How to test for partially predictable chaos
HHow to test for partially predictable chaos
Hendrik Wernecke , Bulcs ´u S ´andor , and Claudius Gros Institute for Theoretical Physics, Goethe University Frankfurt, Germany * [email protected] ABSTRACT
For a chaotic system pairs of initially close-by trajectories become eventually fully uncorrelated on the attracting set. Thisprocess of decorrelation may split into an initial exponential decrease, characterized by the maximal Lyapunov exponent, and asubsequent diffusive process on the chaotic attractor causing the final loss of predictability. The time scales of both processescan be either of the same or of very different orders of magnitude. In the latter case the two trajectories linger within a finitebut small distance (with respect to the overall extent of the attractor) for exceedingly long times and therefore remain partiallypredictable.Tests for distinguishing chaos from laminar flow widely use the time evolution of inter-orbital correlations as an indicator.Standard tests however yield mostly ambiguous results when it comes to distinguish partially predictable chaos and laminarflow, which are characterized respectively by attractors of fractally broadened braids and limit cycles. For a resolution weintroduce a novel 0-1 indicator for chaos based on the cross-distance scaling of pairs of initially close trajectories, showing thatthis test robustly discriminates chaos, including partially predictable chaos, from laminar flow. One can use furthermore thefinite time cross-correlation of pairs of initially close trajectories to distinguish, for a complete classification, also between strongand partially predictable chaos. We are thus able to identify laminar flow as well as strong and partially predictable chaos in a0-1 manner solely from the properties of pairs of trajectories.
Introduction
One characteristic aspect of deterministic chaos is the exponential sensitivity of the dynamics to initial conditions . Thissensitivity leads to an effective breakdown of predictability as the result of the eventual decorrelation of any given pair oftrajectories. The decorrelation occurring on a chaotic attracting set is measured commonly by the maximal Lyapunov exponent ,with a positive value measuring the effective rate of the decorrelation of initially arbitrary close pairs of trajectories. Otherstandard tests for chaos, such as the correlation dimension or the spectral analysis of the auto-correlation function , also relyon correlation measures.Inter-orbital correlations fully decay on chaotic attractors in the limit of long times. This fact does however not preclude theexistence of other types of predictable correlations. E. g. it is well known that the sequence x n produced by the logistic map x n + = rx n ( − x n ) will never decrease twice in a row for 2 < r <
4. In this case we can hence predict with 100% confidencethat x n + will be larger than x n , if x n was smaller than x n − , even if the system is chaotic. It has also been noted that correlationsmay persist for specific chaotic systems for extended (but finite) times, especially in systems characterized by multiple timescales or strong periodic drivings . The respective cross-correlation C of initially close pairs of trajectories can hence beused as a measure for predictability.In the following we will introduce, distinguish and discuss two types of chaotic behavior, denoted partially predictablechaos (PPC) and strong chaos respectively, which differ with respect to what happens for time scales larger than the Lyapunovprediction time T λ . • Strong chaos : Predictability vanishes, approaching zero on a time scale of T λ . • PPC : The first decorrelation occurring on a time scale of T λ does not destroy, in this case, all pair-wise correlations. Thecross-correlation C will retain a finite value even for t (cid:29) T λ , vanishing only for exceedingly long times.Partial predictability may occur whenever the attracting set is characterized by a non-trivial topology. This is genericallythe case for the chaotic state close to a period-doubling transition, when the trajectories wander chaotically around previouslystable limit cycles within closed braids . Our observation of partial predictability may hence be especially of relevance fornatural systems having a tendency to self-organize close to criticality , as it implies that the system will hover continuouslyclose to the transition between laminar and chaotic flow . Another example of PPC is the case of phase locked chaos observedin driven Josephson junctions . a r X i v : . [ n li n . C D ] F e b igure 1. Strong chaos (red, for ρ < ρ p ≈ . ρ p < ρ < ρ C ≈ .
96) andregular flow (blue, for ρ C < ρ ), for the Lorenz system (1). Shown are (from top to bottom), z n from the Poincar´e section x = z n ), the maximal Lyapunov exponent λ m , the cross-distance scaling exponent ν , see Eq. (2), and thecross-correlation C ( t = ) , as defined by Eq. (3). PD1 and PD2 denote examples of period doubling bifurcations and SSBa bifurcation spontaneously breaking the symmetry ( x , y , z ) ↔ ( − x , − y , z ) of (1).It is generically a challenge to distinguish between partially predictable chaos and laminar flow on the basis of the maximalLyapunov exponent, which is positive but small for PPC and hence difficult to evaluate numerically. We therefore introducehere a novel test for chaos based on the cross-distance scaling of pairs of trajectories. It discriminates chaos, including PPC andstrong chaos, unambiguously in a 0 − , for which we find all three types of dynamical regimes: strongchaos, PPC and laminar flow.Thereafter we combine the indicator for chaos with another correlation indicator acting as an effective 0 − Results
We start by considering with x = ( x , y , z ) ,˙ x = σ ( y − x ) , ˙ y = x ( ρ − z ) − y , ˙ z = xy − β z (1)the Lorenz system , which has long been used for studying the interplay between predictability and chaos . We select with β = / σ =
10 standard parameter settings, retaining ρ as the bifurcation parameter.As an overview we present in Fig. 1 the phases of the Lorenz system for a typical parameter window ρ ∈ [ . , . ] . Atransition between two types of chaotic regions is observed to occur at ρ p ≈ .
