Regular and chaotic transport of discrete solitons in asymmetric potentials

Regular and chaotic transport of discrete solitons in asymmetric potentials J. Cuevas,1B. Sánchez-Rey,1and M. Salerno2 1Departamento de Física Aplicada I, E. U. P., Universidad de Sevilla, Virgen de África 7, 41011 Sevilla, Spain 2Dipartimento di Fisica “E.R. Caianiello,” CNISM and INFN–Gruppo Collegato di Salerno, Università di Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy 共Received 24 March 2010; published 20 July 2010兲 Ratchet dynamics of topological solitons of the forced and damped discrete double sine-Gordon system are studied. Directed transport occurring both in regular and in chaotic regions of the phase space and its dependence on damping, amplitude, and frequency of the driving, asymmetry parameter, and coupling constant, has been extensively investigated. We show that the passage from ratchet phase-locked regime to chaotic ratchets occurs via a period doubling route to chaos and that, quite surprisingly, pinned states can exist inside phase locking and chaotic transport regions for intermediate values of the coupling constant. The possibility to control chaotic discrete soliton ratchets by means of both small subharmonic signals and more general periodic drivings, has also been investigated. DOI: 10.1103/PhysRevE.82.016604 PACS number共s兲: 05.45.Yv, 63.20.Ry I. INTRODUCTION Ratchet dynamics in nonlinear systems are presently attracting a great deal of interest due to their relevance in several fields, including molecular motors and Josephson junctions 关1,2兴. For point particles ratchets the directed motion achieved in presence of damping and deterministic forces of zero average, is a direct consequence of the breaking of the spatiotemporal symmetries in the system 关3–6兴. The resulting net motion of the particle is usually phase locked to the external driving 关7,8兴and the deterministic ratchet dynamics is quite robust to exist also in presence of noise. Ratchet motion occurring in infinite-dimensional continuous nonlinear systems with solitonlike solutions 共soliton ratchets兲have been extensively investigated in the past years. This has been done both in the case of spatially asymmetric potentials and single harmonic driving 关9,10兴and for symmetric potentials and temporarily asymmetric bihharmonic forces 关6,11–13兴. Experimental implementations of soliton ratchets in long Josephson junctions were also done in 关14兴 using time-asymmetric biharmonic currents, and in 关15,16兴 using asymmetric magnetic fields or spatially asymmetric currents. The mechanism underlying continuous soliton ratchets, also known as internal mode mechanism, is ascribed to the coupling between the soliton internal degree of freedom 共internal mode兲and the translational mode of the soliton 关10兴. This coupling is induced mainly in the presence of damping and allows an asymmetric transfer of energy from the external driving to the internal mode and to the soliton center of mass which generates the directed motion. Thus, opposite to point particle ratchets for which the action of the force is always direct on the center of mass, in soliton ratchets the force acts indirectly on the center of mass and involves the vibrational mode of the solitons in an essential manner 共a direct action may exist in the overdamped limit where the phenomena becomes negligibly small 关9兴兲. This is quite natural if one thinks that the soliton is not a rigid object but it can vibrate while moving, just like a droplet of liquid does when dropping. This internal mode mechanism has been confirmed to be of general validity for continuous dissipative soliton systems 关6,10,17–20兴. The possibility of soliton ratchets in continuous Hamiltonian systems has been recently demonstrated for the case of the nonlinear Schrödinger equation with asymmetric potentials rapidly flashing in time 关21兴. In contrast to the continuous case, discrete soliton ratchets 共DSR兲in nonlinear lattices have been much less investigated. Detailed studies exist only for ratchetlike behaviors in a dissipative discrete sine-Gordon chain driven by a biharmonic force which breaks the time symmetry 关22,23兴, and for the case of a conservative discrete nonlinear Schrödinger equation describing a one-dimensional 共1D兲array of coupled waveguide optical resonators 关24兴. In both cases the interplay between discreteness and nonlinearity introduce new features in the phenomenon such as: existence of depinning thresholds for transport, staircase like dependence of the mean drift velocity on system parameters, diffusive