Maximal Stochastic Transport in the Lorenz Equations
MMaximal stochastic transport in the Lorenz equations
Sahil Agarwal
1, 2 and J. S. Wettlaufer
1, 2, 3 Program in Applied Mathematics, Yale University, New Haven, USA Mathematical Institute, University of Oxford, Oxford, UK Nordita, Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden ∗ (Dated: October 18, 2018)We calculate the stochastic upper bounds for the Lorenz equations using an extension of thebackground method. In analogy with Rayleigh-B´enard convection the upper bounds are for heattransport versus Rayleigh number. As might be expected, the stochastic upper bounds are largerthan the deterministic counterpart of Souza and Doering [1], but their variation with noise amplitudeexhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increasemonotonically with noise amplitude. However, in the chaotic regime this monotonicity dependson the number of realizations in the ensemble; at a particular Rayleigh number the bound mayincrease or decrease with noise amplitude. The origin of this behavior is the coupling between thenoise and unstable periodic orbits, the degree of which depends on the degree to which the ensemblerepresents the ergodic set. This is confirmed by examining the close returns plots of the full solutionsto the stochastic equations and the numerical convergence of the noise correlations. The numericalconvergence of both the ensemble and time averages of the noise correlations is sufficiently slow thatit is the limiting aspect of the realization of these bounds. Finally, we note that the full solutionsof the stochastic equations demonstrate that the effect of noise is equivalent to the effect of chaos. I. INTRODUCTION
Noise is an integral part of any physical system. Itcan be ascribed to fluctuations arising from intermittentforcing, observational uncertainties, interference from ex-ternal sources or unresolved physics. In circumstanceswhere noise acts to destroy a signal of interest, it isviewed as a nuisance. However, it can also be the casethat fluctuations act to stabilize a system, examples ofwhich include noise-induced optical multi-stability [2],asymmetric double well potentials [3], plant ecosystems[4], population dynamics [5], and in electron-electron in-teractions in quantum systems [6]. Curiously, it has re-cently been shown that noise can have positive effects oncognitive functions such as learning and memory [7]. Fi-nally, a key issue arising when examining observationaldata is whether fluctuations are intrinsic or due to exter-nal forcing, which can be confounded by temporal mul-tifractality [e.g., 8].Given the breadth of settings in which the effects ofnoise manifest themselves on dynamical systems, it ap-pears prudent to examine such matters in a well stud-ied and yet broadly relevant system. Thus, we studythe influence of noise in the Lorenz system [9], which isan archetype of deterministic nonlinear dynamics. More-over, Souza and Doering [1] have recently determined themaximal (upper bounds) transport in the Lorenz equa-tions, thereby providing us with a rigorous test bed forstochastic extensions. In § II we describe the stochasticLorenz model, followed by the derivation of the stochas-tic upper bounds in § III. We interpret the core resultsand their implications in § IV before concluding. ∗ [email protected] II. STOCHASTIC LORENZ MODEL
