2152 Hillside Road, Storrs, CT 06269, USA
Received July 18, 2017
Two non-harmonic canonical-dissipative limit cycle oscillators are considered that oscillate in one-dimensional
Smorodinsky-Winternitz potentials. It is shown that the standard approach of the canonical-dissipative frame-
work to introduce dissipative forces leads naturally to a coupling force between the oscillators that establishes
synchronization. The non-harmonic character of the limit cycles in the context of anchoring, the phase differ-
ence between the synchronized oscillators, and the degree of synchronization are studied in detail.
Key words: canonical-dissipative systems, Smorodinsky-Winternitz potentials, synchronization, anchoring
PACS: 05.45.-a, 05.40.Jc, 87.19.rs
Mass-spring systems are a fundamental topic of classical mechanics and solid state physics. A
sophisticated theoretical framework is available to explain the oscillatory dynamics of macroscopic
particles connected by springs and the oscillatory vibration of molecules interacting by spring-like
forces. Given the success of this field of physics concerned with the inanimate world, the question
naturally arises whether its scope can be broadened to take the life science into account [1]. Since a key
feature of living systems is their ability to move by themselves, the question can be asked from a slightly
different perspective. Can the concepts of classical mechanics be generalized to self-mobile, so-called
active [2–4], systems? A well-studied class of active systems both in the animate and inanimate world
are self-oscillators [5]. A theoretical framework that bridges the research fields of classical mechanics
and self-oscillators is the theory of canonical-dissipative (CD) systems [6–11]. The reason for this is
that a CD system can exhibit attractors that, on the one hand, are stable and in doing so reflect non-
conservative, dissipative system components but, on the other hand, are defined in their respective phase
spaces by the dynamics of conservative systems. In fact, in a series of recent experimental studies it has
been shown that the CD approach can be applied to human self-oscillators, that is, humans producing
oscillatory single limb movements [12–14]. Importantly, this line of research has been generalized to the
non-harmonic case [15]. In general, human rhythmic limb movements exhibit non-harmonic components
and in particular can show a so-called anchoring phenomenon. Anchoring means that a limb movement
slows down during a particular short period of the cyclic activity in a more pronounced way than in
the harmonic case. Non-harmonic self-oscillator models are promising candidates to capture human
non-harmonic rhythmic activity including the anchoring phenomenon. However, humans and animals
are known to coordinate their activities, in general, and movement patterns, in particular [16, 17]. As far
as the CD approach is concerned, for synchronization with zero phase lag and 180 degrees phase lag, a
four-variable CD model has been proposed recently [18]. Here, we proposed a more general model for
two active particles that is motivated by the so-called SET model for swarming [19] and assumes that
active particles are coupled via their angular momentum values (see also [20]). In order to address the
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
44001-1
https://doi.org/10.5488/CMP.20.44001
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
S. Mongkolsakulvong, T.D. Frank
non-harmonic case, we consider self-oscillations in Smorodinsky-Winternitz potentials [21–23]. These
potentials play an important role in physics as confinement potentials [24, 25]. In the context of the CD
approach, the four-dimensional, two particle Smorodinsky-Winternitz systems should be considered as
benchmark systems because they feature three invariants rather than only two. The first two invariants
are the particle Hamiltonian energy functions. The third invariant is an appropriately adjusted angular
momentum [21].
Recall that the standard CD oscillator in a one-dimensional space with coordinate q and momentum
p is defined by [8, 10, 11, 26]
d
dt
q =
p
m
,
d
dt
p = −kq −
∂g
∂p
, g =
γ
2
(H − B)2, (1)
where m denotes mass, k is the spring constant and H = p2/(2m)+ kq2/2 corresponds to the Hamiltonian
energy in the conservative case in which the function g is neglected. The function g describes the
dissipative mechanism. The parameter γ > 0 is the coupling parameter of oscillator with the dissipative
mechanism. Note that −∂g/∂p = −γp(H − B)/m such that the dissipative mechanism is composed of
a negative friction term (i.e., pumping mechanism) +γBp/m with pumping parameter B > 0 and a
nonlinear friction term −γpH/m. The amplitude dynamics can be obtained using standard techniques.
