Download Relation between chaos probability and zero

Cent. Eur. J. Phys. • 7(4) • 2009 • 738-746
DOI: 10.2478/s11534-008-0159-1
Central European Journal of Physics
Relation between chaos probability and zero-point
number of the Melnikov function for a Bose–Einstein
condensate
Research Article
Qianquan Zhu, Wenhua Hai∗ , Shiguang Rong
Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of the Ministry of Education,
and Department of Physics, Hunan Normal University, Changsha 410081, China
Received 31 August 2008; accepted 1 December 2008
Abstract:
We investigate an attractive Bose–Einstein condensate perturbed by a weak traveling optical superlattice.
It is demonstrated that under a stochastic initial set and in a given parameter region solitonic chaos appears
with a certain probability that is tightly related to the zero-point number of the Melnikov function; the latter
depends on the potential parameters. Effects of the lattice depths and wave vectors on the chaos probability
are studied analytically and numerically, and different chaotic regions of the parameter space are found.
The results suggest a feasible method for strengthening or weakening chaos by modulating the potential
parameters experimentally.
PACS (2008): 03.75.Lm, 05.45.Ac, 03.75.Kk, 05.45.Gg
Keywords:
Bose-Einstein condensate • chaotic soliton • Melnikov function • chaos probability • chaotic region
© Versita Warsaw and Springer-Verlag Berlin Heidelberg.
1.
Introduction
A Bose–Einstein condensate (BEC) held in an optical lattice has been a topic of intense investigations [1–3]. There
has also been an increasing interest in the study of BECs
loaded into an optical superlattice [4–6]. On the other
hand, moving optical lattices have been investigated recently, and they have been applied to modulate BECs experimentally [7, 8]. In this context, various remarkable results such as lensing effects [8], gap soliton generation [9],
and transient and stationary chaos [10] have been shown
∗
E-mail: [email protected]
738
successfully. A further physical motivations is, of course,
the behavior of a BEC in a moving optical superlattice.
In mean-field theory, the dynamical behavior of BEC systems is governed by the nonlinear Schrödinger equation
(NLSE) with external potentials [11], also known as the
Gross–Pitaevskii equation (GPE) [12, 13], which can describe many nonlinear effects, such as solitons [14–17],
chaos [18, 19], and the enhancement of quantum reflection
[20]. Chaotic soliton behavior, which attracts extensive
interest, has been studied theoretically using the NLSE
with a periodic perturbation [21–23]. Recently, a few remarkable works have paid attention to chaotic dynamics of
solitons in BEC systems, including chaos and energy exchange [24], discrete solitons and chaotic dynamics in an
array of BECs [25], and bright matter-wave soliton colli-
Qianquan Zhu, Wenhua Hai, Shiguang Rong
sions [26]. The combination of optical lattices with a BEC
is shown to provide an ideal system to study chaos. In
previous works, one has discussed spatial chaos in BEC
systems with a static superlattice [27] and spatio-temporal
chaos of BECs in a moving optical lattice [10, 28, 29] by
using the well-known Melnikov method [30, 31]. The Melnikov method is a widely employed analytical tool for
providing a chaos criterion because the Melnikov function measures the distance between stable and unstable
manifolds in a Poincaré section, and the simple-zero Melnikov function indicates the existence of Smale–horseshoe
chaos [30, 31]. This method has also been used successfully in many different systems [32–36] and can always
give the chaotic and regular regions in parameter space.
However, if we plot the Poincaré section for stochastic
initial and boundary conditions and fixed parameters in
a chaotic region, chaos does not always appear; there is
only a certain probability for its appearance. This chaos
probability may play a very important role for possible
applications of chaos, since in a realistic chaotic system
the initial and boundary conditions cannot be determined
accurately. The dynamical instability of a system just
comes from stochastic initial perturbations [31]. Instabilities of BECs have been investigated [37, 38], and the
relation between chaos and certain types of instabilities
has also been discussed [39]. The sensitivity to initial
conditions is a common property of unstable motion and
chaos. However, chaos in the considered system must satisfy the analytical criterion based on the Melnikov function, which may not be obeyed by nonchaotic unstable
motions. Many works have focused on suppressing chaos
that leads to zero chaos probability. In some realistic applications, such as secure chaos-based cryptosystems [40],
higher chaoticity is desired [41]. This, in turn, calls for
higher chaos probability.
