Dynamical and transport properties in a family of intermittent area-preserving maps
Roberto Artuso,*Lucia Cavallasca,†and Giampaolo Cristadoro‡
Center for Nonlinear and Complex Systems and Dipartimento di Fisica e Matematica, Università degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy and Sezione di Milano, INFN, Via Celoria 16, 20133 Milano, Italy
共Received 19 December 2007; published 11 April 2008兲
We introduce a family of area-preserving maps representing a共nontrivial兲 two-dimensional extension of the Pomeau-Manneville family in one dimension. We analyze the long-time behavior of recurrence time distribu- tions and correlations, providing analytical and numerical estimates. We study the transport properties of a suitable lift and use a probabilistic argument to derive the full spectrum of transport moments. Finally, the dynamical effects of a stochastic perturbation are considered.
DOI:10.1103/PhysRevE.77.046206 PACS number共s兲: 05.45.⫺a
I. INTRODUCTION
Hamiltonian systems are generically not fully hyperbolic 关1兴: for example, the phase space of typical area-preserving maps reveals the coexistence of chaotic trajectories and is- lands of regular motion 共periodic or quasiperiodic trajecto- ries兲 关2兴. Even when we are concerned with statistical prop- erties of motion on the chaotic component, we cannot neglect the presence of regular structures. They deeply influ- ence chaotic motion as, whenever trajectories come close to integrable islands, they stick there for some time, and irregu- lar dynamics is thus punctuated by laminar segments where the system mimics an integrable one. This intermittent be- havior has strong influence on the long-time properties of quantities like correlation decay or recurrence time statistics, which typically present a power-law tail关3–7兴. In unbounded systems, intermittency influences transport properties, gener- ating anomalous diffusion processes共see 关8兴 and references therein兲, in contrast to the normal diffusion observed for fully hyperbolic systems关9兴.
While much effort is still devoted to fully understanding the general picture of mixed phase space, the situation sim- plifies if we let the islands of regular motion shrink to zero:
even the presence of a single marginally stable fixed point can produce intermittentlike behavior关10–12兴. In one dimen- sion this corresponds to the Pomeau-Manneville maps on the unit interval关13兴,
xn+1=兩xn+ xnz兩mod 1, z⬎ 1, 共1兲 which represent one of the few examples of non-fully- hyperbolic systems for which analytic results can be ob- tained with a variety of different techniques 共see, for in- stance,关14–21兴兲. Such maps present a polynomial decay of correlations and recurrence time statistics with exponents that depend on the intermittency parameter z 关15,16,22兴.
Moreover, a proper lift on the real line can generate anoma- lous diffusion and the set of transport moments typically shows a two-scale structure关23–26兴.
The situation is much less satisfactory in more than one dimension. Rigorous results were derived for specific cases, where it was possible to give precise bounds on the rate of mixing 关12,27兴. Here we study a family of area-preserving maps with a neutral fixed point. This family depends on a parameter that governs stability properties of the fixed point, in analogy with the Pomeau-Manneville maps.
We introduce the two-dimensional family of area- preserving maps in Sec. II, where we also discuss the dy- namical features we will look at. In Sec. III we will consider the unstable manifold of the neutral fixed point, and provide simple estimates that will be pivotal in predicting the decay of survival probabilities. Section IV contains extensive in- vestigations on survival probability and correlation functions decay. By lifting the map on an unbounded phase space we then analyze transport properties in Sec. V. The role of a stochastic perturbation is then studied in Sec. VI, while we present our conclusions in Sec. VII.