72, together with a transition via a cascade ofperiod doubling (halving) bifurcations from chaos to laminar flow at ρ C ≈ .
96. The dynamics of the intermediate region −
25 0 25 50 x z ( a ) ρ = ρ = 180 . − −
25 0 25 50 x ( b ) ρ = ρ = 180 . − −
25 0 25 50 x ( c ) ρ = ρ ∗ = 180 . − −
25 0 25 50 x ( d ) ρ = ρ = 181 . . . Figure 2.
Sample trajectories of the Lorenz system (1) projected to the x − z plane. The phases, from left to right, ρ = ρ (ergodic motion), ρ = ρ and ρ = ρ ∗ (partially predictable chaos) and ρ = ρ (limit cycle), are indicated in Fig. 1. The arrowsin the respective insets indicate the width of the (fractal) braids in the partially predictable phase. The respective standarddeviations s of the attractors are s ∈ [ . , . ] for ρ ∈ [ ρ , ρ ] .( ρ ∈ [ ρ p , ρ C ] , green) between strong chaos ( ρ < ρ p , red) and laminar flow ( ρ > ρ C , blue), is governed by PPC. A spontaneoussymmetry-breaking bifurcation (SSB) is additionally shown. Chaos-chaos transitions involving phase space explosions, likethe one occurring at ρ p between partially predictable and strong chaos (which is in part intermittent ), have been studiedpreviously in the context of quadratic maps and driven Josephson junctions . They are due to the collision of an unstablemanifold with the attracting chaotic set, an interior crisis which is a typical example of a global bifurcation .In Fig. 2 we present the projections to the x − z plane of the respective attracting sets for ( a ) strong chaos, ( b ) and ( c )PPC, and ( d ) laminar flow. Partially predictable chaos is at times difficult to distinguish visually from laminar flow (compareFigs. 2 ( c ) and ( d )), we hence provide the respective blow-ups in the insets. The partially predictable chaotic attractors can bethought as fractally broadened limit cycles, viz as braids.The maximal Lyapunov exponent λ m presented in Fig. 1 ( b ) has been evaluated by extracting the initial slope of thelogarithmic distance (cid:104) ln ( | x ( t ) − x ( t ) | ) (cid:105) of two trajectories, as averaged over 10 pairs with initial distances of δ = − ,when plotted as a function of time. | . . . | denotes here the Euclidean distance and (cid:104) . . . (cid:105) the average over initial conditions on theattractor sampled with the natural distribution (the natural invariant measure ). We used in addition 10 pairs of trajectoriesfor the cross-correlation C ( t = ) , see Eq. (3). The choice of t =
200 has been made in order to ensure that we neither haveto deal with initial effects nor with numerical inaccuracies, the latter due to the chaotic nature of the flow.We have also evaluated a scaling exponent ν (discussed further below, see Eq. (2)), which characterizes the scaling of thelong-term distance between two trajectories. Our choice to favor the average logarithmic distance for computing the maximalLyapunov exponent over more sophisticated methods is motivated by a conceptual computational aspect: in this way all threeindicators presented here, i. e. the maximal Lyapunov exponent λ m , the cross-correlation C and the cross-distance scalingexponent ν , can be evaluated from the time evolution of initially close-by trajectories. In the Methods section we compare thisapproach for computing the maximal Lyapunov exponent to the results obtained by Benettin’s method .Two fundamental time scales determine the initial dynamics. The first is the quasi-period τ , which is the average time atrajectory needs to come back to the same intersection of the braid with the Poincar´e plane. It is comparable to the period ofthe limit cycle and we find τ (cid:39) . ρ C < ρ < ρ p . The second time scale is theLyapunov prediction time T λ = ln ( | x − x | / δ ) / λ m , which is the time it takes for two exponentially diverging trajectories Table 1.
The average maximal Lyapunov exponent λ m together with the second- and third largest Lyapunov exponent, λ and λ (compare Fig. 7). The respective Lyapunov prediction times T λ have been evaluated for the two sets of initial distances δ used in Fig. 3. The values of ρ used are indicated in Fig. 1. λ m λ λ T λ ( δ ) δ = − δ = − ρ = .
70 1.18 0.00 -14.85 3 . . ρ = .
78 0.47 0.00 -14.14 9 . . ρ ∗ = .
95 0.14 0.00 -13.81 32 . . ρ = .
10 0.00 -0.60 -13.07 - -
10 20 30 40 50 t − − − − d ( t ) ρ ρ ρ T λ ( δ ) ( a ) − − − − − − δ − − − − d ( t ≫ T λ ) ρ ρ ρ ∗ ρ ( b ) . δ . . δ . . δ . . δ . Figure 3.