and even intermittent ratchetlike dynamics 关22兴. The possibility to enhance DSR transport by introducing phase disorder into the asymmetric biharmonic periodic driving was also considered 关23兴. It is interesting, however, to investigate the phenomenon of DSR in the case in which the potential is asymmetric in the field variable and the system is driven by harmonic forces of zero mean. In this case the discrete solitons have an internal degree of freedom even in absence of damping and driving, a fact which could lead to qualitative different behaviors. The aim of this paper is twofold. From one side, we extend previous investigations on DSR to the case of asymmetric potentials. For this we consider the case of a double sine- Gordon chain with potential function given by V共x兲=C−1 4 ␲ 2cos共2 ␲ x兲+␭ 8 ␲ 2sin共4 ␲ x兲,␭苸关0,1兴, 共1兲 where ␭is an asymmetry parameter and Cis a constant that fixes the zero of the potential. Therefore the system we are going to consider belongs to the Frenkel-Kontorova family 关25兴and can be visualized as a chain of mechanical double PHYSICAL REVIEW E 82, 016604 共2010兲 1539-3755/2010/82共1兲/016604共11兲©2010 The American Physical Society016604-1 pendula connected by a gear of ratio 1/2 with nearestneighbor interactions and with the periodic on-site doublesine potential in Eq. 共1兲due to the gravity 关26兴. The dependence of the transport on the damping constant, amplitude and frequency of the driving force, coupling constant and asymmetry parameter, is investigated in detail. As a result we show that in contrast to DSR driven by asymmetric forces of zero mean 关22兴, kink transport in the present system can be quite effective not only in correspondence of regular 共phaselocked兲orbits, but also on chaotic trajectories. From the other side, we shall investigate the DSR chaotic transport in some detail both with respect to the onset of chaos and with respect to the possibility of controlling the chaotic ratchet trajectories using subharmonic signals which do not break the time symmetry in the system. To this regard, we introduce a suitable map in the space of the kink coordinates by means of which we construct bifurcation patterns. Using this approach, we show that the chaos onset occurs via period doubling and that the chaotic region can be strongly reduced in favor of periodic phase-locked orbits in presence of small subharmonic signals. The complete suppression of chaos is also investigated by means of elliptic driving functions with modulus taken as a free parameter to achieve optimal control. For the regions of parameters we have investigated, we show that the full chaos suppression can be implemented in any practical context by means of two properly chosen harmonic signals. The paper is organized as follows. In Sec. II, we introduce the model equations and discuss the main properties of the static kink solutions, including their Peirls-Nabarro barrier and the small oscillation problems. In Sec. III, we investigate the dependence of the discrete ratchet dynamics, both phaselocked and chaotic, on the system parameters. In Sec. IV,we concentrate on the ratchet dynamics associated to chaotic orbits and on their stabilization by means of small subharmonic modulations of the harmonic driving. We study the onset of chaos by means of a suitable bifurcation map and show that chaos occurs via period doubling. Finally, in Sec. V, we summarize the main conclusions of the paper. II. MODEL EQUATION The chain of double pendula associated to potential Eq. 共1兲is described in a dimensionless form by the equation 关26兴 x ¨n+ ␣ x ˙n+V⬘共xn兲+ ␬ 共2xn−xn+1 −xn−1兲= ␥ +F共t兲+ ␰ n共t兲, 共2兲 with n苸关−N,N兴,xnbeing the displacement of the nth pendulum with respect to its equilibrium position, ␥ a constant bias term, ␣ the damping coefficient, ␬ the coupling constant which measures the discreteness of the lattice, F共t兲=␧ 2 ␲ cos共 ␻ t+ ␪ 兲,共3兲 an external symmetric in time periodic driving force of strength ␧, frequency ␻ and ␰ n共t兲a stochastic term of zero average and autocorrelation function fixed as 具 ␰ n共t兲 ␰ m共t⬘兲典 = ␣ D ␲ ␦ nm ␦ 共t−t⬘兲,共4兲 and modeling the effect of a thermal bath in contact with the system. In the following we will investigate mainly deterministic discrete ratchets in absence of the dc bias ␥ =0 and in presence of the ac driving of zero average Eq. 共3兲.We shall introduce external noise in the system only to check that the directed motion is associated to single stable attractors, a fact which permits to avoid averaging on initial conditions 共e.g., initial phase ␪ of the driving兲. In Fig. 1共a兲we show the potential chain in Eq. 