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Fractional Standard Map

Mark Edelman∗,a,b,1, Vasily E. Tarasova,c

aCourant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USAbDepartment of Physics, Stern College at Yeshiva University, 245 Lexington Ave, New York, NY 10016, USA

cSkobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119991, Russia

Abstract

Properties of the phase space of the standard map with memoryare investigated. This map was obtained from akicked fractional differential equation. Depending on the value of the map parameter and the fractional order of thederivative in the original differential equation, this nonlinear dynamical system demonstrates attractors (fixed points,stable periodic trajectories, slow converging and slow diverging trajectories, ballistic trajectories, and fractal-likestructures) and/or chaotic trajectories. At least one type of fractal-like sticky attractors was observed.

Key words: discrete map, fractional differential equation, attractorPACS: 05.45.Pq, 45.10.Hj

1. Introduction

The standard map (SM) can be derived from the dif-ferential equation describing kicked rotator. The de-scription of many physical systems and effects (Fermiacceleration, comet dynamics, etc.) can be reduced tothe studying of the SM [1]. The SM provides the sim-plest model of the universal generic area preserving mapand it is one of the most widely studied maps. Thetopics examined include fixed points, elementary struc-tures of islands and a chaotic sea, and fractional kinetics[1, 2, 3].

It was recently realized that many physical systems,including systems of oscillators with long range inter-action [4, 5], non-Markovian systems with memory ([6]Ch.10, [7, 8, 9, 10, 11]), fractal media [12], etc., can bedescribed by the fractional differential equations (FDE)[6, 13, 14]. As with the usual differential equations,the reduction of FDEs to the corresponding maps canprovide a valuable tool for the analysis of the proper-ties of the original systems. As in the case of the SM,the fractional standard map (FSM), derived in [15] fromthe fractional differential equation describing a kickedsystem, is perhaps the best candidate to start a generalinvestigation of the properties of maps which can be ob-tained from FDEs.

∗Corresponding authorEmail address: edelman@cims.nyu.edu (Mark Edelman)

As it was shown in [15], maps that can be derivedfrom FDEs are of the type of discrete maps with mem-ory. One-dimensional maps with memory, in which thepresent state of evolution depends on all past states,studied previously [16, 17, 18, 19, 20, 21] were notderived from differential equations. Most results wereobtained for the generalizations of the logistic map.

In the physical systems the transition from integer or-der time derivatives to fractional (of a lesser order) in-troduces additional damping and is similar in appear-ance to additional friction [6, 22]. Accordingly, in thecase of the FSM we may expect transformation of the is-lands of stability and the accelerator mode islands intoattractors (points, attracting trajectories, strange attrac-tors). Because the damping in systems with fractionalderivatives is based on the internal causes different fromthe external forces of friction [22, 23], the correspond-ing attractors are also different from the attractors ofthe regular systems with friction and are called frac-tional attractors [22]. Even in one-dimensional cases[16, 17, 18, 19, 20, 21] most of the results were ob-tained numerically. An additional dimension makes theproblem even more complex and most of the results inthe present paper were obtained numerically.

2. FSM, initial conditions

The standard map in the form

pn+1 = pn−K sinxn, xn+1 = xn+pn+1 (mod 2π)(1)Preprint submitted to Physics Letters A November 24, 2009

can be derived from the differential equation

x + K sin(x)∞∑

n=0

δ

( tT− n)

= 0. (2)

By replacing the second-order time derivative in eq.(2) with the Riemann-Liouville derivative0Dαt one ob-tains a fractional equation of the motion in the form

0Dαt x+K sin(x)∞∑

n=0

δ

( tT−n)

= 0, (1 < α ≤ 2), (3)

where0Dαt x(t) = Dn

t 0In−αt x(t) =

1Γ(n − α)

dn

dtn

∫ t

0

x(τ)dτ(t − τ)α−n+1

(n−1 < α ≤ n), (4)

Dnt = dn/dtn, and0Iαt is a fractional integral. The initial

conditions for (3) are

(0Dα−1t x)(0+) = p1,

(0Dα−2t x)(0+) = b. (5)

The Cauchy type problem (3) and (5) is equivalent to theVolterra integral equation of the second kind [24, 25, 26]

x(t) =p1

Γ(α)tα−1 +

bΓ(α − 1)

tα−2

−KΓ(α)

∫ t

0

sin[x(τ)]∑∞

n=0 δ

(

τ

T − n)

dτ

(t − τ)1−α. (6)

Defining the momentum as

p(t) = 0Dα−1t x(t), (7)

and performing integration in (6) one can derive theequation for the FSM in the form (for the thoroughderivation see [26])

pn+1 = pn − K sinxn, (8)

xn+1 =1Γ(α)

n∑

i=0

pi+1Vα(n − i + 1)+

bΓ(α − 1)

(n + 1)α−2, (mod 2π), (9)

where

Vα(m) = mα−1− (m − 1)α−1

. (10)

Here it is assumed thatT = 1 and 1< α ≤ 2. The formof eq. (9) which provides a more clear correspondence

with the SM (α = 2) in the caseb = 0 is presented inSec. 4 (eq. (31)).

