Abstract Using a technique derived from nonlinear control theory, we demonstrate that two identical inertial ratchets transporting particles in two directions can be synchronized such that both ratchets transport particles in a desired direction. This novel approach to control of directed transport is applicable when there are multiple co-existing attractors in phase space transporting particles in different directions. Numerical simulations are employed to illustrate the approach

J A Laoye

U E Vincent

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Physics Letters A 363 (2007) 91–95
www.elsevier.com/locate/pla
Synchronization and control of directed transport
in chaotic ratchets via active control
U.E. Vincent ∗, J.A. Laoye
Department of Physics, Olabisi Onabanjo University, P.M.B. 2002 Ago-Iwoye, Nigeria
Received 18 August 2006; received in revised form 24 October 2006; accepted 25 October 2006
Available online 10 November 2006
Communicated by C.R. Doering
Abstract
Using a technique derived from nonlinear control theory, we demonstrate that two identical inertial ratchets transporting particles in two
directions can be synchronized such that both ratchets transport particles in a desired direction. This novel approach to control of directed transport
is applicable when there are multiple co-existing attractors in phase space transporting particles in different directions. Numerical simulations are
employed to illustrate the approach.
© 2006 Elsevier B.V. All rights reserved.
PACS: 05.45.Pq; 05.45.Ac; 05.40.Fb; 05.60.Cd
Keywords: Deterministic ratchets; Chaotic transport; Synchronization; Active Control1. Introduction
Recently, there has been an increased interest in the study
of transport phenomena of nonlinear systems that can extract
usable work from unbiased nonequilibrium fluctuations. These
so-called ratchets (or Brownian ratchets) [1,2] can be modeled
by a Brownian particle undergoing random walk on a peri-
odic asymmetric potential and being acted upon by an external
time-dependent force of zero average. Research activities in this
area is partly motivated by the challenge to model and control
some biological processes at both micro and macro scales as
found in transport of ion channels and muscle operations re-
spectively [3]. Another source of motivation is the potential
for technological applications aimed at devising mechanisms
for sorting, separating, pumping and controlling tiny particles
at nanoscales and micro scales (see Refs. [2,4] and references
therein). Outstanding experimental realization of some of these
devices have been carried out. Specifically, the control of mo-
tion of vortices in superconductors [5], particles in asymmetric
* Corresponding author.
E-mail address: ue_vincent@yahoo.com (U.E. Vincent).0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2006.10.086silicon pores [6], charged particles through artificial pores [7],
among others, have been reported recently.
Several attempts have been made to understand the genera-
tion of unidirectional motion from nonequilibrium fluctuations.
The vast majority of the models described in the literature con-
siders the overdamped cases in which the effect of the inertial
term is neglected [1,8]. Recently, ratchet models wherein the
inertial term is considered have been extensively investigated
since it was first studied by Jung et al. [9]. These ratchets pos-
sess, in general, a classical chaotic dynamics that modifies sig-
nificantly the transport properties [9,10]. For instance, current
reversal and multiple current reversals have been attributed to
changes in the bifurcation structure. In addition, the implication
of chaotic dynamics in deterministic ratchets has been recently
addressed in the quantum domain, together with the possible
connection with quantum chaos [11].
In a different context, Savel’ev et al. [12] examined the trans-
port properties of binary mixture of interacting particles and
showed that attracting or repelling interaction among identi-
cal particles can result in the amplification (inversion) of their
net current. This is potentially useful for enhancing and regu-
lating transport (e.g. through synthetic ion channels) and sep-
aration of repelling particles. Interaction among identical and
92 U.E. Vincent, J.A. Laoye / Physics Letters A 363 (2007) 91–95nonidentical particles can lead to synchronized dynamics when
a threshold is reached. Synchronization of two coupled chaotic
ratchets has been recently investigated [13] and it is believed
that the synchronization of coupled ratchets could provide some
information regarding the transport properties of inertia ratch-
ets [13]. Synchronization phenomena in coupled or driven non-
linear oscillator are in general of fundamental importance in
nonlinear dynamics and have been extensively investigated both
theoretically and experimentally since the seminal work of Pec-
ora and Carroll in 1990 [14]. Chaos synchronization is closely
related to the observer problem in control theory [15]. The prob-
lem may be treated as the design of control law for full chaotic
observer (the slave system) using the known information of the
plant (the master system) so as to ensure that the controlled
receiver synchronizes with the plant. Hence, the slave chaotic
system traces the dynamics of the master in the course of time.
Various techniques have been proposed for achieving stable
synchronization between identical and non-identical systems.