We put q = A(t) exp(iωt)+c.c. withω2 = k/m, where A denotes the complex-valued oscillator amplitude
that is related to the real-valued amplitude r(t) and the oscillator phase φ(t) like A(t) = r(t) exp[iφ(t)]/2.
Here and in what follows, c.c. denotes the complex conjugate expression. Assuming that γ is a small
perturbation parameter, by means of the slowly varying amplitude approximation and the rotating wave
approximation [27], we obtain, in lowest order of γ, the following amplitude dynamics
d
dt
A = −γω2 A
(
|A|2 −
B
2mω2
)
. (2)
Note that higher order correction terms in γ can be obtained using alternative techniques (see e.g., [28]).
From equation (2) it follows that dr/dt = −γω2r[r2 − 2B/(mω2)]/4 and dφ/dt = 0. The solution reads
r(t) =
√
2Br2
0/[mω
2r2
0 + (2B − mω2r2
0 ) exp(−γBt/m)]with r0 = r(t = 0). Figure 1 (a) shows a simulation
of the self-oscillator (2) and the analytical solution r(t). From equations (2) and the analytical solution
r(t) it follows that γ (in combination with the factor B/m) determines the time scale of the amplitude
dynamics A(t) and r(t), respectively. Importantly, in the long time limit, r approaches rst =
√
2B/(mω2)
and H converges to the pumping parameter B. That is, B acts as a fixed point value or target value for the
energy dynamics H(t).
Let us generalize the single-oscillator case to a model of two coupled self-oscillators. Each self-
oscillator oscillates in a one-dimensional space and is subjected to the force of a Smorodinsky-Winternitz
potential. Let us describe the oscillator coordinates q1 and q2 and momenta p1 and p2 by means of the
vectors q = (q1, q2) and p = (p1, p2). The Smorodinsky-Winternitz potentials read [21]
V1(q1) = k
q2
1
2
+
α
2q2
1
, V2(q2) = k
q2
2
2
+
β
2q2
2
(3)
with k > 0 and α, β > 0. For α = β = 0, the potentials correspond to parabolic potentials and k can
be interpreted as a spring constant. For α, β > 0, the potentials exhibit minima at q1 = ±(α/k)1/4 and
q2 = ±(β/k)1/4 and exhibit repulsive singularities at q1 = 0 and q2 = 0. By contrast, for q1 → ±∞ and
q2 → ±∞ they increase like parabolic potentials. Therefore, for α, β > 0, the potentials are asymmetric
with respect to their minima. Let us define the dynamics of the two oscillators in the conservative case
by means of the Hamiltonian dynamics
d
dt
q = ∂Htot
∂p ,
d
dt
p = −∂Htot
∂q , Htot = H1 + H2 , H1 =
p2
1
2m
+ V1 , H2 =
p2
2
2m
+ V2 . (4)
In equation (4), the functions H1, H2 and Htot correspond to the Hamiltonian energy functions of
the individual oscillators and the total energy of the two-oscillators system. It can be shown that the
44001-2
Synchronization with anchoring in Smorodinsky-Winternitz potentials
Figure 1. Panel (a): Solution q(t) (solid line) of equation (1) obtained by a numerical Euler forward (EF)
solutionmethod. The analytical solution r(t) is shown aswell (dashed line). Parameters in a.u.:m = k = 1,
γ = 0.1, B = 3. EF time step: τ = 0.01. Initial value: q(0) = 0.1, p(0) = 0⇒ r(0) = 0.1. Panel (b): Phase
difference ψ as function of η (solid lines) as predicted by our theoretical considerations for the harmonic
case (see text). Circles denote simulation results obtained by solving equation (6) numerically (EF).
