Anomalous transport induced by nonhyperbolicity

S. R. Lopes1,∗ J. D. Szezech Jr2, R. F. Pereira3, A. A. Bertolazzo1,4, and R. L. Viana11Departamento de Fısica, Universidade Federal do Parana, Curitiba, PR, Brazil

2Instituto de Fısica, Universidade de Sao Paulo, Sao Paulo, SP, Brazil3Programa de Pos-Graduacao em Ciencias/Fısica, Univ. Est. de Ponta Grossa, PR, Brazil4Instituto de Fısica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil

(Dated: May 10, 2012)

In this paper we study how deterministic features presented by a system can be used to performdirect transport in a quasi-symmetric potential and weak dissipative system. We show that thepresence of nonhyperbolic regions around acceleration areas of the phase space plays an importantrole in the acceleration of particles giving rise to direct transport in the system. Such effect can beobserved for a large interval of the weak asymmetric potential parameter allowing the possibility toobtain useful work from unbiased nonequilibrium fluctuation in real systems even in a presence ofa quasi-symmetric potential.

I. INTRODUCTION

Anomalous transport is an emerging field in physicsand, generally speaking, refers to nonequilibrium pro-cesses that cannot be described by using standard meth-ods of statistical physics. The investigation of anoma-lous transport processes requires a combination of con-cept and methods of diverse disciplines, like stochastictheory, dynamical systems theory and disordered sys-tems [1]. Anomalous transport occurs in a wide realm ofphysical systems ranging from a microscopic level (suchas conducting electrons) to a macroscopic scale (as inglobal atmospheric events). One of the phenomena inthis category is anomalous diffusion, for which the mean-squared-displacement increases with time as a power-lawtµ, where µ 6= 1 [2]. In some cases even analytical resultsfor the anomalous diffusion can be obtained [3]There is a growing interest in anomalous transport

properties of nonlinear systems presenting nonequilib-rium fluctuations, the ratchet systems [4–7]. For thesesystems the second principle of thermodynamics does notprohibit the transport provided we do not have any spaceor time symmetries forbidding it [5, 8]. Ratchet systemsoccur in a variety of physical problems, like unidirectionaltransport in molecular motors [9, 10], micro particles seg-regation in colloidal solutions [11] and transport in quan-tum and nanoscale systems [5, 9, 12]. Recently transportin spatially periodic potential influenced by periodic un-biased external forces was proved to be possible for thecases of large (phase space without islands) and smallexternal force amplitude(phase space with islands) [13].Previous works have studied ratchet systems with a

mixed phase space and weak dissipation. For such sys-tems it has been shown that the ratchet effect resultsfrom a connection between weak dissipation, chaotic dif-fusion, and ballistic transport due to presence of periodicislands [5, 6]. The latter, although occurring in the con-servative case only, still affects the dynamics in the weak

∗ lopes@fisica.ufpr.br

dissipative case since island centers become attractingfixed points in the weak dissipative case [14]. For someclasses of problems it has been shown that in the ab-sent of islands anomalous transport cannot be achievedand the transport currents due to the ratchet effect forweakly dissipative systems are related to the existence ofisoperiodic stable structures [5, 6].

In these previous investigations it has been focusedthe strongly asymmetric case, for which the magnitudeof potential asymmetry is comparable with the originalpotential energy of the system. A somewhat differentmechanism of transport that occurs in either directionand is based on the spontaneously established resonancebetween an ac drive and a periodic potential is also re-ported in the literature [15, 16]. Such phenomena gener-ate transport, without any asymmetry and possible ap-plications can be found in chains of ions trapped by ametallic surface with an external ac electric field play-ing the role of the drive or a chain of Auxons in a longJosephson junction with a regular lattice of narrow inho-mogeneities [17]

In this paper we show that it is not necessary to havestrong asymmetry, in the sense that sizeable ratchet cur-rents can be obtained in weakly dissipative systems withslightly asymmetric potentials. In fact, we claim that thepresence of ratchet currents is influenced not so muchby the potential asymmetry, but rather by the existenceof strongly nonhyperbolic regions in the phase space ofweakly dissipative systems. By a hyperbolic region S wemean a set for which the tangent phase space in eachpoint splits continuously into stable (SM) and an unsta-ble (UM) manifolds which are invariant under the systemdynamics: infinitesimal displacements in the stable (un-stable) direction decay exponentially as time increasesforward (backward) [18]. In addition, it is required thatthe angles between the stable and unstable directions areuniformly bounded away from zero. On the other hand,the nonhyperbolic term will be used here to denote re-gions where we observe (almost) tangencies between sta-ble and unstable manifolds of saddle points (or orbits)embedded in the chaotic region.