2.
Analytical chaotic soliton
The BEC system considered here is loaded into a moving superlattice [42] consisting of primary and secondary
moving optical lattices of the form
Vop (ζ) = V1 cos2 (kζ) + V2 cos2 (γkζ).
(1)
In this formula, we refer to V1 cos2 (kζ) as the primary lattice with V1 and k corresponding to its amplitude and wave
vector, and to V2 cos2 (γkζ) as the secondary one with V2
being its amplitude and γ the ratio of the two laser wave
vectors. The spatio-temporal variable ζ = x + vL t contains the velocity of the traveling lattice vL = δ/(2k) with
δ being the frequency difference between the two counterpropagating laser beams producing the first lattice. The
laser frequencies and amplitudes can be controlled independently by using acousto-optical modulators [7].
For the quasi-one-dimensional (quasi-1D) BEC loaded in
the superlattice potential Vop (ζ) the transverse wave function is approximately in the ground state of a harmonic oscillator of radial frequency ωr , and the system is governed
by the time-dependent 1D GPE
i~
~2 ∂2 Ψ
∂Ψ
=−
+ g1d |Ψ|2 Ψ + Vop (ζ)Ψ.
∂t
2m ∂x 2
(2)
Here, Ψ and m are the macroscopic quantum wave function and the mass, g1d = 2~ωr as characterizes the 1D
interatomic interaction strength [43] and is related to the
s-wave scattering length as , where as > 0 corresponds to
a repulsive interaction and as < 0 denotes an attractive
interaction. In order to get a simple description, we study
a traveling-wave solution of the form
Ψ = R(ζ)ei(αx+βt) ,
(3)
where R(ζ) is a real function of ζ and α and β represent
two undetermined real constants. Inserting Eq. (3) into
Eq. (2), we easily obtain
Given that the existence of chaos is associated with the
simple zero of the Melnikov function, what is the relation
between chaos probability and zero-point number of the
Melnikov function, and how can we get a higher chaos
probability? In this paper, we will present a detailed
analysis of this issue by investigating a chaotic soliton
solution in a BEC system perturbed by a moving optical superlattice consisting of two traveling lattices of the
same velocity but with different amplitudes and wave vectors. Our results suggest a feasible method for reducing or
strengthening chaoticity experimentally. Such a method
can be extended to control spatial chaos with zero traveling velocity and temporal chaos in other systems.
d2 R
+ DR − g1 R 3 ,
dξ 2
= [V˜1 cos2 (kξ) + V˜2 cos2 (γkξ)]R,
(4)
and (2α̃ + v)dR/dξ = 0. The latter formula confines the
constant α to α = −mvL /~. Here we have used the dimensionless variables and parameters
4as
2mvL
α
1
, v=
, α̃ =
= − v,
k0 l2r
~k0
k0
2
~β
V
V
1
2
β̃ =
, D = −(α̃ 2 + β̃), V˜1 =
, V˜2 =
, (5)
Er0
Er0
Er0
ξ = k0 ζ, g1 =
739
Relation between chaos probability and zero-point number of the Melnikov function for a Bose–Einstein condensate
2 2
where k0 denotes a suitable unit
p for k, Er0 = ~ k0 /(2m)
is the recoil energy, and lr = ~/(mωr ) the radial length
of the harmonic oscillator. It is well known that the
driven Duffing-like equation (4) can describe chaotic behavior [32]. So it is very hard to find an exact solution
of this equation. However, when the driving strengths are
weak enough, we can treat the chaotic system by the direct
perturbation approach [33–35] and the Melnikov-function
method [30, 31].