II. THE MODEL
We define the following one-parameter family of maps T␥共x,y兲:T2→T2, whereT2=关−,兲2 共with torus topology兲:
T␥共x,y兲 =再x + f␥共x兲 + y on T,
y + f␥共x兲 on T,冎 共2兲
with
f␥共x兲 =sgn共x兲冏x冏␥, ␥⬎ 1. 共3兲
The map T␥is area preserving for every choice of the impul- sive force f共x兲. The Jacobian matrix is
J␥共x,y兲 =冉1 + ff␥
⬘
共x兲␥⬘
共x兲 11冊 共4兲so we have det J␥共x,y兲=1 and Tr J␥共x,y兲=2+ f␥
⬘
共x兲⬎2 for x僆关−,兲/兵0其: the map is everywhere hyperbolic except on the line x = 0 and thus the fixed point at the origin is margin- ally stable共parabolic兲 for every␥. The parameter␥ changes the ⬙stickiness⬙ of the origin, in analogy with the intermit- tency exponent z appearing in the Pomeau-Manneville maps of Eq. 共1兲. We point out that the choice f共x兲=x−sin共x兲*roberto.artuso@uninsubria.it
†lucia.cavallasca@uninsubria.it
‡giampaolo.cristadoro@uninsubria.it
␥ 共 兲. We note explicitly that for our model f僆Ckwith k =关␥− 1兴 共where 关·兴 denotes the integer part兲 unless␥is an odd integer, in which case f僆C⬁关28兴.
A. Dynamical indicators
In order to get information about the dynamical properties of the systems it is often useful to employ time statistics 关29兴. We choose a set ⍀ including the parabolic fixed point and then define a partition of⍀ into disjoint sets ⍀n, each representing the set of points that leave⍀ in exactly n itera- tions. The survival probability p⍀共n兲 is the fraction of initial conditions in ⍀ that are still in ⍀ after n iterations. The behavior of p⍀共n兲 generally depends on the choice of the measurei with which we distribute initial conditions over
⍀, which may be quite different from the invariant measure.
In the present case the invariant measure is the Lebesgue one
, which also represents the most natural way to spread ini- tial conditions over⍀, soi= and
p⍀共n兲 = 1
共⍀兲兺
k⬎n共⍀n兲. 共5兲
We may also define the waiting time distribution 共or resi- dence time statistics兲 over ⍀,⍀共n兲, as the probability that once a trajectory enters the set⍀ it stays there for exactly n time steps.⍀共n兲 is computed by running a long trajectory and recording residence times in⍀: ⍀共n兲 is just the prob- ability distribution of such residence times. In our case
⍀共n兲 = 1
共⍀兲关共⍀n兲 −共⍀n+1兲兴 共6兲 where ergodicity and the property that the map preserves the Lebesgue measure have been used. We point out that, in general, while p⍀共n兲 depends upon an arbitrary choice of the distribution of initial conditions,⍀共n兲 does not.
From Eqs.共5兲 and 共6兲 we see that the asymptotics of these quantities are tightly related 关29兴: in particular, if the mea- sure of the sets⍀ndecays according to a power law
共⍀n兲 ⬃ n−␣−1, 共7兲
we get
p⍀共n兲 ⬃ n−␣ 共8兲
and
⍀共n兲 ⬃ n−␣−2. 共9兲
Besides their intrinsic interest, these relations have remark- able links with the problem of establishing the mixing rates for the system关4,5,30,31兴. We briefly recall it with a simple argument: suppose we consider an observable A that remains fully correlated for portions of trajectories within ⍀, and otherwise completely decorrelated 共due to randomness of motion outside of⍀兲; then we may easily show that 关31兴
m=n兺 兺
k=m
⍀ 共10兲 so that the exponents of the power-law decay of survival probability and correlations should coincide,
CAA共n兲 ⬃ n−␣. 共11兲
We observe that the validity of Eq. 共10兲 has been carefully scrutinized numerically for chaotic billiards 关31,32兴, and even validated in rigorous estimates of polynomial mixing speed for one-dimensional 共1D兲 intermittent systems 关15兴 共see also 关33兴兲, but indications of its possible failure have also been suggested关34兴.