2

II. THE MODEL

Chaotic orbits of dissipative two-dimensional map-pings are often nonhyperbolic since the SM and UM un-stable are tangent in infinitely many points. As a repre-sentative illustration of this effect we consider a periodi-cally kicked rotor subjected to a harmonic potential func-tion, whose dynamics is two-dimensional. The dynamicsof a periodically kicked rotor with small dissipation andpotential asymmetry can be described in a cylindricalphase space (−∞×∞)×[0, 2π), whose discrete-time vari-ables pn and xn are respectively the momentum and theangular position of the rotor just after the nth kick, withthe dynamics given by the following dissipative asymmet-ric kicked rotor map (DAKRM) [5]:

pn+1 = (1− γ)pn +K[sin(xn) + a sin(2xn + π/2)],(1)

xn+1 = xn + pn+1, (2)

where K is related to the kick strength, 0 ≤ γ ≤ 1 isa dissipation coefficient, and a is the symmetry-breakingparameter of the system. The conservative (γ = 0) andsymmetric (a = 0) limits yield the well-known Chirikov-Taylor map [19]. In the following we will keep the dissi-pation small enough (namely γ = 2 × 10−4) in order tohighlight the effect of the periodic islands of the conser-vative case. Moreover, the asymmetry parameter a willbe kept small so as to emphasize the role of the nonhy-perbolic phase space regions on the anomalous transport.The conservative and asymmetric (a 6= 0) case has

two fixed points (we call P1) given by pR,L = 0 andxR,L = π − sin−1 ΘR,L(a,K) where ΘR,L(a,K) =(1−

√1 + 8a2 ± 16πa/K

)/4a, (+ and − mean right

and left respectively) which are stable centers in the fol-lowing parameter intervals: 0 < a < 1/4, and 6.40 <K < 7.20. These two P1 points are the centers of two res-onant islands that are actually accelerator modes. Thereare also two (left and right) period-3 fixed points (P3)related to secondary resonances around the P1 islands.In the weak dissipative case, the two P1 points become

stable foci, their basins of attraction present a complexstructure. The chaotic region in the conservative mapbecomes a chaotic transient in the weakly dissipative sit-uation. At K ≈ 6.92 the right P3 points collides withthe right fixed point (pR, xR) by a bifurcation. At thebifurcation point the attraction basin of P1 engulfs theSM of the P3 and turns to be accessible to points in alarge phase space region.

III. THE NONHYPERBOLICITY ROLE IN THETRANSPORT

The vicinity of the fixed points plays a key role inthe anomalous transport mechanism, in the same wayas the islands do for the conservative case. More pre-cisely, the wide accessibility of this vicinity near the bi-furcation is responsible for large ratchet currents, just as

FIG. 1. (color online) 2-D histograms (top) and momentumprobability distributions (bottom) for the DAKRM with γ =0.0002, a = 0.005, and K = 6.40 panels (a,e); K = 6.92panels (b,f); K = 6.96 panels (c,g); K = 7.00 panels (d,h).

the role of the accelerator modes in the non-dissipativemap. Figs. 1(a-d) depict 2-D histograms for 5000 or-bits (each orbit containing 103 points) of the DAKRMfrom initial conditions chosen in the phase plane region0 < x < 2π, −π < p < π, as well as the correspondingmomentum probability distributions σ(p) (Figs. 1(e-h)).For K = 6.40 there is a quasi-symmetric situation, theneighborhood of the two P1 fixed points (left and right)being seldom visited [Fig. 1(a)]. Since the left (right)region is responsible for a positive (negative) increase ofthe transport, we observe that for this parameter valuethe momentum distribution function is nearly symmet-ric, with a Gaussian shape [Fig. 1(e)], resulting in a nulltransport.