The Lorenz model is a Galerkin-modal truncationof the equations for Rayleigh-B´enard convection withstress-free boundary conditions on the upper and lowerboundaries. It acts as a rich toy model of low-dimensionalchaos and since it’s origin extensive studies have beenmade spanning a wide range of areas [e.g., 10]. Of par-ticular relevance here, is using the system as a model forheat transport in high Rayleigh number turbulent con-vection [1].The stochastic form of the Lorenz system is describedby the following coupled nonlinear ordinary differentialequations, ddt X = σ ( Y − X ) + A ξ ,ddt Y = X ( ρ − Z ) − Y + A ξ , (1) ddt Z = XY − βZ + A ξ where X describes the intensity of convective motion, Y the temperature difference between ascending and de-scending flow and Z the deviation from linearity of thevertical temperature profile. The control parameters are σ the Prandtl Number , ρ the Rayleigh Number and β adomain geometric factor. The A i are the noise ampli-tudes and ξ i are the noise processes. Clearly, the deter-ministic system has A i = 0.This type of additive noise may appear, for exam-ple, in observational errors, when the errors do not de-pend on the system state or as a model of sub-grid scaleprocesses approximated by noise associated with unex-plained physics [11]. In multiplicative noise the systemhas an explicitly state dependent noise process. a r X i v : . [ n li n . C D ] A ug Although real noise will always have a finite time cor-relation, taking the limit that the noise correlation goesto zero as ∆ t →
0, serves as a good approximation forthe noise forcing. This is the white noise limit of colorednoise forcing. White noise forcing ξ ( t ) is defined by anautocorrelation function written as (cid:104) ξ ( t ) ξ ( s ) (cid:105) = 2 Dδ ( t − s ) , (2)where, t − s is the time lag, D is the amplitude of thenoise, (cid:104)•(cid:105) represents the time average and δ ( r ) is theDirac delta-function. III. STOCHASTIC MAXIMAL TRANSPORT
Initiated by the work of Louis Howard [12], maximiz-ing the transport of a quantity such as heat or mass is acore organizing principle in modern studies of dissipativesystems. In this spirit Souza and Doering [1] studied thetransport in the deterministic Lorenz equations and de-termined the upper bound, which depends on the exactsteady solutions X s , Y s , as lim T →∞ (cid:104) XY (cid:105) T = X s Y s = β ( ρ − X s = Y s = ± (cid:112) β ( ρ −
1) for ρ ≥ ρ is varied.Let X = x, Y = ρy, Z = ρz and A = A = A = A inthe system of equations 1, which transform to ddt x = σ ( ρy − x ) + Aξ ,ddt y = x (1 − z ) − y + Aρ ξ , (3) ddt z = xy − βz + Aρ ξ . In the next two sub-sections, we calculate the stochas-tic upper bound of equations 3 using both
Itˆo and
Stratonovich calculi.
A. Itˆo Calculus Framework
Now, knowing that the state variables ( x, y, z ) in theLorenz system are bounded [1, 13], and following the ap-proach of Souza and Doering [1] for this stochastic sys-tem, the long time averages of x , ( y + z ) and − z can be written as0 = −(cid:104) x (cid:105) T + ρ (cid:104) xy (cid:105) T + A σ + Aσ (cid:104) xξ (cid:105) T + O ( T − ) , (4)0 = − (cid:104) y (cid:105) T + (cid:104) xy (cid:105) T − β (cid:104) z (cid:105) T + A ρ + Aρ (cid:104) yξ (cid:105) T + Aρ (cid:104) zξ (cid:105) T + O ( T − ) , (5) 0 = −(cid:104) xy (cid:105) T + β (cid:104) z (cid:105) T + O ( T − ) , (6)where, the terms A σ in Eq. 4 and A ρ in Eq. 5 are aconsequence of Itˆo’s lemma .Now, let z = z + λ ( t ), where z = r − r is time-independent [1], and equations 5 and 6 now become,0 = − (cid:104) y (cid:105) T + (cid:104) xy (cid:105) T − βz − βz (cid:104) λ (cid:105) t − β (cid:104) λ (cid:105) T + A ρ + Aρ (cid:104) yξ (cid:105) T + Aρ (cid:104) λξ (cid:105) T + O ( T − ) , and (7)0 = −(cid:104) xy (cid:105) T + βz + β (cid:104) λ (cid:105) T + O ( T − ) . (8)Therefore, equation (7) +2 z × (8) becomes,0 = − (cid:104) y (cid:105) T + (1 − z ) (cid:104) xy (cid:105) T + βz − β (cid:104) λ (cid:105) T + A ρ + Aρ (cid:104) yξ (cid:105) T + Aρ (cid:104) λξ (cid:105) T + O ( T − ) . (9)Now adding ρ × (4) to ρ × (9) gives0 = − ρ (cid:104) y (cid:105) T + ρ (1 − z ) (cid:104) xy (cid:105) T + ρβz − ρβ (cid:104) λ (cid:105) T − ρ (cid:104) x (cid:105) T + (cid:104) xy (cid:105) T + Aρσ (cid:104) xξ (cid:105) T + A ρ + A ρσ + A (cid:104) yξ (cid:105) T + A (cid:104) λξ (cid:105) T + O ( T − ) , (10)and adding ( ρ − (cid:104) xy (cid:105) T to both sides gives,( ρ − (cid:104) xy (cid:105) T = ρβz o + A (cid:20) (cid:104) yξ (cid:105) T + (cid:104) λξ (cid:105) T + 1 σρ (cid:104) xξ (cid:105) T (cid:21) − (cid:42)(cid:18) x √ ρ − √ ρy (cid:19) + ρβλ (cid:43) T + A (cid:20) ρ + 12 ρσ (cid:21) + O ( T − ) . (11)We thus arrive at( ρ − (cid:104) xy (cid:105) T ≤ ρβz o + A (cid:20) ρ + 12 ρσ (cid:21) + A (cid:20) (cid:104) yξ (cid:105) T + (cid:104) λξ (cid:105) T + 1 σρ (cid:104) xξ (cid:105) T (cid:21) + O ( T − ) . (12)Comparing equation 12 above with equation 19 fromSouza and Doering [1], we see an additional term dueto the stochastic forcinglim T →∞ (cid:104) XY (cid:105) T = lim T →∞ ρ (cid:104) xy (cid:105) T ≤ β ( ρ −
1) + A ρ − (cid:20) σ (cid:21) + ρAρ − (cid:20) ρ (cid:104) Y ξ (cid:105) T + (cid:104) λξ (cid:105) T + 1 σρ (cid:104) Xξ (cid:105) T (cid:21) , (13)which shows that the stochastic upper bound transcendsthe deterministic upper bound. B. Stratonovich Calculus Framework
In this framework, the equations analogous to 4, 5 and6 are0 = −(cid:104) x (cid:105) T + ρ (cid:104) xy (cid:105) T + Aσ (cid:104) xξ (cid:105) T + O ( T − ) , (14)0 = − (cid:104) y (cid:105) T + (cid:104) xy (cid:105) T − β (cid:104) z (cid:105) T + Aρ (cid:104) yξ (cid:105) T + Aρ (cid:104) zξ (cid:105) T + O ( T − ) , (15)0 = −(cid:104) xy (cid:105) T + β (cid:104) z (cid:105) T + O ( T − ) , (16)Again letting z = z + λ ( t ), where z = r − r , equations15 and 16 now become,0 = − (cid:104) y (cid:105) T + (cid:104) xy (cid:105) T − βz − βz (cid:104) λ (cid:105) t − β (cid:104) λ (cid:105) T + Aρ (cid:104) yξ (cid:105) T + Aρ (cid:104) λξ (cid:105) T + O ( T − ) , (17)0 = −(cid:104) xy (cid:105) T + βz + β (cid:104) λ (cid:105) T + O ( T − ) , (18)and hence 17 +2 z ×
18 becomes,0 = − (cid:104) y (cid:105) T + (1 − z ) (cid:104) xy (cid:105) T + βz − β (cid:104) λ (cid:105) T + Aρ (cid:104) yξ (cid:105) T + Aρ (cid:104) λξ (cid:105) T + O ( T − ) . (19)Now adding ρ × (14) to ρ × (19) we find0 = − ρ (cid:104) y (cid:105) T + ρ (1 − z ) (cid:104) xy (cid:105) T + ρβz − ρβ (cid:104) λ (cid:105) T − ρ (cid:104) x (cid:105) T + (cid:104) xy (cid:105) T + Aρσ (cid:104) xξ (cid:105) T + A (cid:104) yξ (cid:105) T + A (cid:104) λξ (cid:105) T + O ( T − ) . (20)Finally, adding ( ρ − (cid:104) xy (cid:105) T to both sides gives( ρ − (cid:104) xy (cid:105) T = ρβz o + A (cid:20) (cid:104) yξ (cid:105) T + (cid:104) λξ (cid:105) T + 1 σρ (cid:104) xξ (cid:105) T (cid:21) − (cid:42)(cid:18) x √ ρ − √ ρy (cid:19) + ρβλ (cid:43) T + O ( T − ) . (21)We thus arrive at( ρ − (cid:104) xy (cid:105) T ≤ ρβz o + A (cid:20) (cid:104) yξ (cid:105) T + (cid:104) λξ (cid:105) T + 1 σρ (cid:104) xξ (cid:105) T (cid:21) + O ( T − ) (22)Now, comparing Eq. 22 above with Eq. 19 from Souzaand Doering [1], we see an additional term due to thestochastic forcing, which, as expected from the previoussection, increases the upper bound;lim T →∞ (cid:104) XY (cid:105) T = lim T →∞ ρ (cid:104) xy (cid:105) T ≤ β ( ρ − ρAρ − (cid:20) ρ (cid:104) Y ξ (cid:105) T + (cid:104) λξ (cid:105) T + 1 σρ (cid:104) Xξ (cid:105) T (cid:21) (23) Due to the fact that the noise is additive, the upper-bounds from Itˆo and