共1兲for the case ␭=0.5. Notice that minima of the potential are located at n− ␦ 共␭兲, being ␦ 共␭兲=1 2 ␲ sin−1关1−冑1+8␭2 4␭兴. Since a discrete kink 共antikink兲can be thought as a string in space extending from one absolute minimum of the potential to an adjacent one, we will assume ␭苸关0,1兴to avoid the appearance of relative minima in the potential. For large coupling constant ␬ , approximated analytical expressions for kinks and antikinks can be obtained from the corresponding exact solutions of the continuum limit derived in 关10兴共notice that for large ␬ −1 −0.5 0 0.5 1 0 0.01 0.02 0.03 0.04 0.05 0. 06 x V ( x ) (a) −10 −5 0 5 1 0 0 0.5 1 n x n (b) −10 −5 0 5 1 0 0 0.1 0.2 0.3 0.4 n ∇nxn (c) FIG. 1. 共Color online兲共a兲Comparison between the symmetric potential for ␭=0 共continuous line兲and an asymmetric double sine- Gordon potential 共dashed line兲for ␭=0.5. 共b兲Stationary discrete kink solutions of the unperturbed double sine-Gordon chain corresponding to ␭=0 共circles joined by a continuous line兲and ␭=0.5 共squares joined by a continuous line兲. The discrete spatial derivatives of the kink profiles are also shown in 共c兲with corresponding symbols. The value of the coupling constant ␬ is fixed to 1. CUEVAS, SÁNCHEZ-REY, AND SALERNO PHYSICAL REVIEW E 82, 016604 共2010兲 016604-2 the discrete chain is well approximated by the continuous double sine-Gordon equation兲,as xn ⫾= ␦ 共␭兲+2 ␲ tan−1 冦 sgn共␭兲AB A−1− ␩ sinh 冋 ⫾⌫n 2冑AB 兩␭兩 册 冧 ,共5兲 with A=冑1+8␭2, ␩ =2␭冑2共1+A兲,B=冑2共4␭2−1+A兲, and ⌫n=n−Vt 冑1−V2共the plus and minus signs refer to the kink and antikink solutions, respectively兲. For small values of ␬ the kink solutions must be found numerically. In the panel 共b兲of Fig. 1we depict two discrete kinks obtained numerically for the case of symmetric 共␭=0兲and asymmetric 共␭=0.5兲poten- tials. Notice that in the symmetric case the kink is symmetric around the center 关this is clearly seen from the derivative profile shown in panel 1共c兲兴 while in the case of asymmetric potential the kink is asymmetric and the width is a little shrunk. A. Peierls-Nabarro barrier and depinning A main feature of DSR with respect to continuous soliton ratchets is the presence in the discrete case of the Peirls- Nabarro 共P-N兲barrier, which introduces a depinning threshold for kink transport. To investigate the dependence of this threshold on the system parameters we consider static kink solutions of the unperturbed system solving numerically the nonlinear system of equations V⬘共xn兲+ ␬ 共2xn−xn+1 −xn−1兲=0, n苸关−N,N兴,共6兲 with the kink boundary condition xN−x−N=+1. Starting from a stable kink state we can obtain other 共unstable兲kink profiles whose center of mass is shifted with respect to the stable state, and then find the dependence of the kink energy on the position of its center Xc=兺 n=−N n=N nxn+1 −xn−1 2共7兲 In Fig. 2共a兲we show the resulting P-N relief for three different values of the asymmetry parameter: ␭=0, ␭=0.4, and ␭=0.9. Note that while for ␭=0 the PN potential possess the inversion center symmetry VPN共X兲=VPN共−X兲this is not true for the case ␭⫽0, due to the asymmetry of the potential in Eq. 共1兲. In all cases the P-N potential is a periodic function: VPN共X+X0兲=VPN共X兲, with period X0equal to 1. The difference between the maxima and minima potential energies, ⌬EPN, represents the activation energy necessary for putting the kink in motion. In panels 共b兲and 共c兲of Fig. 2we have also plotted the dependence of the activation energy and of the kink width defined as 关27兴 WK=1+2 冑兺 n=−N n=N 共n−Xc兲2兩xn+1 −xn−1兩 2,共8兲 on the asymmetry parameter ␭, respectively. One can see that while the activation energy is an increasing monotonic function of ␭, the kink width decreases with ␭in agreement with what observed for the kink profiles in Fig. 1. It is worth noting that the above symmetry properties of the P-N potential play important role for the existence of directed kink currents in the system. In the lowest approximation, indeed, we have that the dynamics of the kink in Eq. 共5兲can be described in terms of an effective point-particle equation of the type 关6兴 d2X共t兲 dt2+ ␣ X ˙共t兲+VPN ⬘共X兲+F ˜ 共t兲=0, 共9兲 where X共t兲and X ˙共t兲are collective coordinates denoting the position and the velocity of the kink, respectively, and F ˜ 共t兲is an effective driving force proportional to F共t兲in Eq. 共3兲.In order to have directed transport for the kink one must break all symmetries that change sign to the kink velocity and leave the equation of motion invariant 关5,6兴共the presence of such symmetries would imply the presence in the phase space of kink trajectories with opposite velocities, a fact which is obviously incompatible with ratchets兲. Since the P-N potential in our case does not have the center inversion symmetry and since the force is symmetric in time the only −1.5 −1 −0.5 0 0.5 1 1. 