It is interesting to remark that showing that the power-law decays of p⍀共n兲 and ⍀共n兲 differ by 2 关Eqs. 共8兲 and 共9兲兴 employs the fact that the Lebesgue measure is the invariant one for the system. Actually, for the map of Eq.共1兲, where the invariant measure is not uniform, they differ by 1 if we choose the initial conditions uniformly distributed with Le- besgue measure共while the exponent of the waiting time dis- tribution coincides with the one ruling the length of the seg- ments⍀n关13兴兲. A relationship coinciding with Eqs. 共8兲 and 共9兲 holds instead for another intermittent map, introduced by Pikovsky关24兴 共some features of this map are also described in关35兴兲:
xn+1= f˜z共xn兲, 共12兲 where f˜z is an 共odd兲 circle map, again dependent on an in- termittency parameter z, implicitly defined on the torus T
=关−1,1兲 by
x =冦2z1关1 + f˜z共x兲兴z, 0⬍ x ⬍ 1/共2z兲, f˜z共x兲 + 1
2z关1 − f˜z共x兲兴z, 1/共2z兲 ⬍ x ⬍ 1.冧 共13兲
A key feature that the map of Eq.共12兲 shares with our model is that the invariant distribution is smooth, coinciding with the Lebesgue measure, as can be checked by inspection of the form of the Perron-Frobenius operator.
For the map under consideration it is possible to get an estimate of the exponent ␣ by studying the invariant mani- folds of the parabolic fixed point.
III. INVARIANT MANIFOLDS
A typical trajectory staying for a long time in⍀ 共again we take as⍀ a region including the parabolic fixed point兲 enters
⍀ close to the stable manifold, escaping along the unstable one共Fig.1兲. In particular, we are going to discuss how tra- jectories escape by following the unstable manifold of the marginal fixed point. For the odd symmetry of f␥共x兲, we can restrict the analysis to the first quadrant.
Let us call (x , y共x兲) the graph of the unstable manifold in the neighborhood of the origin and suppose that, very close to the marginally stable fixed point, y共x兲⯝x. With the
choice of Eq.共3兲 we have that f␥共x兲⯝a x␥and the Jacobian matrix关Eq. 共4兲兴 of the map is
J␥共x,y兲 =冉1 + bxbx␥−1␥−1 11冊, 共14兲
whose eigenvalues are written to leading order as ⫾
= 1⫾x共␥−1兲/2.
The vector (1 , y
⬘
共x兲), i.e., 共1, x−1兲, tangent to the un- stable manifold satisfiesJ␥共x,y兲冉 x1−1冊⯝ 共1 + x共␥−1兲/2兲冉 1x−1冊, 共15兲
i.e.,
1 + bx␥−1+x−1⯝ 1 +x共␥−1兲/2,
b x␥−1+x−1⯝x−1+x共␥−1兲/2+共−1兲, 共16兲 and from these equations, remembering that xⰆ1 and ␥
⬎1, we obtain
=␥+ 1
2 . 共17兲
We note explicitly that, for␥= 3, this result is in agreement with the case f = x − sin共x兲 derived in 关12兴. We can now study the dynamics restricted to the unstable manifold. Let us de- note byᐉ the arc length coordinate along the manifold; for small x we get
ᐉ共x兲 =冕xdx冑1 +关dy共x兲/dx兴2⯝ x. 共18兲
Denote byᐉn the coordinate ᐉ at a point (xn, y共xn兲) and by ᐉn+1the coordinate along the manifold of T␥(xn, y共xn兲),
ᐉn+1=ᐉn+ h共ᐉn兲. 共19兲 By using Eqs.共18兲 and 共2兲 we get
h共ᐉ兲 ⯝dᐉ
dt共ᐉ兲 =dᐉ dx
dx
dt共ᐉ兲 ⯝ 关y共ᐉ兲 + x␥共ᐉ兲兴 = ᐉ+ᐉ␥. 共20兲 From Eq.共17兲 and by remembering that␥⬎1, we obtain, via a continuous time approximation关36,37兴,
h共ᐉ兲 ⯝ ᐉ= dᐉ
dt. 共21兲