Simulation parameters in a.u.: m = 1, k = (2π)2 ⇒ ω = 2π (i.e., oscillator period equal to 1 time
unit), α = β = 0, B1 = 3, B2 = 5, B3 was varied in the range [−0.5, 1.5], τ = 0.001. Various initial
conditions were used. Panel (c): Trajectories q1(t) and q2(t) of equation (6) for a representative simulation
trial used to generate the numerical results in panel (b). Top and middle sub-panels show transient and
long term dynamics. Bottom sub-panel shows the synchronized state with a fixed phase difference. Here:
B1 = 1.0⇒ η = 0.66 , ψ = 54◦. Panels (d) and (e): Phase portraits p1 versus q1 [panel (d)] and p2 versus
q2 [panel (e)] obtained in the non-harmonic case by solving equation (6) numerically (EF). Parameters
in a.u.: m = 1, k = (2π)2 ⇒ ω = 4π (i.e. oscillator period equal to 0.5 time units), α = 3, β = 2,
γ1,2 = 0.1, γ3 = 0.2, B1 = H1,min + 1, B2 = H2,min + 5, B3 = S3,min + 3, τ = 0.001. Initial conditions:
p1 = 5.0, p2 = 0.3, q1 = (α/k)0.25+0.5, q2 = (β/k)0.25+0.1. The circles show the predicted limit cycles
obtained by solving numerically (EF) the evolution equations of the corresponding isolated, conservative
oscillators. Panel (f): Maximal cross-correlation as function of γ3 obtained by solving equation (14)
numerically (stochastic EF [29]). Averages of 10 trials are shown. Trajectories of 10000 (circles) and
30000 (squares) time units were used in each trial. The maximal cross correlation scores for γ3 = 0 are
by-chance values that decay to zero when trajectory length goes to infinity. Parameters in a.u.: D = 0.02,
all other parameters except for γ3 as in panels (d) and (e). τ = 0.001.
dynamics (4) exhibits three invariants Sj with j = 1, 2, 3 given by [21]
S1 = H1 , S2 = H2 , S3 =
L2
m
+ (q2
1 + q2
2)
(
α
q2
1
+
β
q2
2
)
, (5)
where L denotes the angular momentum L = p2q1 − p1q2. In line with the CD oscillator (1), we define
the CD case like
d
dt
q = ∂H
∂p ,
d
dt
p = −∂H
∂q −
∂gtot
∂p , gtot =
3∑
j=1
gj , gj = γj
1
2
(
Sj − Bj
)2
, (6)
where γj > 0 are the coupling constants. B1,2 > 0 are the pumping parameters. B3 is a target value for S3.
Importantly, since Sj are invariants of the conservative dynamics (4), for the dissipative dynamics (6)
it follows that dgtot/dt = −(∂gtot/∂p)2 > 0. In view of the boundedness of gtot (i.e., gtot > 0), gtot is a
Lyapunov function that becomes stationary in the long term limit. This in turn implies that ∂gtot/∂p = 0
such that equation (6) reduces to equation (4). In total, for t →∞, the system (6) converges to an attractor
that corresponds to a solution of the conservative system (4).
44001-3
S. Mongkolsakulvong, T.D. Frank
Let us consider the harmonic case defined by α = β = 0. Using qk = Ak(t) exp(iωt) + c.c. with
ω2 = k/m again and Ak(t) = rk(t) exp[iφk(t)]/2 and assuming that γj are small perturbation parameters,
we obtain
d
dt
A1 = −γ1ω
2 A1
(
|A1 |
2 −
B1
2mω2
)
− i
γ3
m2ω
A2 U(A1, A2),
d
dt
A2 = −γ2ω
2 A2
(
|A2 |
2 −
B2
2mω2
)
+ i
γ3
m2ω
A1 U(A1, A2) (7)
with U(A1, A2) = L(L2/m − B3) and L = 2miω(A∗1 A2 − A1 A∗2). In terms of the real-valued amplitudes r1
and r2 and the phase difference ψ = φ1 − φ2, we obtain
d
dt
r1 = −
γ1ω
2
4
r1
(
r2
1 −
2B1
mω2
)
−
γ3r2
m2ω
sin(ψ)U(ψ, r1, r2), (8)
d
dt
r2 = −
γ2ω
2
4
r2
(
r2
2 −
2B2
mω2
)
−
γ3r1
m2ω2 sin(ψ)U(ψ, r1, r2) (9)
with U(ψ, r1, r2) = mωr1r2 sin(ψ)(L2/m − B3), L = mωr1r2 sin(ψ), and
d
dt
ψ = −
γ3
m2ω
(
r2
1 + r2
2
r1r2
)
cos(ψ)U(ψ, r1, r2). (10)
A detailed stability analysis based on equations (8)–(10) for the case that both oscillators are excited like
B1, B2 > 0, shows that the target level B3 defines the location of the aforementioned attractor. By rescaling
B3 we obtain the location parameter η = mω2B3/(4B1B2). It can be shown that for B3 6 0⇒ η 6 0, the
two-oscillators system exhibits a stable attractor characterized by S3 = 0 ⇒ ψ = 0◦ ∨ ψ = 180◦ and
H1,2 = B1,2. This implies that gtot → γ3B2
3/2 > 0 for t → ∞. For B3 > 0, we distinguish between two
cases. If η ∈ [0, 1], then the limit cycle attractor is characterized by H1,2 = B1,2 again but with S3 = B3.