Distance scaling of initially close-by trajectories. ( a ) The distance d ( t ) of initially close-by pairs of trajectories,averaged over 10 initial conditions, with initial distances d ( ) = δ = − and δ = − (black bullets). For the regularflows ( ρ = ρ = δ . The strongly chaotic attractor ( ρ = ρ = .
7, redlines) approaches the maximal distance independently of the initial distance within the respective Lyapunov prediction time T λ .In the case of PPC ( ρ = ρ = .
78, green lines) the distances reach a quasi stationary plateau that is independent of theinitial distance. For comparison we marked the Lyapunov prediction times T λ at the respective curves by arrows. ( b ) Thescaling behavior, see Eq. (2), of the averaged long-term distance d ( t = ) . The results (circles) are for the strongly chaoticphase ( ρ = ρ , top), for the partially predictable chaos ( ρ = ρ and ρ = ρ ∗ , middle) and for a limit cycle ( ρ = ρ = . δ to reach a given finite distance | x − x | . For these two distances we used δ = − and | x − x | ∼ .
001 respectively. At the latter distance a finite amount of predictability is lost, viz the cross-correlation C , asdefined by Eq. (3), starts to deviate from unity. Given the values of the maximal Lyapunov exponent λ m presented in Fig. 1we obtain T λ ≈
10 and T λ ≈
25 for ρ = ρ = .
70 and ρ = ρ = .
78 respectively (cf. also Table 1). The initial loss ofpredictability occurs hence after a few cycles around the braid.
Cross-distance scaling
A large body of work has shown that strange attractors are relatively difficult to characterize in detail, even for low-dimensional dynamical systems. For systems with a higher dimension, such as autonomous neural networks or climate models,it may even be a challenge to robustly distinguish laminar from chaotic flows. Here we propose that the scaling of the long-termdistance d ( t (cid:29) T λ ) of two trajectories, d ( t (cid:29) T λ ) ∝ δ ν , d ( t ) = (cid:104)| x ( t ) − x ( t ) |(cid:105) , (2)may be used as a reliable indicator for chaos, where we denote with d ( t = ) = δ the initial distance, and with ν thecross-distance scaling exponent.For an illustration of how the long-term distance d ( t (cid:29) T λ ) depends on the initial distance δ , we show in Fig. 3 ( a )the time evolution of the distance d between pairs of trajectories, considering initial distances δ = − and δ = − , asaveraged over 10 pairs. For strong chaos (red curves, ρ = ρ ) and PPC (green curves, ρ = ρ ) the long-term distance doesnot depend on the initial distance. The scaling exponent thus vanishes, ν =
0, for chaotic motion. The initial slope of thecurves reflects the exponential divergence of chaotic trajectories within the time scale of the Lyapunov prediction time T λ . Forthe laminar flow (blue curves, ρ = ρ ) the long-term distance depends on the other side on δ , leading to a non-zero scalingexponent, ν (cid:54) = b ) we have evaluated for every ρ considered the long-term distance d ( t = ) startingfrom initial distances δ ∈ [ − , − ] , averaging each time over 10 pairs of trajectories. We note that the scaling exponent ν can be extracted reliably from a linear regression of the data in a log-log plot when the initial distance δ is smaller than thedistance of two neighboring attractors or parts of the same attractor. Additionally we note that the choice of t =
200 wasselected such that t (cid:29) T λ holds for the interval of ρ considered (cf. Table 1). Close to the period doubling transition to chaos,viz for ρ (cid:46) ρ C , the maximal Lyapunov exponent becomes very small ( λ m (cid:46) − ) and the Lyapunov prediction times large( T λ > t (cid:29) T λ would be needed.The linear scaling ν = ρ = ρ ) in Fig. 3 ( b ) stems from the fact that any two orbits attractedby a limit cycle follow each other perpetually, with the average final separation being proportional to the initial separation. This
20 40 60 80 100 t . . . . . . C ( t ) exp. div. diff. loss( a ) ρ = ρ = 180 .
70 0 50 100 150 200 250 300 350 400 t . . . . . . C ( t ) ( b ) ρ = ρ = 180 . t − − − C ( t ) T λ t − − − C ( t ) T λ Figure 4.
The cross-correlation C for pairs of trajectories with an initial distance δ = − and averaged over 10 pairsover time. The data is for ( a ) strong chaos ( ρ = ρ = .
70) and ( b ) partially predictable chaos ( ρ = ρ = . C ≈ − . · − e λ C t , λ C = .
17 (inset in ( a )) and as1 − . · − e λ C t , λ C = .