If we fix the boundary of⍀ at a scale L, we can then evalu- ate the time needed to reach the boundary as a function of the arc lengthᐉ along the manifold, by employing the standard argument of关36,37兴:
T共ᐉ兲 = 2
␥− 1共ᐉ−共␥−1兲/2− L−共␥−1兲/2兲; 共22兲 this implies the following scaling for the inverse function:
ᐉ共T兲 ⬃ T−2/共␥−1兲. 共23兲 We now arrive at the crucial point: we estimate p⍀共n兲 as the area of a rectangle having one vertex at the origin共the para- bolic fixed point兲, and another at a point on the unstable manifold共x¯,y¯兲 that exits ⍀ in n steps 共see Fig.2兲:
p⍀共n兲 ⯝ x¯y¯ ⯝ ᐉ共n兲+1⯝ 共n−2/共␥−1兲兲+1= n−共␥+3兲/共␥−1兲. 共24兲 Thus we have an estimate of the power-law decay of the survival probability as a function of the intermittency param- eter␥ as
p⍀共n兲 ⬃ n−共␥+3兲/共␥−1兲, 共25兲 and for the waiting time distribution as well关Eq. 共9兲兴:
⍀共n兲 ⬃ n−共3␥+1兲/共␥−1兲. 共26兲 In view of the argument we earlier mentioned关see Eq. 共11兲兴, the estimate of Eq. 共25兲 suggests the same decay law for 共auto兲correlation functions,
-1 -0.5 0 0.5 x 1
-0.5 -0.25 0 0.25 0.5
y
FIG. 1.共Color online兲 Portion of phase space of the map of Eq.
共2兲 for ␥=3 close to the marginal fixed point, together with the graph of its unstable manifold关continuous 共red兲 line兴.
FIG. 2. 共Color online兲 A few ⍀n 共once we set ⍀ as the first quadrant xⱖ0, yⱖ0兲.
CAA共n兲 ⬃ n−共␥+3兲/共␥−1兲. 共27兲 The next section will present several numerical simulations concerning these quantities.
IV. ASYMPTOTIC DECAYS
We start by considering the survival probability: Fig. 3 shows two examples of numerically computed p⍀共n兲. The numerical data exhibit an excellent agreement with analytic estimates over a wide range of intermittency parameters, as shown in Fig.4, which also provides clear indications of the validity of our estimate for the asymptotic decay of the wait- ing time distribution.
We already mentioned that various arguments support the expectation that correlation functions should decay as the
survival probability关Eq. 共27兲兴, so we proceed to scrutinize this prediction by running extensive direct numerical simu- lations on autocorrelation functions; as we are dealing with an ergodic 共and mixing 关12兴兲 system, autocorrelation func- tions can be evaluated in terms of phase space averages共in- stead of temporal averages兲:
CAA共n兲 =冕Md共z兲A共T␥nz兲A共z兲 −冉冕Md共z兲A共z兲冊2,
共28兲
where A is a smooth function on the phase spaceM andis the invariant Lebesgue measure. From a numerical point of view, it is known that often Monte Carlo evaluation of Eq.
共28兲 cannot be pushed too far, as the statistical error is of order 1/冑N in the number of initial conditions. Generally we also expect an 共exponential兲 transient in the decay 关10,31,38兴; the transient time t¯ might depend on both ␥and the choice of phase space function A关6兴. We also remark that the smoothness of the function A plays a fundamental role;
as a matter of fact, we may obtain arbitrarily slow correlation decay even for Anosov maps by using integrable nonsmooth functions 关39兴, or, from a mathematical point of view, we may have that the degree of smoothness determines the es- sential spectral radius of the Perron-Frobenius operator关40兴.