5 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 V PN x (a) 0 0.2 0.4 0.6 0.8 1 0 0.005 0.01 0.015 ∆EPN λ (b) 0 0.2 0.4 0.6 0.8 1 2.5 3 3.5 WK λ (c) FIG. 2. 共Color online兲共a兲Peierls-Nabarro relief for ␭=0 共continuous line兲,␭=0.4 共dashed line兲and ␭=0.9 共dashed-dotted line兲. 共b兲Peierls-Nabarro energy versus lambda. 共c兲Dependence of the kink width on the asymmetry parameter. REGULAR AND CHAOTIC TRANSPORT OF DISCRETE…PHYSICAL REVIEW E 82, 016604 共2010兲 016604-3 symmetry which could link trajectories with opposite velocities is the one which is indeed an exact symmetry of Eq. 共9兲 in absence of damping. In presence of damping, however, the time inversion symmetry is broken and so are all the symmetries of Eq. 共9兲which link orbits with opposite velocities. This simple point particle symmetry argument shows how important is the damping in our system to establish directed transport. It should be remarked, however, that kinks are not exactly point particles but extended objects with internal structure, which also plays an important role for directed transport, as we shall discuss in the following. B. Internal mode of discrete kinks As previously mentioned, discrete kinks of the double sine-Gordon system possess an oscillation of the kink profile 共internal mode兲even in the pure Hamiltonian limit. To investigate the existence of internal modes we have linearized the unperturbed system Eq. 共6兲around the static kink solution. The numerical spectrum obtained is shown in Fig. 3共a兲as a function of ␭. One can see that, apart the zero mode existing for all values of ␭depicted in panel 共c兲, an internal mode emerges for ␭⫽0 below the phonon band. Its shape, plotted in panel 共b兲, shows a spatial asymmetry induced by the asymmetry of the potential. This mode plays a crucial role in the appearance of directed motion through its coupling with the translation of the kink. It is worth noting that the existence of that internal mode also depends on the value of the coupling constant ␬ . In Fig. 4it is shown that for a fixed value of ␭the internal mode disappears for low enough coupling. In principle one can predict that no kink net motion must exist below that critical coupling. As we will see in next section this prediction is confirmed by simulations corroborating the relevance of the internal mode in the phenomenon. III. PHASE-LOCKED AND CHAOTIC DSR As mentioned before, directed kink motion is expected when symmetries of Eq. 共2兲that relate kink solutions with opposite velocities are broken 关5,6兴. This is the case for the double-sine Gordon potential Eq. 共1兲which violates the symmetry V共x兲=V共−x兲. In order to verify the existence of kink transport and to study the dependence of the phenomenon on system parameters, we have performed numerical integration of Eq. 共2兲by using a fourth-order Runge-Kutta method with constant step size ⌬t=T/200, being Tthe period of the external ac force. Our system is composed of 400 particles and in our simulations static kink was taken as initial condition and aperiodic boundary conditions xN−x−N=1 were also adopted. Three main dynamical regimes are observed: 共i兲no net motion; 共ii兲phase-locked directed motion; and 共iii兲chaotic directed motion. Figure 5shows two examples of both kinds of movement. In all cases we checked that the phenomenon does not depends on the initial phase ␪ of the driving force Eq. 共3兲, e.g., it is independent on the choice of the initial conditions so that one can properly talk about ratchet effect. In the phase-locked dynamics, the kink travels ksites during lperiods of the force, and therefore its velocity is given by 具v典=k l ␻ 2 ␲ .共10兲 In this regime oscillations on the kink profile are perfectly synchronized with the motion of the kink center of mass so that the kink profile is completely reproduced after lperiods except for a shift in space. This suggests the existence of a coupling between internal and translational modes of the kink, similarly as for the continuous case. In the chaotic regime the kink jumps backward and forward in an unpredictable manner and the directed motion can only be observed on a large time scale. In this regime the kink motion is of diffusive type and resembles the one of a Brownian particle. We remark that the directionality of the motion in both cases is preserved also in presence of a noise term of type Eq. 