04 (inset in ( b )), as indicated by the respective solid lines. The linear decrease of C forintermediate times corresponding to a diffusive loss of predictability is ∝ ( − . t ) in ( a ) and ∝ ( − . · − t ) in ( b ), asindicated by the respective dashed lines.relation can be motivated analytically using a local approximation to the attracting set in the normal form of limit cycles (cf. theMethods section).For chaotic phases the long-term average distance settles on the other hand to a finite value determined by the extent of theattracting set, independently of the initial distance δ , leading to a vanishing scaling exponent ν =
0. As observed in Fig. 3 ( a )the time needed for strong chaos to reach long-term stationarity in d ( t ) is proportional to the Lyapunov prediction time T λ .For PPC the long-term limit is however only reached for t > T PPC , where the decorrelation time T PPC , i. e. the time that a pair oftrajectories needs to get fully uncorrelated, is significantly longer than both the quasi-period τ and the Lyapunov predictiontime T λ . The scaling exponent ν ≈ t ∈ [ T λ , T PPC ] . We remarkthat d ( t → ∞ ) is however determined by the overall extent of the attracting set in the limit of large times. For times t > T λ ,right after the initial exponential decorrelation, the typical separation of two orbits d ( t ) is of the order of the braid width (cf.insets in Fig. 2 ( b ), ( c )).In Fig. 1 ( c ) the cross-distance scaling exponent ν for the entire range of ρ considered here is shown. We note, that thetransition from chaos to laminar flow occurring at ρ C ≈ .
96 is accompanied by a sudden jump in ν from zero to one. This isquite remarkable, as the corresponding maximal Lyapunov exponent λ m , also shown in Fig. 1, becomes, on the other hand,continuously smaller when approaching ρ C from the chaotic side.The cross-distance scaling is a robust 0 − . For a further evaluation weapplied it to the chaotic states found in previously studied neural networks , which we generalized in size (with up to 300dimensional phase spaces). We also examined the three-dimensional Shilnikov attractor (cf. Appendix A), as it is similar tothe Lorenz system, albeit with all degrees of freedom evolving on the same time scale. For both systems the test presented hereworked without problems.For a comparison we applied the Gottwald 0 − to the three different dynamical regimes of the Lorenz systempresented above (cf. Fig. 2). Using Gottwald’s method we were able to classify regular motion ρ = ρ and strong chaos ρ = ρ correctly, but not partially predictable chaos. For ρ = ρ even an exceedingly long run time, t = , did not provide a clearresult. Cross-correlation of initially close trajectories
An important point for real-world applications are the long-term repercussions of variations in the initial conditions. Forconcreteness we consider with C ( t ) = (cid:104) ( x ( t ) − µ ) · ( x ( t ) − µ ) (cid:105) / s , (3)the cross-correlation function of two bounded and initially close-by trajectories x ( t ) and x ( t ) . Here (cid:104) . . . (cid:105) denotes an averageover initial conditions on the attractor sampled with the natural distribution, µ the center of gravity, and s the average extent of he attracting set, µ = lim T → ∞ T (cid:90) TT x ( t ) d t , s = lim T → ∞ T (cid:90) TT [ x ( t ) − µ ] d t . (4)The cross-correlation is normalized to unity for close-by trajectories, i. e. for | x ( t ) − x ( t ) | →
0. For chaotic attracting sets thecross-correlation C vanishes in the long-term limit t → ∞ , with a finite C (cid:54) = D ( t ) = (cid:104) [ x ( t ) − x ( t )] (cid:105) between two trajectories,which leads, when using (3), to D ( t ) = s [ − C ( t )] . (5)For large cross-correlations C → s of the attracting region,in the sense that D (cid:28) s .It is evident from Fig. 2, that the overall shape of the attractor changes little across the transition from laminar flow ( ρ = ρ )to chaos ( ρ = ρ ∗ ), and that the previously one dimensional attracting state (the limit cycles) does broaden to a closed chaoticbraid. This behavior can also be viewed as chaotic wandering around limit cycles .In Fig. 4 we present the time evolution of the cross-correlation C for the case of strong chaos, ρ = ρ in ( a ), and partiallypredictable chaos, ρ = ρ in ( b ). We note that C remains close to full predictability, C (cid:39)
1, within the respective timescales of the Lyapunov prediction time T λ . We have included in both panels of Fig. 4 fits to the cross-correlations of the form1 − c exp ( λ C t ) , an approximation resulting from (5), where λ C ≥ λ m29 ( c is a fit parameter). C decreases linearly for timeslarger than T λ , saturating eventually to zero when full decorrelation is achieved : D ( t ) / s = − C ( t ) ∝ (cid:26) e λ C t for t < T λ t for t > T λ . (6)For strong chaos both the exponential divergence and the diffusive loss of predictability happen on the same time scale. Weremark here that the evolution of the cross-correlation presented in Fig. 4 for the case of strong chaos bears a surprisingsimilarity with the measured relative accuracy of weather forecasting over a period of two weeks .For partially predictable attractors, ρ = ρ , we find qualitatively the same behavior as for strong chaos, at least as matter ofprinciple, with a dramatic separation of time scales setting however in beyond the initial phase of exponential divergence. Theslope of the linear decrease is, as evident from Fig. 4 ( b ), three orders of magnitude smaller in the partially predictable case(1 . · − instead of 4 · − ).PPC can be found also in systems controlled by a single microscopic time scale (cf. Appendix A). We hence attribute theemergence of PPC to the fact, that the attractor is topologically equivalent, for ρ = ρ , to elongated closed braids. Comparingthe braid width from the insets in Fig. 2 to the linear distance d ( t = ) in Fig. 3 ( b ) (5 and 0 . . . ρ = ρ and ρ = ρ ∗ respectively), we find that the initial exponential divergence occurs dominantly perpendicular to thebraid. Once the separation of two trajectories has reached the braid width it can increase further only along the braid, which isin turn a diffusive process and hence slow. This means that the chaotic flow remains partially predictable for remarkable longtimes compared to the Lyapunov prediction time T λ . From the linear fit in Fig. 4 ( b ) we estimate that it takes T PPC ≈ untilcorrelations vanish effectively for the partially predictable case ρ = ρ .The cross-correlation C ( t = ) shown in the Fig. 1 ( d ) vanishes for ρ < ρ p , which is hence a phase in which predictabilityis lost for times larger than the Lyapunov prediction time T λ . The ergodicity of pairs of trajectories is however broken forintermediate times T λ < t < T PPC in the PPC phase realized for ρ p < ρ < ρ C . Partially predictable chaos is hence characterizedboth by positive Lyapunov exponents λ m > C ( t (cid:29) T λ ) (cid:54) =