Stratonovich calculi should beequivalent. This is indeed the case because for the
Itˆo result, (cid:104) Xξ (cid:105) = (cid:104) Y ξ (cid:105) = (cid:104) λξ (cid:105) = 0, whereas for the Stratonovich result (cid:104) Xξ (cid:105) = (cid:104) Y ξ (cid:105) = A and (cid:104) λξ (cid:105) = A ρ .Therefore, when we take the ensemble average of equa-tions 23 and 13 we obtain (cid:104) lim T →∞ (cid:104) XY (cid:105) T (cid:105) ≤ β ( ρ −
1) + A ρ − (cid:20) σ (cid:21) (24)We plot equation 24 in Fig. 2, wherein the lines show theanalytic solution and the solid circles denote the numeri-cal solution, taking the ensemble average of equation 23,all as a function of noise amplitude A . ρ
10 20 30 40 50 60 70 80 90 100 h XY i T A = 1A = 2A = 3A = 4A = 5A = 6A = 7A = 8A = 9A = 10DeterministicUpper Bound ρ h XY i T FIG. 1. lim T →∞ (cid:104) XY (cid:105) T , the transport from a single real-ization of the stochastic Lorenz attractor (equation 1), as afunction of ρ and noise amplitude A , with the solid black lineshowing the deterministic upper bound [1]. The inset showsthe increased transport for ρ near the transition to chaos; ρ c = 24 .
74, beyond which the solutions cross below the deter-ministic upper bound.
IV. RESULTS & INTERPRETATION
It is often the case that for the nonlinear dynamicalsystems found in nature, we only have a single time se-ries. Thus, it is a natural question to ask about theproperties of the stochastic upper bound both for the en-semble average and for a small number of realizations.Whereas deterministic chaos acts to decrease the trans-port in the system [1], here we find that it can also beindistinguishable from noise. In Fig. 1 we show individ-ual realizations (one for each of 10 amplitudes A ) of thetransport as a function of ρ . These exhibit two importantfeatures. (1) For ρ below the deterministic transition tochaos ( ρ c = 24 . ρ above ρ c theycross below it. (2) Independent of ρ , there is no system-atic dependence of the solutions on the noise amplitude. ρ
10 20 30 40 50 60 70 80 90 100 h h XY i T i A = 2A = 4A = 6A = 8A = 10A = 0A = 2A = 4A = 6A = 8A = 10
FIG. 2. (cid:104) lim T →∞ (cid:104) XY (cid:105) T (cid:105) as a function of ρ and noise ampli-tude A , with black line showing the upper bound in the deter-ministic case [1], colored lines showing the analytical solutionfrom equation 24 and solid circles showing the numerical so-lution as the ensemble average in equation 23. Taken together these features show that the impact ofnoise differs substantially depending on whether the de-terministic dynamics is chaotic or non-chaotic. Clearly,the role of noise is indistinguishable from the role ofchaotic dynamics and in individual realizations a givennoise amplitude couples with various unstable periodicorbits, which we discuss in more detail below.The analytical solution from equation 24 and the nu-merical solution (taking the ensemble average in equa-tion 23) of the stochastic upper bound (SUB) are shownin Fig. 2. Firstly, we see the increase in the SUB as thenoise amplitude A increases. Secondly, for fixed ampli-tude and values of ρ < ρ c , the origin of the increase inthe SUB are the two terms proportional to 1 /ρ . As ρ in-creases