共4兲, as one can see from Fig. 5. This is a consequence of the fact that in the phase space there is only 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 λ Ω (a) −20 −15 −10 −5 0 5 10 15 2 0 0 0.2 0.4 0.6 Z ero mo d e n (c) −20 −15 −10 −5 0 5 10 15 2 0 −0.5 0 0.5 Internal mode n (b) FIG. 3. 共Color online兲共a兲Small oscillations frequency spectrum versus ␭. The wide band denotes the continuum spectrum, the curve which splits off from the band corresponds to the internal mode while the bottom line refers to the translational mode 共zero mode兲. The profiles of the internal mode and of the translational mode at ␭=0.46 are shown in panels 共b兲and 共c兲, respectively. The value of the coupling constant ␬ is fixed to 1. CUEVAS, SÁNCHEZ-REY, AND SALERNO PHYSICAL REVIEW E 82, 016604 共2010兲 016604-4 one attractor corresponding to the net motion of the kink 共in all investigated cases we have not found phenomena of hysteresis signaling the presence of more attractors for the same set of parameters兲. This allows to concentrate on the deterministic phenomenon and compute the mean kink velocity by averaging on time but not on the initial conditions. Although in principle the spatial symmetry of our system is broken for any value of ␭⫽0, a distinctive feature of discrete systems is the existence of a depinning threshold above which the transport can occur and below which the kink is pinned to a lattice site. We have used to compute it a dc bias ␥ instead of an ac force. The result is plotted in Fig. 6as a function of the asymmetry parameter ␭. The depinning threshold increases monotonously with ␭in accordance with the rise of the P-N barrier shown in Fig. 2. For a fixed amplitude ␧of the ac force the dependence of the mean kink velocity on ␭is shown in Fig. 7. It is interesting to note that even for driving amplitudes above the depinning threshold a minimal value of the asymmetry parameter is required for the directed motion to start. By further increasing ␭a rather complex scenario appears. Chaotic transport plays a dominant role. Notice that the mean drift velocity on chaotic orbits can be comparable in some cases with the one achieved in the phase-locked regime. Kink dynamics is very sensitive to parameter values and small changes of them give raise to sudden changes of the dynamical regime or can even cause a current inversion. This last finding is actually a striking result since in the continuous case the sign of ␭completely determines the sense of net motion 关19兴. Nevertheless, we have checked that the symmetry property 具v共␭兲典=−具v共−␭兲典 existing in the continuous case is preserved in the case of phaselocked dynamics. The window of mobility widens of course if we increase ␧and more resonances appear 关see Fig. 7共b兲兴 but the scenario is qualitatively the same. In Fig. 8we have shown the distribution of regular, chaotic and no transport regimes in the 共␧,␭兲plane. We see that whereas the regular kink transport occurs mainly at small values of ␭共e.g., in the region ␭⬍0.3, 0.3⬍␧⬍0.5兲, chaotic directed motion be- 0 2000 4000 6000 8000 10000 0 20 40 60 80 100 120 140 160 180 200 t X c (a) 6500 7000 7500 8000 8500 9000 100 110 120 130 140 150 0 2000 4000 6000 8000 10000 12000 0 20 40 60 80 100 120 140 Xc t (b) FIG. 5. 共Color online兲共a兲the upper blue line represents phaselocked ratchet dynamics of the kink center of mass in the absence of noise for ␭=0.46 and ␧=0.25. The inset shows the perfect synchronization of the motion in a smaller time scale. When some noise 共D=10−4兲is added the direction of motion is preserved 共bottom red line兲.共b兲Chaotic ratchet dynamics for ␭=0.46 and ␧=0.24. Upper blue line corresponds to the deterministic case while bottom red line corresponds to stochastic dynamics with D=10−4. In all cases, ␣ =0.1, ␻ =0.1, and ␬ =1. 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0 . 2 5 λ γ th FIG. 6. 共Color online兲Depinning threshold for a dc bias ␥ as a function of the asymmetry parameter ␭. Parameters: ␣ =0.1, ␻ =0.1, ␧=0, and ␬ =1. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4 .5 κ Ω FIG. 4. 