0. This notion ofpredictability does naturally not exclude the possibility of finding additional finite time windows of predictability due to thepresence of periods of quasi-laminar flow embedded in the overall chaotic time evolution . Measuring the fractal dimension with the box-counting method we find that the attractors in the PPC phase have fractal dimensions slightly larger than two, asusual for the Lorenz system .The exceedingly slow loss of predictability occurring for ρ = ρ can be observed also in systems in which all definingdynamical parameters are of the same order of magnitude (cf. Appendix A). The magnitude of the respective diffusion coefficientis hence only indirectly related, for the case of the Lorenz system, to the relative size of β , σ and ρ in (1). We also note that theneutral flow along the braids, i. e. the flow along the attractor which is characterized by a vanishing average λ = (cid:104) λ ( l ) (cid:105) = λ ( l ) , is highly dispersive (cf. Fig. 7 in the Methods section). Additionally weremark that PPC manifests itself in a linear decrease of the amplitudes of the periodic oscillation of the auto-correlation function(cf. Methods section). iscussion We have proposed here a new 0 − − and that the system is stable in this state against finiteperturbations . The notion of partial predictability implies macroscopic predictability in terms of coarse grained predictions.Taking the case of weather forecasting, which is plagued notoriously by chaotic instabilities , it may hence be possible topredict with confidence the formation of a low pressure area, to give an example, but not its exact extension and depth.We have shown here that partial predictability is not a consequence of varying local Lyapunov exponents on the attractingset and that the averaged Lyapunov exponents in terms of the Lyapunov spectrum yield prediction times which are orders ofmagnitude smaller than the time scales observed for partial predictability. Partial predictability is essentially a consequence oftopological constraints, e. g. when chaotic braids arise from a previous period doubling transition. PPC is hence expected to befound for a wide range of systems, such as enzyme reactions and models of asset pricing . In this context we point out thatindications for partially predictable chaos have been found recently in the phase space of the sensorimotor loop of simulatedself-organized robots . It would be interesting to investigate in further studies whether the concept of partial predictability,which does not require multiple time scales per se , could be generalized to time dependent snapshot or pullback attractors arising in stochastic and/or driven chaotic systems.The dynamical regimes discussed here – strong chaos, PPC and laminar flow – can be distinguished when combining the0 − C ( t ) ,which may be finite (for PPC and laminar flow) or zero (for strong chaos). We also stress that the three indicators – globalmaximal Lyapunov exponent, cross-correlation and cross-distance scaling – examined in this work rely on the evolution ofinitially close pairs of trajectories. These indicators can hence be evaluated by a straightforward manipulation of the datawithout the need to investigate further the nature of the attracting set. It is possible to automatize the computation of thereexamined 0 − Methods
All computations that involved solving Eq. (1) were performed using a Runge-Kutta-Fehlberg algorithm of order 4/5 andstep size ∆ t = − . Testing the accuracy of the results by systematically varying ∆ t we found that the limitations due to thechaotic nature of the motion allow for reliable results for integration times up to t ∼ Derivation of the cross-distance scaling for limit cycles
Above we showed that the long-term distance d ∞ = lim t → ∞ d ( t ) for of two initially close-by trajectories scales linearly with theinitial distance δ whenever the dynamics settles in an attracting limit cycle. For an analytic understanding of this observationwe consider the two dimensional normal form for limit cycles in polar coordinates ( ϕ , r ) ,˙ ϕ = Ω ( r ) , ˙ r = r ( Γ − r ) , (7)where the time evolution of the angle ϕ is described by an arbitrary smooth function Ω ( r ) of the radius r ,Expanding Eq. (7) to first order around the limit cycle r ( t ) ≡ Γ , viz using r = Γ + ε , we find˙ ϕ = Ω ( Γ ) + Ω (cid:48) ( Γ ) ε , ˙ ε = − Γ ε (8)for the behavior in the close neighborhood of the limit cycle, with Ω (cid:48) = d Ω / d r denoting the derivative with respect to theradius r . Substituting ( ϕ , ε ) → ( x , y ) and ( Ω , Ω (cid:48) , Γ ) → ( a , b , c ) we then obtain with˙ x = a + b y , ˙ y = − c y (9)the Cartesian normal form of a limit cycle. The parameter a hence represents the base speed of the flow along the limitcycle, b the rate with which the flow parallel to the limit cycle changes with the distance y from the limit cycle, and c the time scale needed to relax to the attractor. The solution ( x ( t ) , y ( t )) of the linearized system with the initial conditions ( x , y )( t = ) = ( x o , y o ) is given by x ( t ) = x o + at + bc y o (cid:0) − e − ct (cid:1) , y ( t ) = y o e − ct . (10) − − − − − δ − − − d ( t ≫ T λ ) ( a ) ρ = ρ ∗ = 180 . d (l)12 d = h d (l)12 i . δ . − − − − − δ − − − ( b ) ρ = ρ = 181 . d (l)12 d = h d (l)12 i . δ . Figure 5.