Regarding the weak periodic terms as a perturbation, the
unperturbed part of Eq. (2) is just the standard NLSE,
which has well-known soliton solutions [44, 45]. For simplicity we only consider the single-soliton solution of the
NLSE in the form of Eq. (3). Such a simple single-soliton
solution can be convenient to investigate the transition
probability from soliton to chaos, which we are concerned
about. Apparently, Eq. (4) is just the same as the parametrically driven frictionless Duffing equation, whose chaotic
features have been extensively studied for the singlelattice case [10, 28]. In the first-order weak superlattice
case, we apply the following direct perturbation expansion
R(ξ) = R0 (ξ) +
∞
X
Ri (ξ), |R0 | |Ri | |Ri+1 |,
(6)
i=1
to Eq. (4) obtaining the zeroth-order and ith-order equations
d2 R0
+ DR0 − g1 R03 = 0,
dξ 2
Z
Ri = h
ε1 (ξ) = [V˜1 cos2 (kξ) + V˜2 cos2 (γkξ)]R0 ,
ε2 (ξ) = [V˜1 cos2 (kξ) + V˜2 cos2 (γkξ)]R1 + 3g1 R0 R12 ,
(9)
i=1
s
√
2D
sech[ −D(ξ + c0 )],
g1
hr g
i
1
1
c0 = √
Ar sech
R0 (ξ0 ) − ξ0 .
2D
−D
R0 (ξ) =
(10)
Here, ξ0 = k0 (x0 + vL t0 ) is a function of the initial time t0
and the boundary coordinate x0 , c0 denote an integration
740
ξ
qi
h εi (ξ) dξ,
(11)
Ii± =
∞
X
i=1
ξ
Z
lim
ξ→±∞
pi
f εi (ξ) dξ = 0.
(13)
If such a condition cannot be satisfied, the perturbed
solution (11) becomes infinity as time is tending to infinity, which makes the soliton solution (10) unstable
under perturbations. Eliminating the constant pi from
P∞
i=1 (Ii+ −Ii− ) = 0 we get the well-known Melnikov chaos
criterion [30, 31]
∞
X
i=1
If the atom–atom interactions are attractive, the system
has a negative s-wave scattering length to make g1 < 0,
such that Eq. (7) with a negative D value has the well
known soliton (homoclinic) solution
pi
Z
f εi (ξ) dξ − f
Generally, applying Eq. (11) to Eq. (6), the corrected part
P∞
i=1 Ri (ξ) is unbounded because of the unboundedness
of h at ξ = ∞. However, using l’Hospital rule, we can
easily obtain the sufficient and necessary boundedness
condition [33–35]
∞
X
with
ξ
where pi and qi are arbitrary constants adjusted by the
ith-order perturbed corrections to the initial conditions
R
and f = dR0 /dξ and h = f f −2 dξ are two linearly independent solutions of Eq. (8) for εi (ξ) = 0,
s
√
√
2
f =
Dsech[ −D(ξ + c0 )] tanh[ −D(ξ + c0 )],
−g1
p
√
√
−2g1
h = −
sech[ −D(ξ + c0 )] tanh[ −D(ξ + c0 )]
3
8(−D) 2
n √
√
× 6 −D(ξ + c0 ) − 4 coth[ −D(ξ + c0 )]
o
√
+ sinh[2 −D(ξ + c0 )] .
(12)
(7)
d2 Ri
+ DRi − 3g1 R02 Ri = εi (ξ), i = 1, 2, 3, · · · (8)
dξ 2
··· .
constant determined by the boundary and initial conditions. Obviously, inserting R0 of Eq. (10) into Eq. (3),
we get the bright single-soliton solution with the same
form as the standard NLSE [44, 45]. Applying Eq. (10)
to Eq. (8), we construct the general expressions of the
ith-order corrections in form of an integral [33–35]
Mi (c0 ) =
∞ Z
X
i=1
+∞
−∞
f εi (ξ) dξ = 0,
(14)
where Mi (c0 ) is the Melnikov function for the ith-order
correction to the soliton solution (10); this ith-order function Mi (c0 ) corresponds to the ith-order small quantity.