共Color online兲Small oscillations spectrum versus ␬ for ␭=0.46. Notice the disappearance of the internal mode associated with the oscillations of the kink width for ␬ ⱕ0.20 REGULAR AND CHAOTIC TRANSPORT OF DISCRETE…PHYSICAL REVIEW E 82, 016604 共2010兲 016604-5 comes dominant at large values of ␭. Notice also the presence of a large pinning region for small values of ␭which extends up to intermediate values of ␧. The influence on the phenomenon of the damping constant, ␣ , has also been investigated. In Fig. 9共a兲, we have plotted a diagram with the phase-locked attractors found on the 共␧, ␣ 兲plane for fixed ␭=0.46. We have used in the numerical calculations a step of 0.01 in the damping coefficient and a thinner step of 0.005 in the amplitude of the driving force. A band structure emerges as ␣ increases and, significantly, no phase-locked attractors are found when ␣ →0, showing the importance of damping in the coupling between the translational an internal modes of the kink 关19,20兴.In panel 共b兲of Fig. 9we show the fraction of phase-locked attractors found for each value of damping; from it, we can observe a non-monotonous behavior of the curve and higher probability of phase locking orbits at intermediate damping. In Fig. 10共a兲, we report chaotic and phase-locked attractors found by scanning the 共␭, ␣ 兲plane by keeping the rest of parameters fixed. Again, chaotic behavior is clearly dominant for low damping while the highest probability to find a regular attractor occurs at intermediate values of the damping, as one can see in panel 共b兲of Fig. 10. Since in the undamped limit the kink mean velocity must go to zero 共due to the 0 0.2 0.4 0.6 0.8 1 −0.04 −0.03 −0.02 −0.01 0 0.01 0 . 02 (1,1) (1,−1) (1,−2) λ <v> (a) 0 0.2 0.4 0.6 0.8 1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 (2,1) (1,2) (3,−1) (1,1) (1,−1) (1,−2) (1,−3) (1,−4) λ <v> (b) FIG. 7. 共Color online兲Dependence of 具v典with respect to ␭for ␧=0.1 共a兲and ␧=0.2 共b兲. Full circles 共crosses兲correspond to phaselocked 共chaotic兲dynamics. Numbers in brackets corresponds to the 共l,k兲pair of Eq. 共10兲. In both panels ␣ =0.1, ␻ =0.1, and ␬ =1. 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 .5 ε λ FIG. 8. 共Color online兲Dynamical regimes of the kink motion for different values of the asymmetry parameter ␭and the driving amplitude ␧. Full circles 共crosses兲correspond to phase-locked 共chaotic兲dynamics. Blank spaces stand for pinned-kink states. Parameters: ␣ =0.1, ␻ =0.1, and ␬ =1. 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 α ε (a) 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 .5 α Fraction o f phase−locked attractors f ound (b) FIG. 9. 共Color online兲共a兲Phase-locked 共full circles兲and chaotic 共crosses兲dynamical regimes of the kink ratchet motion in the 共␧, ␣ 兲 plane. Blank spaces stand for pinned-kink states. Other parameters are: ␭=0.46, ␻ =0.1, and ␬ =1. 共b兲Fraction of phase-locked attractors found in panel 共a兲for each value of the damping. CUEVAS, SÁNCHEZ-REY, AND SALERNO PHYSICAL REVIEW E 82, 016604 共2010兲 016604-6 driving force which is symmetric in time兲, and since for large damping the internal mode mechanism is practically suppressed 共this strongly reducing the drift velocity兲, one can expect transport not only be more probable but also more effective at intermediate values of the damping. This is indeed what is shown in panel 共c兲of Fig. 10 where the modulus of the averaged mean drift velocities achieved on the regular and chaotic attractors are shown as functions of the damping. The fact that at intermediate damping the coupling between internal vibration and center of mass motion becomes optimal also correlates with similar studies performed on continuous 关10,19兴and discrete 关22兴soliton ratchets, as well as, with very recent simulations 关28兴of discrete kink ratchets in an asymmetric potential of polynomial type free of the Peirls-Nabarro barrier. The dependence of the mean kink velocity on the driving frequency is characterized by the alternation of chaos with a series of phase-locking resonances. This is shown in Fig. 11共a兲where we have depicted the kink mean velocity normalized to the driving frequency 具v典T=2 ␲ 具v典/ ␻ so that for phase-locked dynamics 具v典Tcoincides with the ratio of rotation numbers k/l. We do not find the monotonic decay observed in the discrete sine-Gordon model with biharmonic forces 关22兴but, as in that case, kink transport disappears for high-enough frequencies ␻ ⬎0.75. Notice that the largest resonant step is achieved around ␻ =0.5 at the rotation numbers k=−1,l=1, and that at low frequencies 共e.g., ␻ ⬍0.2兲 the largest drift velocity is achieved in correspondence of chaotic trajectories. Also notice in panel 共b兲of Fig. 11 that around ␻ =0.5 the phase-locked attractors cover a large region of the 共␭, ␻ 兲plane which partially intersects the first 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 λ α (a) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 α Fraction of phase−locked attractors found (b) 0 0.2 0.4 0.6 0.8 1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 (1,1) (2,1) α | <v> | (c) FIG. 10. 