We performed explicit calculations of the autocorrelation function for different values of the intermittency parameter␥ and for different observables. We obtained the best results 共i.e., cleanest curves and shortest time t¯兲 for large values of␥ and by using A共x,y兲=e−y2. The choice of the smooth func- tion to use is quite arbitrary; we looked for a function not vanishing in correspondence with the marginal fixed point 共as suggested, for example, in 关6兴兲 and by the special choice of a Gaussian depending on a single variable we could save computational time 共see also 关21兴兲. In Figs. 5 and 6 we present results for␥= 3 and 10.
100 102 104 106 n
10-9 10-6 10-3 pΩ(n)
FIG. 3. 共Color online兲 Survival probabilities for ␥=3 共lower curve兲 and 10 共upper curve兲 together with the power-law decays predicted by Eq. 共25兲. We used 1012initial conditions and set ⍀
=关−1/2,1/2兴2.
1 2 3 4 5 6 γ 7
4 8 12
waiting time distribution survival probability
FIG. 4. 共Color online兲 Exponents of power-law decay for sur- vival probability p⍀共n兲 and waiting time distribution ⍀共n兲: lines refer to analytic estimates of Eq.共25兲 共upper兲 and Eq. 共26兲 共lower兲:
circles come from numerical simulations of the waiting time distri- bution, diamonds from survival probability simulations.
100 101 102 n
10-8 10-6 10-4 10-2
C(n)
FIG. 5. 共Color online兲 Autocorrelation function for the observ- able e−y2and ␥=3. We used 2⫻1010initial conditions共uniformly distributed in the torus cell兲. The predicted decay is shown by the 共red兲 straight line, which has a slope −3.
The case reported in Fig. 5 is important because for a similar 2D mapping共having the same intermittent exponent
␥= 3兲 it was proved in 关12兴 that the decay is faster than n−2, and a class of cross correlation was constructed indicating that the bound is optimal; while survival probability data for the same exponent indicate clearly that the decay we predict 共n−3兲 is numerically well reproduced, data for correlations are less conclusive共see Fig. 5兲. In general, numerical data are more difficult to interpret for low values of␥, and nu- merical fits tend to lie below the predicted exponents 共see Fig.7兲, while for larger values of␥ the accordance with our estimates is much better. Moreover, the agreement improves when the number of initial conditions is increased, so that we expect the discrepancy to be essentially due to numerical limitations共Fig.7兲. Further indications of the similarity be-
tween correlations and survival probability decays will be provided in Sec. VI, when we consider the role of stochastic perturbations.
V. TRANSPORT PROPERTIES
In order to study transport properties, we have to abandon the dynamics restricted to the torus 关Eq. 共2兲兴 by lifting the map in an appropriate way.
For the sake of clarity we introduce a third dimension共say z兲 to describe the motion through elementary cells. We then assign a jumping number +1 to the points belonging to the first quadrant, −1 to the points belonging to the third quad- rant, and 0 to all the other points. This means that a laminar phase of length n will correspond to a jump in the positive direction of n elementary cells Fig.8.
Formally, the lift is given by the following formula:
¯T
␥共x,y,m兲 =再共T␥共x,y兲,m + sgn共x兲兲 for xy ⱖ 0,共T␥共x,y兲,m兲 for xy⬍ 0,冎
共29兲 where m is an integer variable.
Considering successive entrances into the laminar regions as uncorrelated, we can approximate the diffusion process by a continuous time random walk 共CTRW兲 关41,42兴, with the probability distribution of the laminar phases given by the waiting time distribution⍀共n兲. In particular, by making use of the CTRW approach, it is possible to characterize the transport properties of the process in terms of the set of mo- ments of the diffusing variable关43兴:
具兩z共n兲 − z共0兲兩q典 ⯝ n共q兲, 共30兲 which is expected to present a sort of phase transition关25兴.