P
Their summation ∞
i=1 Mi (c0 ) measures the distance between stable and unstable orbits in the Poincaré section.
The simple zeros of this function indicate the existence of
chaos for some c0 values. Under the conditions of Eqs. (13)
and (14), we can call Eq. (11) the “chaotic solution.” Thus,
inserting Eqs. (10) and (11) into Eq. (6), we obtain the
chaotic bright soliton solution, which is the superposition
of soliton and chaotic states and propagates with the same
velocity as the traveling superlattice.
Qianquan Zhu, Wenhua Hai, Shiguang Rong
3. Chaos probabilities in different
chaotic regions
Inspection of Eq. (13) shows that Eq. (14) determines only
some necessary boundedness conditions for higher-order
corrections, which implies relationships among the system
parameters and may lead to chaotic regions in parameter
space. Theoretically, solving the ith-order equations one
by one, we can get the conditions for the onset of Melnikov
chaos. Here, for simplicity and convenience, we only consider the first-order Melnikov function neglecting higher
orders. The details of the analysis are the same as those
in many former works.
Applying ε1 (ξ) given by Eq. (9), R0 given by Eq. (10) and
f given by Eq. (12) to the integrand of Eq. (14) with i = 1,
we obtain the first-order Melnikov function
Figure 1.
Plot of the chaotic regions of V˜2 versus k for the dimensionless parameters γ = 2, D = −2, and V˜1 = 0.03.
√
where XN (c0 ) = V˜1 csch(kπ/ −D) + FN (c0 ) with
kπ 2k π h ˜
M1 (c0 ) = −
V1 csch √
sin(2c0 k)
g1
−D
γkπ i
+V˜2 γ 2 csch √
sin(2c0 γk) = 0,
(15)
−D
2
which can be used to investigate chaotic regions and chaos
probabilities in the first approximation. As we can see, it
is a periodic function of c0 for fixed parameters and rational number γ, so only the discrete zero points c0 = c0j
for j = 1, 2, · · · satisfy the chaos criterion M1 (c0j ) = 0.
Nevertheless, c0 is an integration constant depending on
the initial and boundary conditions and cannot be accurately set in experiments. When we numerically generate
integer N 0 Poincaré sections under stochastic initial and
boundary conditions, we know from Eq. (15) that c0 takes
c0j values with only a certain probability, which results
in the number n0 of chaotic attractors being less than N 0 .
The transition probability from soliton to chaos, i.e., the
chaos probability P, can be defined as the ratio n0 /N 0 .
It is clear that the P = n0 /N 0 value should tend to increase with the number n of c0j in one period of M1 (c0 )
(i.e., the zero-point number of M1 (c0 ) in one period), and
it is always smaller than 1. It should be stressed that the
accuracy of the P value depends on the sample number of
the Poincaré sections N 0 .
Now, let us investigate how the chaos probability depends
on the parameter regions. As simple examples we consider
the cases γ = N ≥ 2 with N being an integer. We rewrite
Eq. (15) in the form
M1 (c0 ) = ηX1 (c0 )XN (c0 ) = 0,
η = −2k 2 π/g1 ,
X1 (c0 ) = sin(2c0 k),
(16)
FN (c0 ) = V˜2 N 2 csch
Nkπ
√
−D
sin(N2c0 k)
.
sin(2c0 k)
Clearly, X1 (c0 ) is a periodic function of c0 with two zero
points in one period; XN (c0 ) is also a periodic function of
c0 but with a different number of zero points in different
parameter regions, which can lead to a different number
n of c0j for M1 (c0j ) = 0 in one period and different chaos
probabilities. For a fixed N, the boundary values of V˜2 between the different regions in the parameter plane (k, V˜2 )
can be denoted by Ṽ2bN . For the sake of clarity, we only
take the cases N = 2, 3, 4 into account.