and A decreases the SUB converges to the DUBfrom above. The increase with A at low ρ is a reflectionof the dependence of the “diameter” of the system in the X − Y plane, other parameters being held constant. Be-cause the diameter decreases as ρ decreases, the relativeinfluence of A on the SUB is larger. We show this for ρ = 2 in Fig. 3. However, we note that, using a differ-ent method Fantuzzi and Goluskin (pers. comm.) find aSUB that does not exhibit the low ρ divergence, asymp-totes to our bound for ρ in the region of typical interest,and also recovers the DUB in the appropriate limit.A more detailed view of the SUB from Eq. 23 is shownin Fig 4. The lower right inset shows that for ρ < ρ c theSUB is a monotonic function of the noise amplitude, asone would intuitively expect from Fig. 3. However, asthe system enters the chaotic regime, this monotonicityis lost to reveal an oscillation with amplitude, as shownin the upper inset of Fig. 4 for noise amplitudes A = 9and A = 10. This oscillatory behavior is due to the cou-pling of noise with chaotic orbits that, depending on theamplitude, can result in different residence times of a tra-jectory in different orbits. Indeed, although for ρ < ρ c , X -10 -5 0 5 10 15 20 Y -20-15-10-505101520 A = 4A = 2A = 0
FIG. 3. XY-space of the stochastic Lorenz attractor (equation1), for three different noise amplitudes. The diameter of theattractor increases with noise amplitude. the realization to realization stochastic upper bounds areconsistent, this is not the case in the chaotic regime dueto the coupling between the noise and the chaotic orbits.In consequence, each realization results in a slightly dif-ferent SUB and hence the bound is not strict; it has adiffuseness that depends on the noise amplitude. Thiscombined effect of noise and the exponential divergenceproperty of chaos allows the noise to perturb the stochas-tic system into a different orbit in each realization. ρ
10 20 30 40 50 60 70 80 90 100 h XY i T ρ
94 96 98 h XY i T A = 9A = 10Deterministic ρ h XY i T FIG. 4. The stochastic upper-bound (circles) from Eq. 23 asa function of ρ and noise amplitude A . The solid black lineis the DUB [1]. The bottom inset shows that the SUB is amonontoic function of A in the non-chaotic regime ( ρ < ρ c ).The top inset shows the oscillations of the SUB between noiseamplitudes A = 9 ,
10 in the chaotic regime ( ρ < ρ c ). The close returns plot of Mindlin and Gilmore [14] canbe used to extract unstable periodic orbits (UPO) froma chaotic time series. Thus, to demonstrate the couplingbetween noise and chaos discussed above in a differentmanner, we show the close returns plot for the stochasticLorenz attractor (equation 1) for A = 9 ,
10 and ρ = 97in Fig. 5 ( a, c ). These plots help us distinguish betweennoise and chaos. Whereas in a noisy system the pointsare more diffuse, in a chaotic system they are more struc-tured, with continuous straight lines defining the UPOs. FIG. 5. Scaled close returns plots (a,c) and the correspondinghistograms of the 2 − norm of these (b,d) for the stochasticLorenz attractor (equation 1) with ρ = 97, where the hori-zontal segments represent the UPOs in the system when theyclose up when embedded in the phase space of the attractor.The scaled index in the time series is i , and α is the scaledperiod of the UPO. ( a, b ) A = 9. ( c, d ) A = 10. To further quantify this structure, in Fig. 5 ( b, d ) weplot the histogram of the 2 − norm of the points from theclose returns plot. The peaks in the histogram show theUPOs in the system, corresponding to the lines in theclose returns plot. Whereas the histogram in Fig. 5( b )has a broad-band structure, that in Fig. 5( d ) revealsprominent peaks at different norms. V. SUMMARY