Symmetry-breaking effects start to be noticeable af-ter K = 6.40 and reach its maximum at the bifurcationK ≈ 6.92, where only the vicinity of the left P1 is scarcelyvisited by orbits of the map [Fig. 1(b)]. This effect istriggered by the bifurcation whereby the right P3 fixedpoint collides with the right P1 point and turns its vicin-ity easily accessible (not shielded), what is reflected inthe asymmetric left tail in the momentum distributionfunction [Fig. 1(f)]. The vicinity of the left P1 is not yetaffected since the collision process did not occur yet forleft P1 and P3. As a result, a net transport current is gen-erated. However, this is more an effect of the bifurcation(due to nonhyperbolicity) than of the symmetry-breakingitself. In other words, if there is weak symmetry-breakingbut no bifurcation (and no shield process), the ratcheteffect will not occur, at least with the magnitude we ob-served in this example.

Not too far from the bifurcation, (K = 6.96), thevicinities of both P1 fixed points become now almostequally visited [Fig. 1(c)] generating negative and pos-

3

5 6 7 8 9 10

-50

0

50

p_a = 0.5a = 0.005a = 0.0005

K

a = 0.005 (noise)(a)

(b)

FIG. 2. (color online) (a) Ensemble averaged net current fordifferent values of the nonlinearity and asymmetry parametersof the DAKRMwith γ = 0.0002. (b) Net transport current (incolorscale) as a function of the asymmetry and nonlinearityparameters.

itive currents, but no net currents. For this case themomentum distribution function is again approximatelysymmetric [Fig. 1(g)] but presenting right and left tails.The situation changes again after the left P1 and P3 fixedpoints collision [Fig. 1(d)], through the same bifurca-tion mechanism described for their right counterparts.The increase in the accessibility of the vicinity of the leftP1 point leads to an asymmetric momentum distributionfunction [Fig. 1(h)], restoring a net transport current. Inthis last case is noticeable that the seldom visited regionis very small, nevertheless it is enough to inhibit nega-tive currents. Once again the nonhyperbolicity of rightregion seems to be more important than the asymmetricsituation itself.The variation of the average net transport current p

with the nonlinearity parameter K is depicted in Fig.2(a) for different values of the asymmetry parameter.The net current is ensemble-averaged over a large num-ber (106) of initial conditions, each of them being followedby a short time (t = 103) to prevent the system to settledown into any fixed point. On varying K we obtain aseries of positive and negative net transport currents re-sulting from the ratchet effect. Curiously the net currentfluctuates less for both very small and large asymmetry,being more sensitive to K for intermediate values of a.As we have seen, the appearance of net currents is due tothe fact that the left and right P3 fixed points (that bifur-cate in pairs for the symmetric case) start to bifurcate atdifferent values of K. For the interval 6.40 < K < 7.00,corresponding to Fig. 1, the right P3 fixed point bifur-cates before the left one, generating a sequence of nega-tive and positive net currents with very large peak values.For K = 6.92 the maximum amplitude of the net trans-port current for a = 0.005 is at least three times largerthan for the a = 0.5 (high asymmetry case). Neverthelessappreciable net currents can also be acquired for asym-metry parameter as small as a = 0.0005 or even smaller,

although the peak values decrease considerably. Such de-crease in the net transport current for small values of a isexpected since for the case of a = 0 no transport can beobserved due to the symmetry of the standard map. Weemphasize that the transport mechanism is a propertyof the phase space and is not related to any asymptoticstate of the system.

By the way of contrast, with a higher asymmetry valueas those in Ref. [5] such large net currents are observedonly for larger nonlinearities (8.5 < K < 10), hencethey are not primarily related to the bifurcations wepresent. In such case the role of the potential asym-metry overcomes the bifurcation mechanism presentedhere. The sensitive dependence of the net transport cur-rent on the nonlinearity parameter in the weak asym-metry case reminds us of a similar behavior for thediffusion coefficient of the conservative and symmetric(Chirikov-Taylor) map, caused by the existence of accel-erator modes [3].