Case 1. In the simplest case N = 2, the analysis is fairly
straightforward. The quantity XN (c0 ) reads
kπ 2kπ X2 (c0 ) = V˜1 csch √
+ 8V˜2 csch √
cos(2c0 k).
−D
−D
(17)
Obviously, the parameter space can be divided into two
regions with different zero-point numbers of X2 (c0 ); the
boundary between them is described by
Ṽ2b2
√
V˜1 csch(kπ/ −D)
V˜1
kπ
√
=
=
cosh √
.
4
8csch(2kπ/ −D)
−D
(18)
In Fig. 1, we show the boundary curve Ṽ2b2 (k) in the
parameter plane (k, V˜2 ) with D = −2 and V˜1 = 0.03.
The region A2 above the boundary curve corresponds to
V˜2 > Ṽ2b2 and the region B2 below the boundary curve is
associated with V˜2 < Ṽ2b2 .
In order to show clearly the different zero-point numbers of
M1 (c0 ) in different regions, we can generate from Eqs. (16)
and (17) plots showing the zero points of the periodic functions X1 (c0 ), X2 (c0 ), and M1 (c0 ) with the period of M1 (c0 ).
741
Relation between chaos probability and zero-point number of the Melnikov function for a Bose–Einstein condensate
range c0 ∈ [0, π), namely for one period of M1 (c0 ), and for
fixed parameters in the region A2 , both X1 (c0 ) and X2 (c0 )
have two zero points, and all of them are not coincident
leading to four zero points of M1 (c0 ). However, in Fig. 2(b)
M1 (c0 ) has the same zero points as X1 (c0 ) because X2 (c0 )
has no zero point in the region B2 . Therefore, the chaos
probability P is predicted to be higher in the region A2
because of the larger zero-point number n.
Figure 2.
Plots of the functions X1 (c0 ) (dashed curve), X2 (c0 ) (dotted
curve), and M1 (c0 ) (solid curve) versus c0 for the parameters γ = 2, D = −2, g1 = −0.5, k = 1, V˜1 = 0.03, and (a)
V˜2 = 0.06 > Ṽ2b2 or (b) V˜2 = 0.02 < Ṽ2b2 , where for each
different function a different amplitude scale is adopted to
show the zero points.
Such plots are depicted in Fig. 2, where sample parameters have been used. In Fig. 2(a), we can see that in the
To confirm numerically the analytical results, we use the
MATHEMATICA code (19) to generate three groups of
Poincaré sections in the equivalent phase space (R, Rξ ) for
each set of parameters used in Figs. 2(a) and 2(b) and for
the random initial conditions {R(ξ0 ) ∈ [0, 3.2], Rξ (ξ0 ) ∈
[−2.2, 2.2]} associated with a suitable range for c0 . The
Poincaré section is a discrete set of phase space points
at every period of the periodic potential. Each group contains N 0 = 500 Poincaré sections. For the parameters in
the region A2 , the average number of chaotic attractors
per group is n0A = 127, and that for the parameters in
the region B2 is n0B = 68. The numerical results indicate
that in the chaotic region B2 the chaos probability reads
P = 68/500 = 0.136, and in the region A2 it becomes
P = 127/500 = 0.254. This implies that the chaos probability is higher in the latter region with more zero points of
M1 (c0 ). This result agrees qualitatively with the analytical
assertion. As these are statistical samples, more Poincaré
sections would generally lead to more exact chaos probabilities. The typical chaotic attractor is shown in Fig. 3(a),
and the closed curves in Figs. 3(b) and 3(c) are the typical
regular orbits from the numerical results.