The scaling behavior of the averaged long-term distance d ( t = ) (colored circles), see Eq. (2), with therespective fits on a log-log scale (fitted for δ < − ) (solid lines, cf. Fig. 3). The distribution of the non-averageddistances d ( l ) ( t = ) , as obtained from 1000 initial conditions, are shown in addition (gray circles). ( a ) For PPC with ρ = ρ ∗ = .
95 one finds, in agreement with Fig. 3, a close to constant cross-distance scaling. ( b ) For a laminar flow with ρ = ρ = .
10 the scaling exponent is ν ≈ δ < − . In both cases the scaling breaks down when asymmetry related close-by attractor starts to attract a fraction of the orbits for δ > − .As we are interested in the behavior of two initially close trajectories (of which both are close to the attractor), we considertwo trajectories starting from ( x o , y o ) and ( x o + δ x , y o + δ y ) . Here δ x and δ y denote the initial distances between the trajectoriesin their respective dimension and δ = ( δ x + δ y ) / the initial Euclidean distance between the trajectories. Both trajectoriesconverge in the long-term limit t → ∞ to the limit cycle y →
0. The Euclidean distance between the trajectories is hence givenby d ( t ) = (cid:0) [ x ( t ) − x ( t )] + [ y ( t ) − y ( t )] (cid:1) / = (cid:32)(cid:20) δ x + bc δ y ( − e − ct ) (cid:21) + δ y e − ct (cid:33) / . (11)The distance approaches a finite value in the long-term limit t → ∞ , which we term the long-term distance d ∞ ( δ x , δ y ) = lim t → ∞ d ( t ) = (cid:12)(cid:12)(cid:12)(cid:12) δ x + bc δ y (cid:12)(cid:12)(cid:12)(cid:12) , (12)where |·| denotes the modulus.Averaging d ∞ ( δ x , δ y ) over a circle C centered around ( x o , y o ) , defined by δ = δ x + δ y , we obtain (cid:104) d ∞ (cid:105) = πδ (cid:73) C d s d ∞ ( δ x , δ y ) = π (cid:18) b c + (cid:19) / δ . (13)The average long-term distance (cid:104) d ∞ (cid:105) is hence proportional to the initial distance δ , with the constant of proportionality2 ( b / c + ) / / π depending through b / c on the properties of the flow close to the limit cycle. The factor b / c can be smalleror larger than unity, implying that the long-term distance of the two trajectories may exceed the initial distance.The normal form (9) describes the local flow close to a limit cycle. For the case of a non-uniform base velocity a = a ( x ) one need to generalize (13) by averaging over full periods. Choice of initial distances
The cross-distance scaling (cf. Eq. (2)) is valid only when the two trajectories considered are attracted by the same attractor.This condition is satisfied for the values of δ considered in Figs 1 and 3, namely δ ∈ [ − , − ] , but not necessarily for largervalues of δ , as illustrated in Fig. 5.For the partially predictable chaotic attractor, with ρ = ρ ∗ = .
95 in Fig. 5 ( a ), we find the expected scaling ν = . d ( t = ) (colored bullets) and initial distances up to δ ≈ − . The gray dotsrepresent the unaveraged cross-distances d ( l ) ( t = ) , as obtained from distinct 1000 initial distances. For δ > − the twoorbits start to end up in distinct attractors, as a second (symmetry related) PPC attractor exists close-by in phase space. After acrossover region δ ∈ [ − , ] one observes a second scaling plateau for δ > ρ . . . . λ m ( a ) Benettinslope − − − ( b ) λ m λ λ ρ − − − − − λ i Figure 6.
