Journal of the Korean Physical Society, Vol. 50, No. 1, January 2007, pp. 243∼248
Control of a Deterministic Inertia Ratchet System via Extended Delay Feedback Woo-Sik Son and Young-Jai Park Department of Physics, Sogang University, Seoul 121-742
Jung-Wan Ryu, Dong-Uk Hwang and Chil-Min Kim∗ National Creative Research Initiative Center for Controlling Optical Chaos, Pai-Chai University, Daejeon 302-735 (Received 15 August 2006) We study the control of a transport property in a deterministic inertia ratchet system via extended delay feedback. We stabilize a chaotic current of a ratchet system to a regular current by controlling an unstable periodic orbit embedded in a chaotic attractor of the uncontrolled system via feedbacking the delayed velocity of a particle. By stabilizing an unstable periodic orbit, we can control the velocity and the direction of a particle, which we desire. PACS numbers: 05.40.-a, 05.45.Gg, 05.45.Pq, 05.60.Cd Keywords: Ratchets, Molecular motor, Chaos control
I. INTRODUCTION
potential with a time delay feedback was also studied [14], and the anticipated synchronization was observed in deterministic inertia ratchet systems coupled by delay feedback [15]. Meanwhile, time-delayed chaotic systems have attracted much attention for the purposes of controlling chaotic behavior [16] and understanding the effect of the time delay in those systems [17]. In the former case, Pyragas proposed a method for stabilizing unstable periodic orbits in a chaotic system [16]. The method utilizes a control signal with a difference between the current state and the previous (delayed by the period of an unstable periodic orbit) state of the system, and it can be easily applied to real experimental situations. The stability and the analytical properties of a delayed feedback system have also been investigated [18]. Modifications of the Pyragas method have been proposed to improve its efficiency [19]. Socolar, Sukow and Gauthier have proposed a new method utilizing information from many previous states of the system, and their method has been called extended time-delay autosynchronization or extended delay feedback (EDF). In this paper, via EDF, we aim to control the transport property of a deterministic inertia ratchet system that is driven by a deterministic driving force considering the effect of the inertia term. We have stabilized a chaotic current to be a regular current. Through the control method, we have obtained the desired velocity and direction among unstable periodic orbits embedded in a chaotic attractor.
In recent years, ratchet systems have been studied theoretically and experimentally in many different fields of science [1], e.g., molecular motors [2], asymmetric superconducting quantum interference devices [3], quantum Brownian motion, [4], Josephson-junction arrays [5], etc. A ratchet system is defined as a system that can afford to make a directional motion of a particle in a periodic structure while, on average, an unbiased force is acting [6]. For this directional motion, named the “ratchet effect,” to be produced, two conditions are necessary [7]. First, a correlated stochastic [8] or a deterministic perturbation [9] is required for the ratchet system to be in a non-equilibrium state. Second, an asymmetric periodic potential is required to break the spatial inversion symmetry. The ratchet effect has been shown to be producible in a spatially periodic potential with a perturbation that is time-asymmetric [10]. For various applications of a ratchet system, many works concerning the control of ratchet dynamics have been presented. In a collective flashing ratchet, the velocity of the center of mass was maximized by feedbacking particle’s velocity [11]. The signal mixing of two driving forces was used to control the transport property in a overdamped ratchet system [12]. The parameter space where regular currents are observed was extended by applying a weak subharmonic driving force in a deterministic inertia ratchet system [13]. The dynamics of an inertia ratchet system on an asymmetric ratchet ∗ E-mail:
[email protected]; Fax: +82-42-520-5821
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Fig. 1. Asymmetric periodic potential V(x) with d = −0.19, δ = sin(2π|d|) + sin(4π|d|), and C = −(sin(2πd) + 0.25 sin(4πd))/4π 2 δ.
II. DETERMINISTIC INERTIA RATCHET SYSTEM The deterministic inertia ratchet system is written as follows: x ¨ + bx˙ + V 0 (x) = a cos(ω0 t),
(1)
where b is the friction coefficient, and ω0 and a are the frequency and the amplitude of the driving force, respectively. V (x) is the asymmetric periodic potential described by ³ ¡ ¢ V (x) = C − sin 2π(x − d) ¡ ¢´ 1 + sin 4π(x − d) /4π 2 δ, (2) 4 where d, δ and C are introduced so that the potential has a minimum at x = 0 with V (0) = 0. In this system, multiple current reversals have been observed to occur as the amplitude of the driving force is varied [20], the current reversal has been observed to be related to a bifurcation from chaotic to regular regime [21], and the type-I intermittency has been observed to be exist in this bifurcation [22]. Fig. 1 shows the asymmetric periodic potential, i.e., the ratchet potential. The deterministic inertia ratchet system shows a regular current in a positive or negative direction and a chaotic current with almost zero averaged velocity, depending on the parameter a for fixed parameters b = 0.1 and ω0 = 0.67 [21–23]. When the deterministic inertia ratchet system shows a regular current, the time required for the particle to move from one well of the potential to another is commensurable with the period of the driving force. Hence, the mean velocity of the regular current is locked with the period of the driving force: n ω0 n n L = = vl , (3) mT m 2π m where L and T are the spatial period of the ratchet potential, and the time period of the driving force, ren is an irreducible fraction (n, m ∈ Z) spectively, and m hvi =
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Fig. 2. Three different trajectories in the deterministic inertia ratchet system for three cases of a at b = 0.1 and ω0 = 0.67: (a) a period-2 positive current at a = 0.074, (b) a period-4 negative current at a = 0.081, and (c) a chaotic current at a = 0.083.