T = π; e[{Rnew− , vnew− }] := {R[T ], v[T ]}/.F latten[NDSolve[{R 0 [ξ] == v[ξ], v 0 [ξ] == g1 R[ξ]3 − DR[ξ]
+(V1 C os[kξ]2 + V2 C os[γkξ]2 )R[ξ], R[0] == Rnew, v[0] == vnew}, {R, v}, {ξ, 0, T }]]; Do[pici = ListPlot
[Drop[Nestlist[e, {Random[Real, {0, 3.2}], Random[Real, {−2.2, 2.2}]}, 3000], 100], {i, 1, 500}]
742
(19)
In order to show the results in a clear and concise manner,
we present Tab. 1, which displays the zero-point numbers
for X1 (c0 ), X2 (c0 ), and M1 (c0 ) in one period π/k of M1 (c0 ),
as well as the corresponding P values in the different
parameter regions A2 and B2 .
Case 2. In the case γ = 3, XN (c0 ) becomes
Making use of Eq. (20), we easily obtain the formula determining the boundary parameters between the different
regions with different zero-point numbers for X3 (c0 ) as
kπ 3kπ X3 (c0 ) = V˜1 csch √
+ 27V˜2 csch √
−D
−D
3kπ −36V˜2 csch √
sin2 (2c0 k).
(20)
−D
(21)
Ṽ2b3
√
V˜1 csch(kπ/ −D)
√
=
9csch(3kπ/ −D)
2kπ i
V˜1 h
=
.
1 + 2 cosh √
9
−D
In Fig. 4, we have used Eq. (21) to plot the boundary curve
Ṽ2b3 (k) in the parameter plane (k, V˜2 ) for the parameters
Qianquan Zhu, Wenhua Hai, Shiguang Rong
Table 1.
Figure 3.
Zero-point numbers for X1 (c0 ), X2 (c0 ), and M1 (c0 ) in one
period of M1 (c0 ), and the corresponding chaos probabilities
P for γ = 2.
Regions
X1 (c0 )
X2 (c0 )
M1 (c0 )
P
A2 : V˜2 > Ṽ2b2
B2 : V˜2 < Ṽ2b2
2
2
4
0.254
2
0
2
0.136
Figure 4.
Plot of the chaotic regions of V˜2 versus k for the dimensionless parameters γ = 3, D = −2, and V˜1 = 0.03.
Figure 5.
Plots of the functions X1 (c0 ) (dashed curve), X3 (c0 ) (dotted
curve), and M1 (c0 ) (solid curve) versus c0 for γ = 3, k =
0.5 with the other parameters the same as in Fig. 2, and
(a) V˜2 = 0.06 > Ṽ2b3 , and (b) V˜2 = 0.01 < Ṽ2b3 . This figure
is plotted in the same manner as Fig. 2.
Typical orbits on the Poincaré section (R, Rξ ) from the
MATHEMATICA code. The confused points in Fig. 3(a)
and the closed curves in Figs. 3(b) and 3(c) correspond
to a representative chaotic attractor and regular orbits, respectively.
γ = 3, D = −2, and V˜1 = 0.03. This plot clearly shows
the different chaotic regions.
In the chaotic region A3 with parameters obeying V˜2 >
Ṽ2b3 , X1 (c0 ) has two and X2 (c0 ) has four zero points in one
period of M1 (c0 ) resulting in six zero points of M1 (c0 ), as
shown in Fig. 5(a). In Fig. 5(b) with parameters satisfying
V˜2 < Ṽ2b3 , M1 (c0 ) keeps the same zero points as X1 (c0 )
because X2 (c0 ) has none. This analysis indicates that the
chaos probability P in the region A3 is higher than that
in the region B3 .
We again adopt the MATHEMATICA code (19) to perform
a numerical verification of our analytical results. The numerical results show an average of 158 chaotic attractors
per group consisting of 500 Poincaré sections for the pa-
743
Relation between chaos probability and zero-point number of the Melnikov function for a Bose–Einstein condensate
Table 2.