A. Continuous time random walk approach
For completeness, we briefly recall the standard theory of continuous time random walks关41–45兴. Generally speaking, a CTRW is a stochastic model in which steps of a simple random walk take place at times ti, following some waiting time distribution. Mathematically, it is asserted that a CTRW is a共non-Markovian兲 process subordinated to a random walk
100 101 102 n 103
10-4 10-3 10-2 10-1 100 C(n)
FIG. 6. 共Color online兲 Autocorrelation function for the observ- able e−y2and␥=10. We used 109initial conditions共uniformly dis- tributed in the torus cell兲. The predicted decay is shown by the 共red兲 straight line, which has a slope −13/9.
3 4 5 6 7 8 9 γ 10
1 2 3 4
4 6 8 10
correlations decay exponent correlations decay exponent (more initial conditions) analytical estimate
FIG. 7. 共Color online兲 Numerical values of power-law decay exponents for the autocorrelation function for the observable e−y2 关共black兲 circles and 共blue兲 triangles兴 together with the analytical estimates关full 共red兲 line兴. Circles were obtained by using 2⫻109 initial conditions while the triangles were obtained for␥=2.5 by using 5⫻1010initial conditions, in the range 3ⱕ␥ⱕ6 by using 2
⫻1010 initial conditions, and in the range 6.5ⱕ␥ⱕ10 by using 1010initial conditions.
FIG. 8. 共Color online兲 Jumping numbers and lifted map.
共r,兲, the probability density function for moving a dis- tance r during a time intervalin a single motion event; the dependence upon r and can be either decoupled 关i.e.,
共r,兲=共r兲㜷共兲兴 or coupled 关e.g.,共r,兲=共兩r兩兲㜷共兲兴.
The object we are interested in is the probability density function P共x,t兲 of being at x at time t; indeed it allows us to obtain the full spectrum of transport moments, through the formula
具x共t兲q典 = 共i兲qL−1冉kqq兩P˜ 共k,u兲兩ˆ k=0冊, 共31兲
where L−1is the inverse Laplace transform and P˜ˆ
denotes the Fourier-Laplace transform, k being the Fourier variable and u the Laplace variable.
Let us introduce 共x,t兲, the probability density function of passing through 共x,t兲, even without stopping at x, in a single motion event,
共x,t兲 = P共兩x兩t兲冕t
⬁
d冕兩x兩⬁dr共r,兲. 共32兲
P共x,t兲 is given by the sum of the probabilities of passing through 共x,t兲, even without stopping at x, in one or more motion events,
P共x,t兲 =共x,t兲 +冕−⬁
⬁
dx
⬘
冕0 td共x
⬘
,兲共x − x⬘
,t −兲 + ¯ . 共33兲 By performing the convolution integrals, the Fourier-Laplace transform of this expression assumes the closed formPˆ
˜ 共k,u兲 = ˜ 共k,u兲ˆ 1 −˜ 共k,u兲ˆ
. 共34兲
A special realization of the CTRW is the so-called velocity model 关42兴: a particle moves at a constant velocity for a given time, then stops and chooses a new direction and a new time of sojourn at random according to given probabili- ties.
Our case belongs to this class, with velocities being⫾1, and
共兩r兩兲 =1
2␦共兩r兩 −兲 and 㜷共兲 ⬃−g, 共35兲 so that
共r,兲 ⬃1
2␦共兩r兩 −兲−g and 共x,t兲 ⬃1
2␦共兩x兩 − t兲t−g+1, 共36兲 where g =3␥−1␥+1, 㜷共兲 being given by the waiting time distri- bution⍀共n兲 of Eq. 共26兲.