The latter relation implies that the phase difference is given by ψ = arcsin(√η), ψ = 180◦ − arcsin(√η),
ψ = arcsin(√η)+180◦, and ψ = 360◦−arcsin(√η). Moreover, gtot → 0 for t →∞. For B3 > 0 and η > 1,
it is impossible to have H1,2 = B1,2 and S3 = B3. Rather, the attractor is given by ψ = 90◦ ∨ ψ = 270◦,
H1,2 = B1,2 and S3 < B3. In addition, we have gtot → γ3(S3 − B3)
2/2 = 8γ3B2
1 B2
2(1− η)
2/(m2ω4) > 0 for
t → ∞. Figure 1 (b) illustrates the attractor location in terms of the phase difference ψ as a function of
the location parameter η. Figure 1 (c) shows, for a representative simulation, the trajectories q1 and q2.
The trajectories demonstrate that the two-oscillators system converges to a stable periodic pattern (limit
cycle).
Let us consider the non-harmonic case with α, β > 0. First we note that the individual oscillators
(i.e., γ1,2,3 = 0) for small amplitudes experience a linearized force of f (qj) = −klinqj with klin = 4k
irrespective of α and β. Consequently, the oscillation frequency is two times the oscillation frequency
of the harmonic case and the period is half the period of the harmonic case. This implies that when
removing the singularities in the potentials Vj by putting α = β = 0, then the oscillation frequency
drops in a discontinuous fashion from 2ω0 to ω0 with ω2
0 = k/m. In general, the oscillation period for the
oscillators j = 1, 2 can be computed from the integralTj = 2
√
m
∫qmax
qmin
{2[Hj(t = 0)−Vj(qj)]}
−1/2dqj with
Hj(t = 0) being the initial energy of oscillator j. The integration limits qmin,max are the turning points
defined byVj = Hj(t = 0). Numerical computations show thatTj is independent of Hj(t = 0) and α and β.
In line with the small amplitude oscillation case, we obtain Tj = T0/2 ∀ Hj(t = 0) > Hj,min, α, β > 0
with T0 = 2π/ω0, where Hj,min denote the minimal energy values H1,min =
√
αk and H2,min =
√
βk.
Let us consider the case γ1,2,3 > 0. For the sake of brevity, we consider only the case in which all
three invariants of the conservative dynamics, H1,2 and S3, converge to their respective target values (i.e.,
H1,2 → B1,2 and S3 → B3) such that gtot → 0. Our first objective is to show that the shapes of the
oscillator limit cycles are distorted compared to the harmonic case. Panels (d) and (e) of figure 1 show
the phase portraits q1, p1 and q2, p2 obtained from a numerical simulation. The trajectories (solid lines)
converge to “egg shaped” limit cycles. The limit cycles are defined by the limit cycles of the corresponding
44001-4
Synchronization with anchoring in Smorodinsky-Winternitz potentials
conservative oscillators with γ1,2,3 = 0. For the example shown in panels (d) and (e), the limit cycles of
the corresponding conservative oscillators are illustrated by circles. Importantly, the limit cycles reveal
an anchoring phenomenon. The dynamics slows down (more pronounced as in the harmonic case) when
the oscillators swing to the right of their potential minimum locations q1 = (α/k)0.25 and q2 = (β/k)0.25,
see figures 1 (d) and (e). By contrast, the dynamics speeds up when the oscillators swing to the left
of their potential minimum locations. This is because the forces are relatively weak on the right-hand
sides [where Vj(qj) are approximatively parabolic potentials] and relatively strong on the left-hand sides
[where Vj(qj) exhibit singularities].