In order to test the robustness of the problem we havesimulated the effect of noise in the dynamics. A smallamount of white noise distributed in the interval (0, 0.1)was introduced in the x dynamics. The results are dis-played in Fig. 2(a). The effect of the nonhyperbolicityin the transport is still observed.

A more complete comparison is shown in Fig. 2(b). Ifthe potential asymmetry is large (a . 0.5) sizeable netcurrents are obtained only for higher values of K. Thisis due to the fact that large asymmetries shift the smallislands present in the symmetric system phase space tosome other intervals of K. On the other hand, if theasymmetry is very weak (a & 0) we obtain large net cur-rents (both positive and negative) for K-values consid-erably lower. Moreover, the net current amplitude doesnot change appreciably along the blue and red lines inFig. 2(b) when a is varied, emphasizing the role of thenonhyperbolic regions as an essential requirement for theratchet effect to arise.

The existence of nonhyperbolic regions in phase space,however small they may be, constitutes a deterministicmechanism underlying anomalous transport in the pro-duction of net currents through a ratchet effect. In or-der to quantify the degree of nonhyperbolicity relatedto the phenomena we describe in this paper, let usconsider an initial condition (p0, x0) and a unit vec-tor v, whose temporal evolution is given by vn+1 =J (pn, xn)vn/|J (pn, xn)vn|, where J (pn, xn) is the Ja-cobian matrix of the DAKRM. For n large enough, v isparallel to the Lyapunov vector u(p, x) associated to themaximum Lyapunov exponent λu of the map orbit be-ginning with (p0, x0). Similarly a backward iteration ofthe same orbit gives us a new vector vn that is parallelto the direction s(p, x), the Lyapunov vector associatedto the minimum Lyapunov exponent λs [20, 21]. For re-gions where λs < 0 < λu the vectors u(p, x) and s(p, x)are tangent to the UM and SM, respectively, of a point(p, x).

The nonhyperbolic degree of a region S can be stud-

4

FIG. 3. (color online) θ(p, x) values evaluated from 105 initialconditions uniformly distributed around (0.2 radius) SR,L

ε andthe distribution function of θ. Blue triangles are right and leftP3 fixed points (inexistent in (a),(b) and (d) panels).

ied computing the local angles between the two manifoldsθ(p, x) = cos−1(|u ·s|), for (p, x) ∈ S [21]. So, θ(p, x) = 0denotes a tangency between UM and SM at (p, x). LetSR,Lε = (p, x) ∈ Ω : |(p − x) − (pR,L, xR,L)| < ε be a

ε-radius neighborhood of right and left P1 fixed points.Results for θ(p, x) and its distribution function ρ(θ) cal-culated in both regions are shown in Fig. 3 for four valuesof the nonlinear parameter K. The dark regions in Fig.3 correspond to strongly nonhyperbolic region boundingthe acceleration region.For K = 6.40, near the fixed points (pR,L, xR,L) there

is a strong nonhyperbolic region which shields the accel-eration area [Figs. 3(a-b)]. In fact, almost all θ values forthe right area (red curve) and left (blue curve) are con-fined in the interval θ < π/16, configuring a strongly non-hyperbolic region around both fixed points [Fig. 3(c)].By way of contrast, when K = 6.92, only the left fixedpoint neighborhood is shielded, resulting in a large neg-ative transport since only the right acceleration region isregularly visited [Figs. 3(d-e)]. The right P1 and P3 fixed

points suffer a bifurcation and all tangencies of SM andUM disappear from the right area, allowing the trajec-tories to visit the acceleration region. Accordingly Fig.3(f) presents different distributions of θ values for leftand right regions. The red curve (right) presents a dis-tribution peak around θ = π/8 considerably greater thanthe blue one, leading to absence of shielding in the rightregion.

In the case of K = 6.96 neither of the areas surround-ing the fixed points are effectively shielded by the tangen-cies of manifolds, resulting in large positive and negativetransport currents, but no net current at all [Figs. 3(g-i)]. This situation, however, is different from the onedisplayed by Figs. 3(a-c), where the regions surroundingboth fixed points were scarcely visited, hence there is nonet current since the positive and negative currents arevery small. Finally, for a higher value of K, only theright acceleration region is shielded, resulting in a pos-itive transport current [Figs. 3(j-k)]. The distributionρ(θ) presents a peak near zero for the red curve, confirm-ing the existence of a shielded right area [Fig. 3(l)].