[23]. The spatial period of the ratchet potential satisfies L = 1, as shown in Fig. 1. Fig. 2 shows regular and chaotic currents of the system. At a = 0.074, the system shows a period-2 positive current with mean velocity hvi = 12 vl , a period-4 negative current with hvi = − 41 vl at a = 0.081, and a chaotic current with almost zero averaged velocity at a = 0.083. In Fig. 3(a), we plot a bifurcation diagram of the stroboscopic recording of particle’s velocity at t = kT , where k is a positive integer and T is the period of the driving force. A bifurcation diagram of the mean velocity
Control of a Deterministic Inertia Ratchet· · · – Woo-Sik Son et al.
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Fig. 4. Unstable period-1 orbits. (the positive, the negative, and the confined current) obtained from the uncontrolled system at a = 0.083.
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Fig. 3. Bifurcation diagrams as a function of a at b = 0.1 and ω0 = 0.67. In the region from a = 0.063 to 0.071, coexisting attractors are found. (a) shows a stroboscopic recording of the particle velocity, and (b) shows the mean velocity of current.
of the system is shown in Fig. 3(b) as a function of the parameter a. When the system shows a chaotic current at a = 0.083, there are various unstable periodic orbits embedded in the chaotic attractor, and all of them satisfy the locking phenomenon as in Eq. ¡ (3). In¢this ¡system, the periodic ¢ orbit is defined by x ˜(t), v(t) = x ˜(t + τ ), v(t + τ ) , where x ˜(t) = x(mod 1), and τ is the time period of the orbit. In Fig. 4, we have shown unstable period-1 orbits located by Newton methods. There are three period-1 orbits; two of them are positive and negative currents, and the other is an oscillating orbit confined in one well of the potential. The mean velocities of the three periodic orbits are hvi = 11 TL = vl , hvi = − 11 TL = −vl , and hvi = 0L 1 T = 0, respectively. By selecting one which is to be stabilized among other orbits, we can easily control the transport property of the deterministic inertia ratchet system.
III. CONTROL BY EXTENDED DELAY FEEDBACK: NUMERICAL RESULT The deterministic inertia ratchet system controlled by using the EDF is described as follows: x ¨ + bx˙ + V 0 (x) = a cos(ω0 t) + F.
(4)
F is an EDF term, i.e., the difference between the particle’s present velocity and the previous velocities delayed
by multiples of the period of the unstable periodic orbit. F is described as follows: ∞ ³ ´ X ¡ m−1 ¢ F = K (1 − R) R x(t ˙ − mτ ) − x(t) ˙ , (5) m=1
where K is the strength of feedback, τ is the delay time, which coincides with the period of the desired unstable periodic orbit, and R (0 ≤ R < 1) is the parameter that adjusts the distribution of each element’s magnitude in the EDF term. When R = 0, the EDF¡ method is the ¢ same as the Pyragas method, i.e., F = K x(t−τ ˙ )−x(t) ˙ . Now, let us consider the stability analysis of a periodic orbit in the presence of the EDF. Let the small deviation from the periodic orbit, ξ0 (t), be δξ(t) = ξ(t) − ξ0 (t). According to the Floquet theory, δξ(t) is expressed as µ ¶ δx(t) δξ(t) = = e(λ+iω)t u(t), (6) δv(t) where λ+iω is the Floquet exponent, and u(t) = u(t+τ ) is an eigenfunction. Then, after the period τ of orbit has passed, the deviation is described as ¡ ¢ δξ(t + τ ) = exp (λ + iω)(t + τ ) u(t + τ ) (7) ¡ ¢ = exp (λ + iω)τ + (λ + iω)t u(t) ¡ ¢ = exp (λ + iω)τ δξ(t) ≡ (Λ + iΩ)δξ(t), where Λ + iΩ is the Floquet multiplier. As including the delay terms, the phase space of the system becomes infinite dimensional and the system has an infinite number of Floquet multipliers. If the largest Floquet multiplier satisfies |Λ1 + iΩ1 | < 1, i.e., the leading Floquet exponent λ1 = (ln |Λ1 + iΩ1 |)/τ is less than zero, then the unstable periodic orbit is stabilized. The time evolution of δξ(t) is given by µ ¶ µ ¶ 0 1 0 0 δ ξ˙ = δξ(t) + (8) −V 00 (x) −b 0 1
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Fig. 6. (Color online) Basins of period-1 orbits: the initial points marked by ¤ (red), 4 (black), and • (black) are the basins of the positive, the negative, and the confined currents, respectively.