Our second objective is to address the synchronization of the oscillators. On the limit cycles, the
oscillators oscillate with the same oscillation frequency of ω = 2ω0, see above. From H1,2 → B1,2 and
S3 → B3 it follows that
p2
1
2m
= B1 − V1(q1),
p2
2
2m
= B2 − V2(q2), B3 =
L2
m
+ (q2
1 + q2
2)
(
α
q2
1
+
β
q2
2
)
︸ ︷︷ ︸
W
. (11)
Using two first equations of (11), the function L occurring in W (defined above) can be expressed as
L(q1, q2) = p2q1 − p1q2 = (−1)nq1
√
2m[B2 − V2(q2)] + (−1)mq2
√
2m[B1 − V1(q1)] (12)
with m, n ∈ {0, 1}. This implies that the last equation of (11) can be written as B3 = W(q1, q2). The
synchronized state is then described by{
m Üq1 = −
d
dq1
V1(q1) ⇒ q1(t)
}
∧
{
B3 = W(q1, q2) ⇒ q2(t)
}
. (13)
That is, the coordinate q2 is given by a nonlinear (implicit) mapping from q1 to q2. Importantly, this
mapping is stable against perturbation because the two-oscillators system is attracted to the state with
gtot = 0 and S1,2,3 = B1,2,3. Therefore, the two oscillators are synchronized. Note that the same argument
holds in the opposite direction. Considering the second oscillator as independent oscillator, the coordinate
q1 of the first oscillator is given by a nonlinear (implicit) mapping from q2 to q1.
Let us illustrate the synchronization of the two oscillators by considering the CD oscillator model (6)
under the impact of fluctuating forces. Using a standard approach for introducing noise terms into CD
systems [6–8, 10], equation (6) becomes
d
dt
q = ∂H
∂p ,
d
dt
p = −∂H
∂q −
∂g
∂p +
√
D
(
Γ1(t)
Γ2(t)
)
, (14)
where Γj(t) are independent Langevin forces [29] normalized to 2 units. The parameter D > 0 is the
diffusion constant. The Langevin equation (14) exhibits a Fokker-Planck equation that can be cast into
the form of a free energy Fokker-Planck equation [26]. The stationary probability density P(q1, q2, p1, p2)
can then be expressed in terms of a Boltzmann function of gtot as P = exp(−gtot/D)/Z0, where Z0 is a
normalization factor [19, 26]. Considering γj as small perturbation parameters, we may introduce the
smallness parameter γ0 and put γj = cjγ0 with cj > 0. Then, P = exp(−g̃tot/θ)/Z0 with g̃tot = gtot/γ0
holds, where θ = D/γ0 can be considered as a non-equilibrium temperature. In this form, the analogy
to equilibrium systems becomes obvious [6]. Importantly, without coupling between the oscillators, that
is, for γ3 = 0 ⇒ c3 = 0, we have P(q1, q2, p1, p2) = P1(q1, p1)P2(q2, p2) with Pj = exp(−gj/D)/Z j
for j = 1, 2, where Z j are normalization factors again. That is, the probability density P(q1, q2, p1, p2)
factorizes. By analogy, the transition probability density (conditional probability density) factorizes.
Therefore, for γ3 = 0, there are no cross-correlations between the oscillators at any time lag.