The zero-point numbers of X1 (c0 ), X3 (c0 ), and M1 (c0 ) in one
period of M1 (c0 ) and the corresponding chaos probabilities
P for γ = 3.
Regions
X1 (c0 )
X3 (c0 )
M1 (c0 )
P
A3 : V˜2 > Ṽ2b3
B3 : V˜2 < Ṽ2b3
2
4
6
0.316
2
0
2
0.248
rameters in the region A3 and an average of 124 for region
B3 resulting in the chaos probabilities of 0.316 and 0.248,
respectively. This is also in agreement with our analysis. As for the γ = 2 case, we summarize our results
in Tab. 2, which shows the zero-point numbers of X1 (c0 ),
X2 (c0 ), and M1 (c0 ) in one period of M1 (c0 ) followed by the
corresponding chaos probabilities for the different regions.
It is apparent that P grows with the zero-point number n
in one period of M1 (c0 ).
Case 3. For the case of γ = 4, XN (c0 ) turns into
kπ 4kπ X4 (c0 ) = V˜1 csch √
+ 64V˜2 csch √
cos(2c0 k) cos(4c0 k)
−D
−D
4kπ 4kπ kπ = 128V˜2 csch √
cos3 (2c0 k) − 64V˜2 csch √
cos(2c0 k) + V˜1 csch √
.
−D
−D
−D
(22)
From the above equation containing the cubic term of
cos(2c0 k), one can obtain three parameter regions with
different zero-point numbers of X4 (c0 ), which are separated by the following two boundaries
Ṽ2b4
0
Ṽ2b4
√
√
3 6V˜1 csch(kπ/ −D)
√
=
,
128csch(4kπ/ −D)
√
V˜1 csch(kπ/ −D)
√
=
.
64csch(4kπ/ −D)
(23)
Figure 6 clearly shows these three parts in the chaotic
region due to the two boundaries for sample parameters.
By solving X4 (c0 ) = 0, we obtain the following conclusions: In the region A4 , the parameters obey V˜2 > Ṽ2b4
and result in four zero points of X4 (c0 ) in one period of
M1 (c0 ), such that M1 (c0 ) has six zero points in one period.
0
In the region B4 , the inequality Ṽ2b4
< V˜2 < Ṽ2b4 leads to
two zero points for X4 (c0 ) and four zero points of M1 (c0 )
in one period of M1 (c0 ). In the region C4 , M1 (c0 ) has the
same two zero points as X1 (c0 ) because X4 (c0 ) 6= 0 due to
0
V˜2 < Ṽ2b4
. To illustrate these analytical results, plots of
these functions are shown in Fig. 7 for the parameter sets
corresponding to the respective three regions. Owing to
the dependence of the zero-point number of M1 (c0 ) on the
chaos probability P, we predict that the chaos probability
in the region A4 is higher than that in the region B4 , and it
is higher in the region B4 than it is in the region C4 . The
analytical results have also been confirmed numerically
by the MATHEMATICA code (19) with the average numbers of chaotic attractors for the three sets of parameters
in Fig. 7 being 120, 97, and 66 per group of 500 Poincaré
744
Figure 6.
Plot of the chaotic regions of V˜2 versus k for the dimensionless parameters γ = 4, D = −2, and V˜1 = 0.03. The dotted curve represents the boundary Ṽ2b4 , and the dashed
0
curve denotes Ṽ2b4
.
sections. The corresponding chaos probabilities are 0.240,
0.194 and 0.132, respectively. The above-mentioned analytical predictions are in rather good agreement with the
numerical results. Tab. 3 gives the zero-point numbers of
X1 (c0 ), X4 (c0 ), and M1 (c0 ) in one period of M1 (c0 ) and the
relevant values of P for the different regions. The results
show again that the highest chaos probability appears in
the region with the largest zero-point number of M1 (c0 ),
and the lowest one is associated with the smallest zeropoint number.