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Fig. 5. Leading Floquet exponents: (a), (b), and (c) exhibit the leading Floquet exponents for the positive, the negative, and the confined current as functions of K for the given R, respectively. ∞ X ¡ ¢ ×K (1 − R) Rm−1 δξ(t − mτ ) − δξ(t) ,
where the matrix in the first term in Eq. (8) is the Jacobian of the uncontrolled system, and the second term is the effect of the EDF. On the other hand, expanding the second term, Eq. (8) can be rewritten as follows: µ
¶ 1 δξ(t) −V 00 (x) −b − K µ ¶ 0 0 + (1 − R)δξ(t − τ ) 0K µ ¶ 0 0 + (1 − R)Rδξ(t − 2τ ) + · · · . 0K
Φ˙ t = AΦt , Φ0 = I.
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The eigenvalue of Φτ defines the Floquet multiplier as follows: det[Φτ − (Λ + iΩ)I] = 0.
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R where the ratio of geometric series satisfies |Λ+iΩ| < 1. Eq. (11) includes information on the unstable periodic orbit. Hence, the Floquet multiplier is related to the eigenvalue problem of the monodromy matrix, Φτ , which satisfies
(13)
The Floquet multipliers have been obtained by numerically solving Eqs. (11)-(13), and the results of the leading Floquet exponents for period-1 orbits are shown in Fig. 5. The period-1 positive current is stabilized at R ≥ 0.6, the negative current is stabilized at R ≥ 0.4, and the confined current can be stabilized at R = 0. This phenomenon is caused by the different degrees of instability of unstable periodic orbits in the uncontrolled system (K = 0). As Fig. 5 shows, λ1 of the positive current at K = 0 is λ1 ' 0.42, λ1 of the negative current is λ1 ' 0.3, and λ1 of the confined current is λ1 ' 0.06. With the larger degree of instability in the uncontrolled system, the unstable periodic orbit is stabilized with a
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positive or the negative currents always overlaps with that of the confined current. Now, we are interested in the specific parameters R = 0.8 and K = 0.67 for which all of the period-1 orbits are stabilized. In Fig. 5, each orbit’s leading Floquet exponents at these parameters are marked by a black circle, and all of them are less than zero, which means that the three stabilized period-1 orbits coexist at the same control parameters K = 0.67, R = 0.8, and τ = T = 2π/ω0 . In Fig. 6, we have investigated the basins of period-1 orbits. We have integrated Eqs. (4) and (5) with the initial condition (x0 , v0 ), where x0 is distributed in one well of the potential, x0 ∈ (−0.38, 0.62), and v0 is confined to the ranges of velocities in the unperturbed system, v0 ∈ (−0.3, 0.3). As Fig. 6 shows, the basins of period-1 orbits are, to the order of their size, the confined, the negative, and the positive currents. In Fig. 7, we have plotted three stabilized period-1 orbits that evolved from different initial conditions. In the uncontrolled system, all of the initial conditions evolve into a chaotic current in the same manner (Fig. 2(c)). However, the controlled system via the EDF shows different periodic orbits for different initial conditions, even though the control parameters of the system, K = 0.67, R = 0.8, and τ = T = 2π/ω0 , are the same. The chaotic current of the deterministic inertia ratchet system is controlled to unstable period-1 orbits for regular currents. To control the unstable periodic orbits, we obtain the leading Floquet exponents for each orbit. However, we do not control the higher-order unstable periodic orbit because it is hard to obtain the leading Floquet exponent for the higher-order period from Eqs. (11)-(13).
0 -0.2 100
IV. CONCLUSIONS
t
Fig. 7. Stabilized period-1 orbits: (a) the positive current with the initial values of (x0 , v0 ) = (−0.35, 0.2), (b) the negative current with (x0 , v0 ) = (0.2, 0.0), and (c) the confined current with (x0 , v0 ) = (0.0, 0.0).
larger R. In all cases, the EDF method (R > 0) is more effective than the Pyragas method (R = 0). The stabilized region of K, in which the desired unstable periodic orbit is stabilized (λ1 < 0), increases with increasing R and if R is not very large, then the minimum of λ1 (K) is deeper with increasing R. The EDF method gives a faster convergent rate of nearby orbits to the desired unstable periodic orbit and makes the control scheme more resistant to noise. Among period-1 orbits, only the confined current can be stabilized with an appropriate parameter space, (K, R), while there is no parameter space in which only either the positive or the negative current is stabilized alone because the stabilized region of the
We study the control of the deterministic inertia ratchet system via extended delay feedback and show control of a chaotic current to a regular one. To obtain the control parameters, the feedback strength K and the distribution of the EDF term R, we solve for the leading Floquet exponent in the presence of extended delay feedback. By using the control parameters, we can stabilize three unstable period-1 orbits, depending on the initial values of the system, the positive, the negative, and the confined currents. Through this study, we can understand that the deterministic inertia ratchet system can be stabilized for a desired orbit.
ACKNOWLEDGMENTS The authors thank Dr. S. Rim for valuable discussions. This study was supported by the Creative Research Initiatives (Center for Controlling Optical Chaos) of MOST/KOSEF.
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