By making use of the Tauberian theorems for the Laplace transform 关46兴 and by applying the CTRW formalism 关45兴 we derive, through Eqs. 共31兲 and 共34兲 the full spectrum of
transport moments. The obtained spectrum of moments 共more precisely, from the previous calculation it is possible to obtain only the even moments, and then to infer that a similar law drives also the behavior of the absolute value of odd moments兲 is
具兩z共n兲 − z共0兲兩q典 ⯝ n共q兲, 共37兲 where the exponent共q兲 has a piecewise linear behavior
共q兲 =再qq −/2␣ if qif q⬍ 2⬎ 2␣␣,,冎 ␣=␥␥+ 3− 1, 共38兲
in agreement with numerical results shown in Fig. 9. The transition at q = 2␣ in the momenta spectrum of Eq. 共38兲 is general in systems manifesting anomalous diffusion关25兴.
As an outcome, we have that共anomalous兲 transport prop- erties fully agree with the power laws we deduced for the waiting time distribution关Eq. 共26兲兴.
VI. NOISE EFFECTS
In order to better understand the link between correlation decay and time statistics 共and to verify, if not rigorously prove, it兲, we consider the effects of a small stochastic per- turbation. The behavior, under the modified dynamics, of the survival probability and of correlation decay may provide further information about the interconnection between them.
At the same time, the dynamical effects of a superimposed noise are interesting by themselves共see Refs. 关16,37,47,48兴兲.
The general expectation is that small-scale stochasticity blurs the behavior in the vicinity of the parabolic fixed point, enhancing the chaotic character of the motion; one then ex- pects a transition to an exponential decay of the survival probability and correlation functions: this intuition is cor- roborated by numerical experiments, reported in Fig.10. We perturb the system by introducing a stochastic noise, adding at each iteration of the map a random vector of the type
=共1,2兲 withiindependent identically distributed共i.i.d兲 in
0 2 4 6 8 q 10
0 2 4 6
ν(q) γ=5γ=11/3
γ=7
FIG. 9. 共Color online兲 Spectrum of the transport moments for different values of the parameter␥. Lines correspond to theoretical predictions of Eq.共38兲; symbols correspond to numerical simula- tions: circles␥=3, diamonds ␥=11/3, squares ␥=5, and triangles
␥=7.
共−⑀,⑀兲. The effects of the perturbation are expected to be dominant in the region of phase space共which will depend on the noise intensity ⑀ and on the stickiness parameter ␥兲 where the deterministic step is small compared to noise.
Based on this assumption, we are able to give an analytical estimate of the crossover time tc共defined as the characteristic time of the asymptotic exponential decay兲, which is in good agreement with numerical simulations.
Following 关47兴, we divide the phase space into two complementary sections: one small region surrounding the fixed point, where the dynamics is dominated by stochastic diffusion, and its complement, far from the fixed point, where the dynamics is dominated by the deterministic cha- otic motion.
Now the main problem is a proper definition of the boundary of such a partition. The criterion suggested in关47兴 consists in defining具Tdeterministic典 as the mean exit time from the region, say ⌬, evaluated with the assumption that the dynamics is only due to the deterministic motion of the un- perturbed map; then we define具trandom典 as the mean exit time evaluated as if the dynamics were only due to diffusion.