Notable among these methods, the active control scheme [16]
has received a considerable attention in the last few years due
to its simplicity and robustness. Applications to various systems
abound, some of which includes the Lorenz, Chen and Lü sys-
tem [17], geophysical system [18], spatiotemporal dynamical
system [19], the so-called unified chaotic attractor [20], elec-
tronic circuits, which model a third-order “jerk” equation [21],
the Bloch equation [22] and most recently in RCL-shunted
Josephson junction [23].
In the present Letter, we employ the active control technique
to first, examine the synchronization behavior of two inertial
ratchets with identical system parameters; both evolving from
different initial conditions. Secondly, we explore the synchro-
nization property to show that the direction of particle transport
in inertial ratchets can be reversed or controlled by using active
control technique.
2. The chaotic ratchet model
Let us consider the one-dimensional problem of a particle
driven by a periodic time-dependent external force under the
influence of an asymmetric potential of the ratchet type [9,10].
The time average of the external force is zero. In the absence of
stochastic noise, the dynamics is exclusively deterministic. The
dimensionless equation of motion for a particle of unit mass
moving in the ratchet potential V (x) is given by (see Ref. [10]
for instance):
(1)ẍ + bẋ + dV (x)
dx
= a cos(ωDt),
where time t has been normalized in the unit of ω−10 , ω0 be-
ing the frequency of the linear motion around the minima of
V (x). b is the damping parameter, while a and ωD are the am-
plitude and frequency of the driving force respectively; V (x) is
the dimensionless potential given by
(2)V (x) = C − 1
4π2δ
[
sin 2π(x − x0) + 0.25 sin 4π(x − x0)
]
.
The constant C  0.0173 and δ  1.600. The potential is
shifted by a value x0 in order that the minimum of V (x) is lo-cated at the origin (see Fig. 1(a)). We note that apart from its
periodicity, the ratchet potential (2) has an infinite number of
potential wells; so that the orbits transport particles from one
well to another. Thus, in the Poincaré section representation,
one can utilize this periodicity to collapse the dynamics to a
unit cell within a phase space region for which −0.5  x  0.5
(Fig. 1(b)).
The extended phase space in which the dynamics is tak-
ing place is three-dimensional, since we are dealing with an
inhomogeneous differential equation with an explicit time de-
pendence. Eq. (1) can be expressed in autonomous form and
then solved numerically using the fourth-order Runge–Kutta al-
gorithm. Since the equation is nonlinear, its solution therefore
allows the possibility of periodic and chaotic orbits. Fig. 1(b)–
(d) shows the chaotic behavior of system (1).
3. Synchronization of inertial ratchets using active control
Let us consider two identical ratchets in a master–slave con-
figuration, such that the master ratchet with the subscript 1 is
to control the slave ratchet with subscript 2. Without losts of
generality, we can express the master ratchet system as
ẋ1 = y1,
(3)ẏ1 = a cos(wDt) − by1 − dV (x1)
dx1
,
and the slave ratchet as
ẋ2 = y2 + u1(t),
(4)ẏ2 = a cos(wDt) − by2 − dV (x2)
dx2
+ u2(t),
where u1(t) and u2(t) are control functions to be determined. In
order to estimate the control functions, we subtract Eq. (3) from
Eq. (4) and defining the error system as the difference between
the master and the slave ratchets:
(5)x3 = x2 − x1; y3 = y2 − y1,
we obtain
ẋ3 = y3 + u1(t),
(6)ẏ3 = −by3 − dV (x2)
dx2
+ dV (x1)
dx1
+ u2(t).
We redefine the control functions to eliminate all items that can-
not be shown in the form x3 and y3:
u1(t) = V1(t),
(7)u2(t) = dV (x2)
dx2
− dV (x1)
dx1
+ V2(t).
Thus,
ẋ3 = y3 + V1(t),
(8)ẏ3 = −by3 + V2(t),
which can be written as [ẋ3, ẏ3]T = M + V (t), where M =( 0 1
0 −b
)
is a feedback matrix and V (t) = [V1(t),V2(t)]T . The
error dynamics (8) is a full state controllable entity so that
U.E. Vincent, J.A. Laoye / Physics Letters A 363 (2007) 91–95 93(a) (b)
(c) (d)
Fig. 1. Ratchet dynamics (a) the ratchet potential, (b) chaotic attractor (in the Poincaré section), (c) intermittent chaotic transport (x-variable), (d) y variable.
a = 0.08092844, b = 0.1, ω = 0.67.Dfeedback gains can be designed to stabilize the error system
[x3, y3]T , such that error signals asymptotically converge to
zero as t → ∞. We choose a constant matrix K which will con-
trol the error dynamics (8) such that:
(9)
(
V1(t)
V2(t)
)
= K
(
x3
y3
)
.