The second initial condition in (5) can be written as

0Dα−2t (0+) = lim

t→0+0I2−α

t x(t) =

limt→0+

1Γ(2− α)

∫ t

0

x(τ)dτ(t − τ)α−1

= b, (1 < α ≤ 2), (11)

which requiresb = 0 in order to have a solution boundedat t = 0 for α < 2. The assumptionb = 0 leads tothe FSM equations which in the limiting caseα = 2coincide with the equations for the standard map underthe conditionx0 = 0.

In this paper the FSM is taken in the form derived in[15] which coincides with (8) and (9) ifb = 0. It is alsoassumed thatx0 = 0 and the results can be compared tothose obtained for the SM withx0 = 0 and arbitraryp0.As a test, for the SM and for the FSM withα = 2 andthe same initial conditions numerical calculations showthat phase portraits look identical.

System of equations (8) and (9) can be consideredeither in a cylindrical phase space (x mod 2π) or in un-bounded phase space. The second case is convenient tostudy transport. The trajectories in the second case areeasily related to the first case. The FSM has no period-icity in p (the SM does) and cannot be considered on atorus.

3. Stable fixed point

The SM has stable fixed points at (0,2πn) for K <Kc = 4. It is easy to see that point (0, 0) is also a fixedpoint for the FSM. Direct computations using (8) and(9) demonstrate that for the small initial values ofp0

there is a clear transition from the convergence to thefixed point to divergence when the value of the param-eter K crosses the curveK = Kc(α) on Fig. 1a fromsmaller to larger values.

The following system describes the evolution of tra-jectories near fixed point (0, 0)

δpn+1 = δpn − Kδxn, (12)

δxn+1 =1Γ(α)

n∑

i=0

δpi+1Vα(n − i + 1). (13)

The solution can be found in the form

δpn = p0

n−1∑

i=0

pn,i

( 2Vαl

)i( VαlK2Γ(α)

)i, (n > 0), (14)

2

1 1.4 1.8 22

3

4

α

K

K=Kc(α)

1 1.4 1.8 21

3

5

7

9

α

S∞, I

∞

S∞

I∞

a b

0 100 200−5

0

5

n

Sn, I

n

In

Sn

0 1 x 104−1

0

1x 10−4

n

Sn

− S ∞

, I∞

−I

n

I − dashed line,S − solid line

c d

−2 −1 0 1 2

−1

0

1

2

p

x −0.1 0 0.1

0

0.5

1

1.5

p

x

e f

Figure 1: Stability of the fixed point(0,0): a). The fixed point is stablebelow the curveK = Kc(α); b). Values ofS∞ and I∞ obtained after20000 iterations of eqs. (22) and (23). Asα → 2 the valuesS∞ andI∞ increase rapidly. Forα = 1.999, S∞ ≈ 276 andI∞ ≈ 552 after20000 iterations; c). An example of the typical evolution ofS∞ andI∞ over the first 200 iterations for 1< α < 2. This particular figurecorresponds toα = 1.8; d). Deviation of the valuesS n and In fromthe valuesS∞ ≈ 2.04337 andI∞ ≈ 3.37416 forα = 1.8 during thefirst 20000 iterations (this type of behavior remains for 1< α < 2); e).Evolution of trajectories withp0 = 1.5+ 0.0005i, 0 ≤ i < 200 for thecaseK = 3,α = 1.9. The line segments correspond to thenth iterationon the set of trajectories with close initial conditions. The evolutionof the trajectories with smallerp0 is similar; f). 105 iterations on bothof two trajectories forK = 2, α = 1.4. The one at the bottom withp0 = 0.3 is a fast converging trajectory. The upper trajectory withp0 = 5.3 is an example of the ASCT in whichp100000≈ 0.042.

δxn =p0

Γ(α)

n−1∑

i=0

xn,i

( 2Vαl

)i( VαlK2Γ(α)

)i

, (n > 0), (15)

The origin of the terms in parentheses, as well as thedefinition

Vαl =

∞∑

k=1

(−1)k+1Vα(k) (16)

will become clear in Sec. 5. Eqs. (12) - (16) lead to thefollowing iterative relationships

xn+1,i = −

n∑

m=i

(n−m+1)α−1xm,i−1, (0 < i ≤ n), (17)

pn+1,i = −

n∑

m=i

xm,i−1, (0 < i < n) (18)

with the initial and boundary conditions

pn+1,n = xn+1,n = (−1)n, pn+1,0 = 1, (19)

xn+1,0 = (n + 1)α−1.

From (17) and (18) it is clear that the series (14) and (15)are alternating and it is natural to apply the Dirichlet’stest to verify their convergence. This can be done byconsidering the totals

S n =

n−1∑

i=0

xn,i

( 2Vαl

)i, (20)

In =

n−1∑

i=0

pn,i

( 2Vαl

)i

. (21)

They obey the following iterative rules

S n = nα−1 −2

Vαl

n−1∑

i=1

(n − i)α−1S i, S 1 = 1, (22)

In = 1−2

Vαl

n−1∑

i=1

S i. (23)

Computer simulations show that values ofS n and In

converge to the values (−1)n+1S∞ and (−1)n+1I∞ de-picted on Fig. 1b. Figs. 1c, 1d show an example of thetypical evolution ofS n andIn over the first 20000 itera-tions. It means that the condition of convergence ofδpn

andδxn is

VαlK2Γ(α)

< 1. (24)

3

Numerical evaluation of the equalityK = 2Γ(α)/Vαl

perfectly reproduces the curve on Fig. 1a obtained bythe direct computations of (8) and (9).