The border of the region is determined by the constraint 具Tdeterministic典 ⯝ 具trandom典. 共39兲 We restrict the analysis to the first quadrant and choose as region⌬ the area defined by the survival probability p⍀共k兲, for some time k. In this way具Tdeterministic
k 典 is given by 具Tdeterministic
k 典 = 1
p⍀共k兲n兺ⱖk⬁ 共⍀n兲共n − k + 1兲. 共40兲 Performing the calculation by substituting the probabilities obtained in the previous sections关Eqs. 共7兲 and 共8兲兴, we get
具Tdeterministic
k 典 ⯝ k. 共41兲
The calculation of具trandom典 is performed as follows. First we approximate p⍀共k兲 as in Sec. III with the rectangular regions of Eq.共24兲, say xkyk; this rectangular region can be exited
along the x or y direction, independently共thanks to the par- ticular form of our noise兲, so that
具trandom
k 典 = min共具trandomk,x 典,具trandomk,y 典兲. 共42兲 From the diffusion equation describing stochastic dynamics 共see 关47,49兴兲, we get
具trandomk,z 典 ⯝zk2
⑀2, 共43兲
where z can be either x or y. Remembering that close to the origin the motion follows dynamics on the unstable manifold and from Eqs.共23兲 and 共24兲 we get
xk⬃ ᐉk⬃ k−2/共␥−1兲,
yk⬃ xk⬃ k−共␥+1兲/共␥−1兲. 共44兲 By substituting Eq.共44兲 in Eq. 共43兲, we obtain
具trandomk 典 ⯝ min冉共k−2/共␥−1兲⑀2 兲2,
共k−共␥+1兲/共␥−1兲兲2
⑀2 冊. 共45兲
Finally, from Eq.共39兲,
k⬃共k−共␥+1兲/共␥−1兲兲2
⑀2 . 共46兲
Writing the characteristic time of the exponential decay as a function of the noise strength⑀and the intermittency param- eter␥, we derive an estimate for the crossover time tc:
tc= k⬃⑀−共␥兲⬃⑀−共2␥−2兲/共3␥+1兲. 共47兲 Eventually we compare the behavior of 共␥兲 with the nu- merical results, for the survival probability and for the cor- relation decay.
Numerical results are obtained in the following way. For each value of␥we consider either the survival probability or the correlation functions for several values of⑀共see Fig.10兲:
the exponential decay rates, which we consistently observe, are then fitted according to a power law in⑀. The nice re-
0 50 100 150 n 200
10-4 10-2 100 C(n)
FIG. 10. 共Color online兲 Correlation decay for noisy dynamics, for ␥=10 and various values of ⑀ 共from top to bottom: ⑀=0.0, 0.005, 0.010, 0.015, 0.020, 0.025, 0.030, 0.035, 0.040, 0.045, and 0.050兲. Each correlation function is computed by considering 3
⫻109initial conditions.
3 4 5 6 7 8 9 γ 10
0.4 0.5 0.6 β(γ)
survival probability correlations theoretical prediction
FIG. 11. 共Color online兲 Comparison between theory and nu- merical results for the function共␥兲 of Eq. 共47兲. Circles, numerical data for the survival probability; squares, numerical data for the correlations; dashed line, theoretical prediction.
good agreement with the analytical result of Eq. 共47兲 共see Fig.11兲. strengthen our belief that the two distributions in the unperturbed case should be driven by the same exponent.
VII. CONCLUSIONS
We considered a one-parameter family of area-preserving maps which generalizes intermittent behavior in two dimen- sions共while admitting a smooth invariant measure兲. This is a paradigmatic example of weak chaos for Hamiltonian maps even if the measure of the “regular” portion of the phase space is zero共only a parabolic fixed point兲: the parameter␥ controls sticking of trajectories to the fixed point. By consid- ering the motion along the unstable manifold close to the fixed point, we are led to estimates for power-law decays of the survival probability and waiting time distribution. Nu- merical computations of survival probabilities and residence time statistics show a very close agreement with predicted
predictions. Our results are further supported by considering transport properties for a lift of the map, and by studying dynamical effects induced by a stochastic perturbation.
While we think that a complete quantitative understand- ing of weak chaos in more than one dimension is still in its infancy, our results fortify the connection of local features 共motion along the unstable manifold close to the fixed point兲 to global dynamical quantities, such as mixing speed, trans- port properties, and response to stochastic perturbations.
ACKNOWLEDGMENTS
This work has been partially supported by MIUR-PRIN projects⬙Transport Properties of Classical and Quantum Sys- tems⬙ and ⬙Quantum Computation with Trapped Particle Ar- rays, Neutral and Charged.⬙ We thank Raffaella Frigerio and Italo Guarneri for sharing their early work on the problem, and Carlangelo Liverani and Sandro Vaienti for useful dis- cussions and information.
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