There are several choices of K that could lead to the stability of
the closed-loop system (8). In principle, the elements of matrix
K could be chosen such that the feedback matrix M has all the
eigenvalues with negative real parts. Here, we choose
(10)K =
( −1 −1
0 b − 1
)
.
With matrix (10), the error system (8) is stable and the close-
loop system has eigenvalues (−1,−1). This choice will lead to
the synchronization of the master–slave ratchets.
4. Numerical results
We first illustrate the synchronization behavior of two iden-
tical chaotic ratchets. In our numerical simulations, we used the
4th-order Runge–Kutta scheme with a time-step of 0.02π/ωD ;
where ωD , the frequency of the external driving force is fixedat 0.67 throughout the Letter. Other parameters of the master–
slave ratchet system were chosen such that the dynamics is
chaotic as shown in Fig. 1. The initial conditions for the master
are x1(0) = y1(0) = 0 while the initial conditions for the slave
are: x2(0) = 0.5, y2(0) = 0.1. When the control is switched off,
the chaotic transport of the master–slave ratchet is as shown in
Fig. 2(a) and when the control is switched on, the two ratchets
are synchronized as shown in Fig. 2(b).
In order to examine the control of directed transport, we
make the following choice of initial conditions: x1(0) = −0.10,
y1(0) = 0.25, x2(0) = 0.43, y2(0) = −0.12 and simulate the
system for a = 0.156 as employed in Ref. [24]. These ini-
tial conditions and parameter setting are of particular interest
as they corresponds to the case where two attractors co-exist
in phase space as already presented in [24]; and the situa-
tion is analogous to a mixture of identical or nonidentical
particles [12]. Here, the master system corresponds to the
chaotic attractor transporting particles in the positive direc-
tion while the slave is the periodic attractor, which transports
particles in the negative direction as shown in Fig. 3(a). It
should be noted that our numerical results shown in Fig. 3(a),
is obviously the same as Fig. 4 in Ref. [24]. It is very easy
to show that the direction of particle transport can be re-
versed or controlled at any given time to follow the direc-
94 U.E. Vincent, J.A. Laoye / Physics Letters A 363 (2007) 91–95(a)
(b)
Fig. 2. Synchronization dynamics of the master–slave ratchets when the control
is (a) switched off (b) switched on at t = 0. x1(0) = y1(0) = 0 and x2(0) = 0.5,
y2(0) = 0.1 and ωD = 0.67. The inset (bottom-right) in (b) is the zoom show-
ing the initial transient (t = 4 s) before complete synchronization is achieved.
The inset (top-left) in (b) is the error dynamics (x3 vs. t ) showing asymptotic
convergence to zero as t → ∞.
tion of another attractor by using the mechanism of active
control. We illustrate this in Fig. 3(b); where the controllers
have been activated at a target time of 50 s. Obviously, the
periodic attractor now follows the direction of the chaotic
attractor, both transporting particles in the positive direc-
tion.
5. Concluding remarks
In summary, we have demonstrated a specific application of
the active control in nonequilibrium statistical physics wherein
the direction of particle transport can be reversed by designing
appropriate active controllers that ensures stable synchroniza-
tion between the master ratchet and the slave ratchet. Thus, by
identifying the directions of particle transport for co-existing
attractors, one can decide a desired direction of particle trans-
port and the time for which a reverse to this direction is re-
quired. With appropriate controllers, any desired direction of
transport can be achieved. Finally, we note that the active con-
trol method has been found effective in synchronizing non-
identical chaotic systems as well as systems with parameter(a)
(b)
Fig. 3. Synchronization dynamics of the master–slave ratchets for initial con-
ditions: x1(0) = −0.10, y1(0) = 0.25, x2(0) = 0.43, y2(0) = −0.12 and
a = 0.156, b = 0.1, ωD = 0.67. In (a) control is switched off while in (b) con-
trol of particle transport to the positive direction in a chaotic manner is achieved
when control is switched on at t = 50. Inset (top-left) in (b) is the error dynam-
ics (x3 vs. t ) while the inset (bottom-right) is the zoom of the initial transient
when control is activated t = 0.
mismatch (see for example Refs. [17,21,25]). The implication
is that, in mixtures of interacting particles, one can employ
the active control technique to control the motion of identi-
cal or nonidentical particles moving in a ratchet potential as
well as the motion of identical particles moving in nonidenti-
cal ratchet potentials. Detailed study of these interesting cases
have already been carried out together with its connection to
particle separation and will be reported elsewhere. As illus-
trated in [16], implementation of the active control scheme
is visible and simple; and could be extended to ratchet de-
vices.
Acknowledgements
We would like to thank Dr. Anatole Kenfack, Max Planck In-
stitute for the Physics of Complex System, Dresden, Germany
for supplying us with relevant literature in the domain. We also
acknowledge useful comments by the reviewers.
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