We calculated the stochastic upper bounds of the heatfor the Lorenz equations using an extension of the back- ground method of Souza and Doering [1] used in the de-terministic system. Whilst one might have expected thatthe stochastic upper bounds transcend their determinis-tic counterpart of [1], their variation with noise amplitudeexhibits rich behavior. In the non-chaotic regime theupper bounds increase monotonically with noise ampli-tude. However, in the chaotic regime this monotonicitydepends on the number of realizations in the ensemble;at a particular Rayleigh number the bound may increaseor decrease with noise amplitude. The origin of this be-havior is the coupling between the noise and unstable pe-riodic orbits, the degree of which depends on the degreeto which the ensemble represents the ergodic set. Thisis confirmed by examining the close returns plots of thefull solutions to the stochastic equations. These solutionsalso demonstrate that the effect of noise is equivalent tothe effect of chaos for a wide range of noise amplitude.Finally, we note that although in
Itˆo -calculus the ana-lytic bound (equation 24) relies on vanishing noise cor-relations ( (cid:104) Xξ (cid:105) = (cid:104) Y ξ (cid:105) = (cid:104) λξ (cid:105) = 0), numerically suchcorrelations never completely vanish [e.g., 15][16], for asthe size and diffuseness of stochastic attractor continuallyincreases, the extent to which ensemble average reachesthe ergodic set remains a concept rather than a practicalreality. ACKNOWLEDGMENTS
SA and JSW acknowledge NASA GrantNNH13ZDA001N-CRYO for support. JSW acknowl-edges the Swedish Research Council and a Royal SocietyWolfson Research Merit Award for support. This workwas completed whilst the authors were at the 2015Geophysical Fluid Dynamics Summer Study Programat the Woods Hole Oceanographic Institution, which issupported by the National Science Foundation and theOffice of Naval Research. We thank C.R. Doering, G.Fantuzzi, D. Goluskin and A. Souza for feedback. [1] A. N. Souza and C. R. Doering, Phys. Lett. A , 518(2015).[2] A. Barb´eroshie, I. Gontsya, Y. N. Nika, and A. K. Ro-taru, Zh. Eksp. Teor. Fiz. , 2655 (1993).[3] R. Mankin, E. Soika, and A. Sauga, WSEAS Trans. Syst. , 239 (2008).[4] P. D’Odorico, F. Laio, and L. Ridolfi, Proc. Natl. Acad.Sci. USA , 10819 (2005).[5] M. Parker, A. Kamenev, and B. Meerson, Phys. Rev.Lett. , 180603 (2011).[6] R. Landauer, Nature , 658 (1998). [7] V. H. Rausch, E. M. Bauch, and N. Bunzeck, J. Cogn.Neuroscience , 1469 (2013).[8] S. Agarwal, W. Moon, and J. S. Wettlaufer, Proc. Roy.Soc. Lond. A , 2416 (2012).[9] E. N. Lorenz, J. Atmos. Sci. , 130 (1963).[10] S. H. Strogatz, Nonlinear Dynamics and Chaos: WithApplications to Physics, Biology, Chemistry, and Engi-neering , 2nd ed. (Westview Press, 2014).[11] H. M. Arnold, I. M. Moroz, and T. N. Palmer, Phil.Trans. R. Soc. A (May 28, 2013).[12] L. N. Howard, Annu. Rev. Fl. Mech. , 473 (1972). [13] C. R. Doering and J. D. Gibbon, Dyn. Stab. Syst. ,255 (1995).[14] G. M. Mindlin and R. Gilmore, Physica D , 229(1992).[15] H. M. Ito, J. Stat. Phys. , 151 (1984).[16] To test time convergence in the Itˆo Case we take the timeaverage of these noise correlations for turnover times of 50and 1550, and find that the magnitude of the correlations are of the same order; the correlations change from ∼ ∼ ∼ ∼∼