Let us show with the help of a stochastic CD model (14) that for γ3 > 0, the two-oscillators
model exhibits a stable synchronized state. To this end, we solved numerically equation (14) for a fixed
value of γ3 and calculated the cross-correlation coefficients Corr
(
q1(t), q2(t − τ)
)
for different time lags
τ ∈ [0,T = T0/2]. We determined the maximal coefficient. Subsequently, we varied γ3. In doing so, we
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S. Mongkolsakulvong, T.D. Frank
obtain the maximal cross-correlation coefficient as a function of the coupling parameter γ3. Figure 1 (f)
summarizes the simulation results. For γ3 = 0, there was a finite by-chance value for the maximal cross-
correlation coefficient that decayed when the simulation duration was increased. As far as the impact of
γ3 is concerned, figure 1 (f) demonstrates that the maximal cross-correlation increased as a function of
γ3 — as predicted. This increase of the maximal cross-correlation coefficient was taken as the evidence
that for γ3, the two oscillators were to some degree synchronized.
Future studies may focus, in particular, on the stochastic aspects of the proposed CD two-oscillators
model. For example, it has been suggested to use the analytical solution for the short-time propagator
to define maximum likelihood estimators that can be used to estimate the model parameters of CD
systems [30]. In fact, for single CD oscillator models in a series of studies, the CD theory has been applied
to the experiments on human rhythmic motor behavior and model parameters have been estimated from
experimental data both in the harmonic [12–14] and non-harmonic case [15].
References
1. Pikovsky A., RosenblumM., Kurths J., Synchronization: a Universal Concept in Nonlinear Sciences, Cambridge
University Press, Cambridge, 2001.
2. Schweitzer F., Brownian Agents and Active Particles, Springer, Berlin, 2003.
3. Frank T.D., Condens. Matter Phys., 2014, 17, 43002, doi:10.5488/CMP.17.43002.
4. Lindner B., Nicola E.M., Eur. Phys. J. Spec. Top., 2008, 157, 43, doi:10.1140/epjst/e2008-00629-7.
5. Jenkins A., Phys. Rep., 2013, 525, 167, doi:10.1016/j.physrep.2012.10.007.
6. Haken H., Z. Phys., 1973, 263, 267, doi:10.1007/BF01391586.
7. Graham R., Haake F., In: Quantum Statistics in Optics and Solid-State Physics. Springer Tracts in Modern
Physics, Vol. 66, Höhler G. (Ed.), Springer-Verlag, Berlin, 1973, 1–97.
8. Ebeling W., Sokolov I.M., Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems,
World Scientific, Singapore, 2004.
9. Mongkolsakulvong S., Frank T.D., Condens. Matter Phys., 2010, 13, 13001, doi:10.5488/CMP.13.13001.
10. Romanczuk P., Bär M., Ebeling W., Lindner B., Schimansky-Geier L., Eur. Phys. J. Spec. Top., 2012, 202, 1,
doi:10.1140/epjst/e2012-01529-y.
11. Ebeling W., Condens. Matter Phys., 2004, 7, 539, doi:10.5488/CMP.7.3.539.
12. Dotov D.G., Frank T.D., Motor Control, 2011, 15, 550, doi:10.1123/mcj.15.4.550.
13. Dotov D.G., Kim S., Frank T.D., Biosystems, 2015, 128, 26, doi:10.1016/j.biosystems.2015.01.002.
14. Kim S., Gordon J.M., Frank T.D., Open Syst. Inf. Dyn., 2015, 22, 1550007, doi:10.1142/S1230161215500079.
15. Gordon J.M., Kim S., Frank T.D., Condens. Matter Phys., 2016, 19, 34001, doi:10.5488/CMP.19.34001.
16. Schmidt R.C., Carello C., Turvey M.T., J. Exp. Psychol.: Hum. Percept. Perform., 1990, 16, 227,
doi:10.1037/0096-1523.16.2.227.