The above three cases show consistently that for γ =
2, 3, 4 the chaotic region of parameter space can be divided into several parts with different chaos probabilities.
Moreover, the number of chaotic regions may increase to-
Qianquan Zhu, Wenhua Hai, Shiguang Rong
gether with γ, and the chaos probability rises with the
increase of the zero-point number of M1 (c0 ). These results have been confirmed numerically.
4.
Discussions and conclusions
In conclusion, we have investigated an attractive BEC system loaded into a weak moving optical superlattice created
by two pairs of counter-propagating laser beams with different amplitudes and wave vectors. Such a system is a
typical chaotic one whose initial and boundary conditions
cannot be determined accurately. It is shown that when we
plot the Poincaré section for stochastic initial and boundary conditions and fixed parameters numerically, chaos
does not always appear but rather with a certain probability. We find the chaotic soliton solution up to the ithorder correction and the corresponding ith-order Melnikov
function. In the first-order approximation, the Melnikov
function M1 (c0 ) is a periodic function containing an integration constant c0 adjusted by the initial and boundary
conditions. It is illustrated numerically that the superlattice can separate the chaotic region into several parts with
different chaos probabilities, which are related to different zero-point numbers of M1 (c0 ). For a fixed integer γ,
which is the ratio of the secondary laser wave vector to the
primary one, the number of chaotic regions may increase
together with γ and the chaos probability rises with the
increase of the zero-point number of M1 (c0 ). Furthermore,
the amplitudes and wave vectors of the two lattices play
very important roles in determining the zero-point number
of M1 (c0 ) and thereby the chaos probability.
Figure 7.
Table 3.
Plot of the functions X1 (c0 ) (dashed curve), X4 (c0 ) (dotted
curve), and M1 (c0 ) (solid curve) versus c0 for the parameter γ = 4 with the other parameters the same as in Fig. 5,
0
and (a) V˜2 = 0.08 > Ṽ2b4 , (b) Ṽ2b4
< V˜2 = 0.04 < Ṽ2b4 ,
0
and (c) V˜2 = 0.008 < Ṽ2b4
. This figure is plotted in the
same manner as Fig. 2.
Zero-point numbers of X1 (c0 ), X4 (c0 ), and M1 (c0 ) in one period of M1 (c0 ) and the corresponding chaos probabilities P
for γ = 4.
Regions
A4 : V˜2 > Ṽ2b4
0
B4 : Ṽ2b4
< V˜2 < Ṽ2b4
˜
C4 : V2 < Ṽ 0
2b4
X1 (c0 )
X4 (c0 )
M1 (c0 )
P
2
6
8
0.240
2
2
4
0.194
2
0
2
0.132
Chaos probability may play a very important role for
possible applications of chaos because of the undetermined initial and boundary conditions of chaotic systems.
Our results suggest a feasible method for weakening or
strengthening chaoticity [41] experimentally: through decreasing or increasing the chaos probability. When we
call for higher (lower) chaotic probability in some practical
applications, we can raise (decrease) the chaos probability by adjusting certain controllable parameters to reach
a higher (lower) chaotic region. These results can be applied to investigate spatial chaos in the corresponding
static BEC system by setting to zero the traveling velocity.
In fact, these results are essential for a time-independent
potential due to the Galilean transformation. The used
method could also easily be extended to the case of temporal chaos in other chaotic systems.
The higher-order Melnikov functions such as M2 (c0 ) and
M3 (c0 ) have been ignored in this paper. For the regularmotion case with M1 (c0 ) 6= 0, such higher-order small
quantities are not important. However, for chaotic mo745
Relation between chaos probability and zero-point number of the Melnikov function for a Bose–Einstein condensate
tion with M1 (c0 ) = 0 the higher-order Melnikov functions
may be useful and warrant further study.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant Nos. 10575034
and 10875039.
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