The source of nonhyperbolicity in these regions is thetangencies between SM and UM of saddle orbits embed-ded in the chaotic region therein. For perpendicular man-ifold crossings an area near the acceleration region (P1

vicinity) will map another area inside the accelerationregion. However if we have tangencies of manifolds, theouter area (outer P3 vicinity) that maps an inner onetends to zero forbidding the trajectory to visit the ac-celeration area [20]. The scenario can be regarded as acounterpart of the Poincare-Birkhoff’ theorem that de-scribes the torus breakdown of a conservative two-degreeof freedom map.

IV. CONCLUSIONS

In conclusion we have shown that anomalous transportdisplayed by a quasi-symmetric potential and weakly dis-sipative system is strongly related to the topology of theacceleration regions around fixed points displayed by thesystem. The presence of nonhyperbolic regions causedby almost parallel UM and SM can inhibit a chaotictrajectory to visit the neighborhood of the accelerationregion surrounding fixed points of the system. Thismechanism is closely related to the scenario describedby the Poincare-Birkhoff theorem in area-preserving two-dimensional maps. This dynamical phenomenon yieldslarge net transport current in some direction even thoughthe potential has an extremely small degree of symmetry-breaking. Hence such net currents can yield usefulwork from unbiased nonequilibrium fluctuation even withquasi-symmetric potentials, which enlarges the realm ofdynamical systems displaying the ratchet effect.

This work is partially supported by CNPq, CAPES,FAPESP and Fundacao Araucaria.

5

[1] R. Klages, G. Radons, and I. M. Sokolov (Eds.), Anoma-lous Transport (Wiley-VCH, Weinheim, 2008).

[2] R. P. Feynman, R. B. Leighton, and M. Sands,The Feyn-man Lectures on Physics (Addison-Wesley, Reading, MA.1996).

[3] R. Venegeroles, Phys. Rev. Lett. 101, 054102 (2008).[4] J. L. Mateos, Phys. Rev. Lett. 84, 258 (2000).[5] L. Wang et al., Phys. Rev. Lett, 99, 244101 (2007).[6] A. Celestino et al., Phys. Rev. Lett. 106, 234101 (2011).[7] N. A. C. Hutchings el al., Phys. Rev. E, 70, 036205

(2004).[8] S. Flach, O. Yevtushenko, and Y. Zolotaryuk Phys. Rev.

Lett, 84, 2358 (2000).[9] R. D. Astumian and P. Hanggi, Phys. Today, 55 No. 11,

33 (2002).[10] C. Veigel and C. F. Schmiidt, Science 325, 826 (2009).[11] J. Rousselet et al., Nature(London) 370, 446 (1994).

[12] H. Linke et al., Europhys. Lett. 44, 341 (1998).[13] X. Leoncini, A. Neishtadt, and A. Vasiliev, Phys. Rev.

E, 79, 026213; D. Makarov et al, Eur. Phys. J. B, 73,571-579 (2010).

[14] U. Feudel et al., Phys. Rev. E, 54, 71 (1996).)[15] L. L. Bonilla and B. A. Malomed, Phys. Rev. B, 43,

11539 (1991).[16] B. A. Malomed, Phys. Rev. A, 45, 4097 (1992).[17] A. V. Ustinov1 and B. A. Malomed Phys. Rev. B, 64,

020302(R) (2001).[18] J. Guckenheimer and P. Holmes, Nonlinear Oscillations,

Dynamical Systems, and Bifurcations of Vector Fields(Springer, New York, 2002).

[19] A. J. Lichtenberg, and M. A. Lieberman, M.A. Regularand Chaotic Dynamics (Springer, Berlin. 1992).

[20] C. Grebogi et al., Phys. Rev. Lett., 65, 1527 (1990).[21] F. Ginelli et al., Phys. Rev. Lett., 99, 130601 (2007).