17. Winfree A.T., The Geometry of Biological Time, 2nd Edn., Springer, Berlin, 2001.
18. Chaikhan P., Frank T.D., Mongkolsakulvong S., Acta Mech., 2016, 227, 2703, doi:10.1007/s00707-016-1642-1.
19. Schweitzer F., Ebeling W., Tilch B., Phys. Rev. E, 2001, 64, 021110, doi:10.1103/PhysRevE.64.021110.
20. Ebeling W., Röpke G., Physica D, 2004, 187, 268, doi:10.1016/j.physd.2003.09.014.
21. Teğmen A., Verçin A., Int. J. Mod. Phys. B, 2004, 19, 393, doi:10.1142/S0217751X04017112.
22. Winternitz P., Smorodinsky Y.A., Uhlíř M., Friš J., Sov. J. Nucl. Phys., 1967, 4, 444.
23. Friš J., Mandrosov V., Smorodinsky Y.A., Uhlíř M., Winternitz P., Phys. Lett., 1965, 16, 354,
doi:10.1016/0031-9163(65)90885-1.
24. Jin M., Xie W., Chen T., Superlattices Microstruct., 2013, 62, 59, doi:10.1016/j.spmi.2013.07.002.
25. Jasmine P.C.L., Peter A.J., Lee C.W., Chem. Phys., 2015, 452, 40, doi:10.1016/j.chemphys.2015.02.013.
26. Frank T.D., Nonlinear Fokker-Planck Equations: Fundamentals and Applications, Springer, Berlin, 2005.
27. Haken H., Light, Vol. 2, North-Holland Physics Publishing, Amsterdam, 1985.
28. O’Malley R.E. (Jr.), Williams D.B., J. Comput. Appl. Math., 2006, 190, 3, doi:10.1016/j.cam.2004.12.043.
29. Risken H., The Fokker-Planck Equation: Methods of Solution and Applications, Springer, Berlin, 1989.
30. Frank T.D., Phys. Lett. A, 2010, 374, 3136, doi:10.1016/j.physleta.2010.05.073.
44001-6
https://doi.org/10.5488/CMP.17.43002
https://doi.org/10.1140/epjst/e2008-00629-7
https://doi.org/10.1016/j.physrep.2012.10.007
https://doi.org/10.1007/BF01391586
https://doi.org/10.5488/CMP.13.13001
https://doi.org/10.1140/epjst/e2012-01529-y
https://doi.org/10.5488/CMP.7.3.539
https://doi.org/10.1123/mcj.15.4.550
https://doi.org/10.1016/j.biosystems.2015.01.002
https://doi.org/10.1142/S1230161215500079
https://doi.org/10.5488/CMP.19.34001
https://doi.org/10.1037/0096-1523.16.2.227
https://doi.org/10.1007/s00707-016-1642-1
https://doi.org/10.1103/PhysRevE.64.021110
https://doi.org/10.1016/j.physd.2003.09.014
https://doi.org/10.1142/S0217751X04017112
https://doi.org/10.1016/0031-9163(65)90885-1
https://doi.org/10.1016/j.spmi.2013.07.002
https://doi.org/10.1016/j.chemphys.2015.02.013
https://doi.org/10.1016/j.cam.2004.12.043
https://doi.org/10.1016/j.physleta.2010.05.073
Synchronization with anchoring in Smorodinsky-Winternitz potentials
Синхронiзацiя та анкерування двох негармонiчних
канонiчно-дисипативних осциляторiв за допомогою
потенцiалiв Смородинського-Вiнтернiца
С.Монгколсакувонг1, Т.Д. Франк2,3
1 Факультет природничих наук, вiддiлення фiзики, унiверситет Касертсарт, Бангкок 10900, Таїланд
2 CESPA, вiддiлення психологiї, Коннектикутський унiверситет, CT 06269, США
3 Вiддiлення фiзики, Коннектикутський унiверситет, CT 06269, США
Розглянуто канонiчно-дисипативнi граничнi цикли для двох негармонiчних осциляторiв, що колива-
ються в одномiрних потенцiалах Смородинського-Вiнтернiца. Показано, що стандартний канонiчно-
дисипативний пiдхiд iз введенням дисипативних сил природно приводить до появи взаємодiї мiж осци-
ляторами, яка синхронiзує їх рух. Детально дослiджено негармонiчний характер граничних циклiв у кон-
текстi анкерування, рiзницю фаз мiж синхронiзованими осциляторами та ступiнь їх синхронiзацiї.
Ключовi слова: канонiчно-дисипативнi системи, потенцiали Смородинського-Вiнтернiца,
синхронiзацiя, анкерування
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