共Color online兲共a兲Phase-locked 共full circles兲and chaotic 共crosses兲dynamical regimes of the kink ratchet motion in the 共␭, ␣ 兲plane. Other parameters are ␧= ␻ =0.1 and ␬ =1. 共b兲Fraction of phase-locked attractors found in panel 共a兲for each value of the damping. 共c兲Modulus of the averaged mean drift kink velocity achieved on the attractors in panel 共a兲for ␭=0.46 as a function of the damping. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −3 −2 −1 0 1 2 3 ω <v>T (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0. 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω λ (b) FIG. 11. 共Color online兲共a兲Dependence of the normalized kink velocity 具v典Ton the driving frequency for ␧=0.1, ␣ =0.1, ␭=0.46, and ␬ =1. 共b兲Phase-locked 共full circles兲and chaotic transport 共crosses兲in the 共␭, ␻ 兲plane. Blank spaces stand for pinned-kink states while the green line represents half the frequency of the unperturbed internal mode at the same value of ␭. REGULAR AND CHAOTIC TRANSPORT OF DISCRETE…PHYSICAL REVIEW E 82, 016604 共2010兲 016604-7 subharmonic curve of the unperturbed internal mode frequency, this indicating a coupling between the driving frequency and the internal mode ⍀I⬇2 ␻ . Lastly, we have investigated the dependence of the ratchet phenomenon on the interaction constant ␬ . In our model this parameter measures the discreteness of the system, corresponding ␬ →⬁to the continuum limit. As expected 共see Fig. 12兲high discreteness prevents propagation of the kink, and the depinning amplitude of the driving decreases as ␬ increases. Just above the depinning amplitude chaotic transport predominates while regular phase-locked motion becomes more important as ␬ increases. Quite surprisingly, wide regions in parameter space in which the kink becomes pinned and the net motion suppressed appears for intermediate values of the interaction constant. This feature was not observed in DSR systems driven by biharmonic forces and indicates that the transport may be influenced by the local dynamics even if the driving amplitude is above the depinning threshold for a fixed value of the coupling constant ␬ . This unexpected transport suppression could be due to anti-resonances with the external driving or to chaos in the kink motion inside the local Peirls-Nabarro potential which prevents the escape. The precise mechanism underlying this phenomenon, however, deserves further investigations. IV. CHAOTIC DSR CONTROL In contrast to continuous soliton ratchet systems which, except for the case of strongly localized inhomogeneities 关29兴, are very robust against chaos, discrete soliton ratchets exhibit chaos particularly in the case of asymmetric potential we are dealing with. We have investigated conditions which allow to reduce the chaotic orbits in favor of the phase locked dynamics. To this regard we have added to the sinusoidal driving Eq. 共3兲a weak subharmonic signal ␩ cos共 ␻ t/2兲in phase with it. Stabilization effects induced by external weak subharmonic signals are known to be a general mechanism for suppressing chaos 关30兴, both in point-particle systems 关31兴and in infinite degree of freedom systems described by partial differential equations 关32兴. The small subharmonic driving is used not to create new ratchetlike orbits but to stabilize regular orbits over larger ranges of the system parameters. In our study we have chosen ␧as bifurcation parameter since, in principle, in a possible experimental implementation of the system, ␧would be easier to control and change than ␭for instance. For simplicity we have focused our study on an interval of ␧where a destabilization of phase locking orbits occurs. In Fig. 13共a兲, we have plotted, for ␩ =0, a variable ymod 1 versus ␧, where yis defined as the stroboscopic map of Xcwith period T=2 ␲ / ␻ 共i.e., Xctaken each period Tafter a transient兲. One can observe phase-locked orbits represented by constant values of ymod 1 for a given ␧. Transition to chaos occurs via a cascade of period doubling bifurcations. In this graph, for the sake of continuity of the variable y, we have used as initial condition for each step the solution obtained at the previous one. Another stroboscopic map for the same interval and ␩ =10−4 is shown in panel 共b兲of Fig. 13. We see that in this case the subharmonic signal, although significantly reduces the dispersion in yin the chaotic region, it does not give the full suppression of the chaotic region. This suggests that other harmonic signals may also be involved in the full chaos suppression. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0. 