The Lyapunov exponents of the Lorenz system for ρ ∈ [ . , . ] . ( a ) Comparing the maximal averageLyapunov exponent λ m obtained using the method of Benettin (solid line) and from the initial slope of the averagedlogarithmic distance (cid:104) ln | x ( t ) − x ( t ) |(cid:105) (dots). The distinct regimes are color coded (red/green/blue for strongchaos/PPC/laminar flow). ( b ) All three average Lyapunov exponents λ i computed by the method of Benettin . (Note thebreak in the vertical scale.)For the cross-distance scaling of the limit cycle, as presented for ρ = ρ = .
10 in Fig. 5 ( b ), we observe an equivalentbehavior. In the limit of small initial distances δ < − we obtain in accordance with Fig. 3 the close to linear scaling ν = . δ > − .We note that attractors are surrounded, by definition, by a possibly small but in any case finite-sized basin of attraction and that the here proposed scaling analysis can be performed generically when considering initial conditions close enough tothe attractor, separated by small initial distances δ . Basins of attraction may however fan out further away from the attractingset into complicated and possibly fractal structures . Global Lyapunov exponent
The results for the maximal average Lyapunov exponent λ m presented above were computed from the averaged logarithmicdistance (cid:104) ln | x ( t ) − x ( t ) |(cid:105) between two initially close-by trajectories over time. Alternatively one may evaluate λ m usingBenettin’s algorithm .In Fig. 6 (left) we compare the average maximal Lyapunov exponent λ m of the Lorenz system in the parameter range ρ ∈ [ . , . ] as obtained by the linear slope of the averaged logarithmic distance between two initially close-by trajectories(colored dots) with the λ m found when using Benettin’s method (solid line). For the latter method λ m is given by the logarithmicratio of an initial deviation from the attractor, here δ = − , and its stretched time evolution. This quantity has been averagedequidistant in time for ∼ points over the respective attractor with an integration time step of d t = − . The data matcheswell for regular motion (blue) and for PPC (green). For strong chaos (red) there is however a non-negligible degree of scattering.Using Benettin’s method we also computed the complete spectrum of Lyapunov exponents ( λ m , λ , λ ) for the Lorenzsystem in the considered parameter range, as depicted in Fig. 6 (right). The largest and the second largest exponents are positiveand zero for both chaotic regimes, as expected, and zero and negative respectively for a limit cycle. Summing up the exponentsleads to ∑ i λ i ≈ − .
67, which is in agreement with the phase space contraction rate − − β − σ = − .
66 of the Lorenzsystem.
Distribution of local Lyapunov exponents
It is of interest to evaluate not only the averaged Lyapunov exponents, as presented in Fig. 6 and Table 1, but the full distributionof local Lyapunov exponents on the attracting set, both for the case of PPC and for strong chaos. For the data presentedin Fig. 7 we computed the local Lyapunov exponents λ ( l ) i as the logarithm of the local expansion coefficient (the ratio oflengths of the orthogonalized deviation vectors after one simulation step and the initial deviation vectors), using Gram-Schmidtorthogonalization in every step. The local Lyapunov exponents were sampled equidistant in time with an integration time stepof d t = − , over a total time T max = , where the length of initial deviation vectors has been set to δ = − after everystep of the simulation.The distribution of local Lyapunov exponents covers a rather wide range of values as compared to the respective globalLyapunov exponents. There is hence no directly evident connection between the distribution of local Lyapunov exponents andthe shape of the chaotic attractor, or to the characteristics of PPC. We emphasize that the local Lyapunov exponents presented in m = 1 . λ = 0 . λ = − . − −
20 0 20 λ (l) i . . . . p ( λ ( l ) i ) ( a ) ρ = ρ = 180 . λ m λ λ λ m = 0 . λ = 0 . λ = − . − −
20 0 20 λ (l) i ( b ) ρ = ρ = 180 . λ m λ λ Figure 7.
The probability distribution p ( λ ( l ) i ) of the local Lyapunov exponents λ ( l ) i for the Lorenz attractor; the globalLyapunov exponents λ i given in the legend and indicated by arrows in the plot are the average of the respective distributions.The standard deviations are s m = . s = . s = . a ) for ρ = ρ = .
70 (strong chaos) and s m = . s = . s = . b ) for ρ = ρ = .
78 (PPC). A direct connection between the functional form of the distributionsof Lyapunov exponents with the dynamics of PPC is not evident. For the PPC the variance of the local neutral exponent λ ( l ) isan order of magnitude larger (5.6 vs. 0.47) than the corresponding maximal Lyapunov exponent.Fig. 7 are obtained from the stretching of the deviation vectors at every time step (not averaged) and that the global Lyapunovexponents λ i = (cid:104) λ ( l ) i (cid:105) correspond to the averages of the corresponding local exponents over the attracting set.Fig. 7 shows, most interestingly, that the neutral flow λ = (cid:104) λ ( l ) (cid:105) = C remains diffusivefor prolonged time scales, as observed in Fig. 4 ( b ), would result from an effective decoupling of a smooth phase and a chaoticradial evolution. Auto-correlations within PPC
Instead of considering the properties of the cross-correlation between two trajectories one may study, alternatively, theautocorrelation function A ( t ) = lim T → ∞ T s (cid:90) TT d t (cid:48) ( x ( t (cid:48) ) − µ ) ( x ( t (cid:48) − t ) − µ ) (14)for the trajectory x ( t ) on the attractor, where µ and s denote, as defined by Eq. (4), respectively the mean and the average extentof the attracting set. In Fig. 8 we present A ( t ) for the PPC discussed in Fig. 4 ( b ), i. e. for ρ = ρ = .