4 κ ε FIG. 12. 共Color online兲Phase-locked 共full circles兲and chaotic transport 共crosses兲in the 共␧, ␬ 兲plane. Blank spaces stand for pinned-kink states. Parameters are: ␧=0.1, ␣ =0.1, ␻ =0.1, and ␭ =0.46. 0.228 0.23 0.232 0.234 0.236 0.238 0.2 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0 . 14 ε y mod 1 (a) 0.228 0.23 0.232 0.234 0.236 0.238 0.24 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 ε y mod 1 (b) FIG. 13. 共Color online兲Bifurcation diagrams for ymod 1 with respect to ␧for ␩ =0 共a兲and ␩ =10−4 共b兲. Variable ymod1is obtained taken the position of the kink center each period of the driving after a transient. Other parameters are: ␭=0.4, ␣ =0.1, and ␻ =0.2. CUEVAS, SÁNCHEZ-REY, AND SALERNO PHYSICAL REVIEW E 82, 016604 共2010兲 016604-8 To find the optimal shape of the small signal for chaos suppression, it is convenient to use a more general driving in the form of a Jacobi elliptic function F共t兲=␧⬘cn 冉 2K ␲␻ t,m 冊 ,共11兲 where ␧⬘is the amplitude of the driving, 0ⱕmⱕ1 is the elliptic modulus and Kis the complete elliptic integral of the first kind K共m兲=兰0 ␲ /2共1−msin2 ␪ 兲−1/2d ␪ 关33兴. This driving allows us to change the periodicity of the signal continuously by means of the modulus mused as a free parameter to optimize the shape. An elliptic driving, apart from being periodic, satisfying the symmetry F共t兲=−F共t+T/2兲and reducing to a sinusoidal driving Eq. 共3兲in the limit m=0, has the advantage of being expressed in terms of sinusoidal functions as F共t兲=2 ␲ m1/2K兺 n=0 ⬁qn+1/2 1+q2n+1cos关共2n+1兲 ␻ t兴,共12兲 with the coefficient of the series depending on the Nome q =e− ␲ K共1−m兲/K共m兲and with only odd multiples of the fundamental frequency entering the sum. Similar drivings were also used to suppress chaotic dynamics of Bose-Einstein condensates loaded into a moving optical lattice 关34兴. The above driving was used to stabilize, for example, the chaotic region of our original system around ␧=0.232 in Fig. 13共a兲. By replacing the sinusoidal subharmonic driving with the elliptic one Eq. 共11兲and starting from m=0, we have increased mlooking for a regularization route to a phaselocking dynamics. In order to properly compare with the sinusoidal case 共m=0兲we have adjusted, while increasing m, the amplitude ␧⬘so as to keep the first Fourier coefficient of the expansion Eq. 共12兲to be ␧, i.e., we performed the regularization by following the path ␧⬘共m兲= 冦 ␧if m=0 ␧m1/2K共1+q兲 2 ␲ q1/2if 0 ⬍m⬍1 冧 .共13兲 The results are depicted in Fig. 14共a兲. We see that as mis increased a progressive regularization occurs. The system undergoes an inverse period doubling sequence of bifurcations until it reaches a periodic orbit around m=0.7. We have chosen this value to compute now a new stroboscopic map with ␧as bifurcation parameter which it is shown in panel 共b兲of Fig. 14. Comparison with Fig. 13共a兲reveals the complete regularization of the system for ␧ⲏ0.2316. An elliptic driving in practical applications could be difficult to implement. We observe, however, that the main contributions in the series expansion Eq. 共12兲come from the first and third harmonics. Thus, for instance, for m=0.7 the coefficients of the third and fifth harmonics are 0.0802␧⬘and 0.006 03␧⬘, respectively, being ␧⬘the amplitude of the first harmonic. In Fig. 15, we have depicted the resulting stroboscopic map when the elliptic function Eq. 共11兲is substituted for the first two terms of its Fourier expansion. Note that the addition of a third harmonic to the fundamental frequency is indeed enough to suppress chaos. Therefore a reasonable option in a possible experimental set-up to control chaos would be to add to the ac driving a third harmonic conveniently adjusted. 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0 . 14 m y mod 1 (a) 0.228 0.23 0.232 0.234 0.236 0.238 0.2 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0 . 14 ε y mod 1 (b) FIG. 14. 共Color online兲共a兲bifurcation diagram for ymod1with respect to parameter mof the elliptic function Eq. 共11兲and ␧ =0.232. The amplitude ␧⬘has been adjusted according to Eq. 共13兲. 共b兲bifurcation diagram as a function of ␧for m=0.7. In both panels parameters are the same as in Fig. 13. 0.228 0.23 0.232 0.234 0.236 0.238 0.2 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0 . 14 ε y mod 1 FIG. 15. 共Color online兲Bifurcation diagram as a function of ␧ obtained for m=0.7 when the elliptic function is replaced with the first two terms of its Fourier expansion. Parameters are the same as in Fig. 13. REGULAR AND CHAOTIC TRANSPORT OF DISCRETE…PHYSICAL REVIEW E 82, 016604 共2010兲 016604-9