78 (using T = ).One observes that the topology of the motion along fractally broadened braids shows up prominently in the autocorrelationfunction, with the quasi-period τ (cid:39) . A ( t ) .The steady loss of predictability observed in Fig. 4 ( b ) for PPC translates into a corresponding linear decrease (as indicatedby the dashed line in Fig. 8 ( b )) of the heights of the local maxima of A ( t ) . Using the autocorrelation function for theinvestigation of the slow decorrelation occurring in partially predictable chaos is hence possible, but plagued by the oscillatorynature of A ( t ) . For this reason we concentrated in this study on the cross-correlation C ( t ) . The initial exponential decrease ofcorrelations is furthermore only visible in the data for C ( t ) , but not for A ( t ) (compare Fig. 4 ( b ) and Fig. 8). Acknowledgments
The authors thank Prof. Tam´as T´el for his suggestions. B. S. also acknowledges useful discussions at the Seminars in StatisticalPhysics in Budapest. Further, the authors acknowledge the financial support of the German research foundation (DFG) and ofStiftung Polytechnische Gesellschaft Frankfurt am Main.
Author contributions statement
All authors contributed equally to the present work. All authors reviewed the manuscript. t − . − . . . . . . . A ( t ) ( a ) t . . . . . . A ( t ) ( b ) Figure 8.
The autocorrelation A ( t ) , as defined by Eq. (14), for ρ = ρ = .
78 (PPC, compare Fig. 4 ( b )). ( a ) Theoscillations resulting from looping around the braids are spaced by τ (cid:39) . b ) On longer timescales the auto-correlationdecreases linearly, with the dashed line corresponding to 0 . − · − t . Additional information
The authors declare no competing financial interests.
A Appendix A: PPC in a system without separation of scales
Our choice of the parameters β = / σ =
10 and ρ >
180 for the Lorenz system (1) resulted in parameters and hencepossibly also in time scales of distinct orders of magnitude. The question then arises if the observed large timescale for the finaldecorrelation process in the partially predictable phase may be a consequence of occurrence of distinct microscopic time scales.In order to rule out this scenario we have investigated with ˙ x = y , ˙ y = z , ˙ z = x − x − y − bz (15)a system for which the defining parameters are with b ∈ [ , ] all of the same order of magnitude. For b = . λ m = . , λ = , λ = − . b > . .The cross-correlation shown in Fig. 9 (right) decays extraordinarily slow (cf. Fig. 4 ( b )), with a slope of the order of 10 − ,implying that the retention of partial predictability cannot be attributed to the occurrence of large intrinsic time scales. Theusual initial drop of the cross-correlation by about 0 .
01% within the Lyapunov prediction time T λ (cid:39)
805 is however present.
B Appendix B: Automation of the testing procedure
As the cross-distance scaling exponent and the finite time cross-correlation can be used as 0 − λ m , the cross-distancescaling exponent ν and the finite time cross-correlation C ( t (cid:29) T λ ) , where the latter two will be practically binary. The threedifferent dynamics can be characterized subsequently by the criteria listed in Table 2.1. First one needs to localize the attractor and estimate an upper bound for the initial distance δ (cid:28) Table 2.
Combining the scaling exponent ν and the level of the cross-correlation at finite time, C ( t ) , allows to classify thethree possible types of dynamics (compare Fig. 1). ν C ( t ) dynamics0 0 strong chaos0 1 PPC1 1 laminar flow . − . . . . x − . − . . . . y ( a ) t . . . . . C ( t ) ( b ) exp. div.diff. loss t − − − − C ( t ) Figure 9.
The attractor of the dynamical system described by Eq. (15), showing partially predictable chaos for b = . a ) The projection to the x − y plane showing chaotic braids. ( b ) The cross-correlation C , which decreases first exponentially(gray shaded region), as indicated by the fit 1 . · − e λ C t , λ C = . ∝ . · − t (dashed line).2. One then computes the average maximal Lyapunov exponent λ m from the slope of the averaged logarithmic distance (cid:104) log | x ( t ) − x ( t ) |(cid:105) , where the average is performed over pairs of trajectories x ( t ) and x ( t ) starting from a fixed initialdistance δ (cid:28)
1. Other methods, like Benettin’s algorithm , may be used alternatively.3. The Lyapunov prediction time T λ is then given by the inverse of the Lyapunov exponent.4. Next, one computes the average Euclidean distance d ( t = T λ ) for a range of initial distances δ (cid:28)
1, from which thescaling exponent ν is extracted using d ( t > T λ ) ∼ δ ν . The flow is chaotic for ν = ν = C ( t = T λ ) for pairs oftrajectories with initial distance δ (cid:28)
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