Because not only the stability problem (12) and (13),but also the original map (8) and (9), contains convolu-tions, the use of generating functions [27], which allowstransformations of sums of products into products ofsums, could be utilized in the investigation of the FSMand some other maps with memory. As an example, inthe particular case of the stability problem (12) and (13),the introduction of the generating functions

Wα(t) =KΓ(α)

∞∑

i=0

[(i + 1)α−1 − iα−1]ti, (25)

X(t) =∞∑

i=0

δxiti, (26)

P(t) =∞∑

i=0

δpiti, (27)

leads to

X(t) =p0Wα(t)

Kt

1− t(

1− Wα(t)) , (28)

P(t) = p01+ Wα(t)

1− t(

1− Wα(t)) . (29)

Now the original problem is reduced to the problem ofthe asymptotic behavior att = 0 of the derivatives ofthe analytic functionsX(t) andP(t), which is still quitecomplex and is not considered in this article.

In the region of the parameter space where the fixedpoint is stable, the fixed point is surrounded by a finitebasin of attraction, whose widthW depends on the val-ues ofK andα. For example, forK = 3 andα = 1.9the width of the basin of attraction is 1.6 < W < 1.7.Simulations of thousands of trajectories withp0 < 1.6performed by the authors, of which only 200 (with1.5 < p0 < 1.6) are presented in Fig. 1e, show onlyconverging trajectories, whereas among 200 trajecto-ries with 1.6 < p0 < 1.7 in Fig 2a there are trajecto-ries converging to the fixed point as well as some tra-jectories converging to attracting slow diverging trajec-tories (ASDT), whose properties will be discussed inthe following section. Trajectories in Fig. 1e convergevery rapidly. In the caseK = 2 andα = 1.4 in addi-tion to the trajectories which converge rapidly and AS-DTs there exist attracting slow converging trajectories(ASCT) (Fig. 1f).

−4 −2 0 2 4−30

0

30

p

x −2 0 20

10

20

p

x

a b

−2 0 2

0

4

8

p

x −1 0 14

8

12

p

x

c d

Figure 2: Phase space with ASDTs: a). The same values of parame-ters as in Fig 1e butp0 = 1.6 + 0.0005i; b). 200 iterations on trajec-tories with p0 = 4+ 0.02i, 0 ≤ i < 500 for the caseK = 2, α = 1.9.Trajectories converging to the fixed point, ASDTs withx = 0, andperiod 4 attracting trajectories are present; c). 2000 iterations on tra-jectories with p0 = 2 + 0.04i, 0 ≤ i < 50 for the caseK = 0.6,α = 1.9. Trajectories converging to the fixed point, period 2 and 3attracting trajectories are present; d).The same values ofparametersas in Fig 1e butp0 = 5+ 0.005i.

4. Attracting slow diverging trajectories (ASDT)

As it can be seen from Fig 2a, the phase portrait on acylinder of the FSM withK = 3 andα = 1.9 containsonly one fixed point and ASDTs approximately equallyspaced along thep-axis. This result corresponds to thefact that the standard map withK = 3 has only onecentral island. More complex structure of the standardmap’s phase space for smaller values ofK (for exam-ple for K = 2 andK = 0.6) can explain more complexstructure of the FSM’s phase space, where periodic at-tracting trajectories with periodsT = 4 (Fig. 2b),T = 2,andT = 3 (Fig. 2c) are present.

Each ASDT has its own basin of attraction (seeFig. 2d). Between those basins two initially close tra-jectories at first diverge, but then converge to the sameor different fixed point or ASDT.

Numerical evaluation shows that for ASDTs whichconverge to trajectories along thep-axis (x → xlim =

0) in the area of stability (which is the same as for thestability of the fixed point) the following holds (for largen see Fig. 3a)

pn = Cn2−α. (30)

The constant C can be easily evaluated for 1.8 < α < 2.Consider an ASDT withxlim = 0, T = 1, and 2πM,

4

2 4 61

2

3

4

log 10

p

log10

n

α=1.9,slope=0.1

α=1.5,slope=0.5

0 50−2

−1

0

1

p−

2πΓ(

α)n2−

α /(α−

1) n

a b

0 0.5 1 1.5x 10

50

0.01

(p+

0.38

−2π

Γ(α)

n2−α /(

α−1)

)/p

n 0 0.5 1 1.5

x 105

0

1 x 10−6

x+

2π(2

−α)Γ

(α)n

1−α /(

α−1)

/K

n

c d

Figure 3: Evaluation of the behavior of the ASDTs: a). Momenta fortwo ASDTs with xn ≈ 2πn in the unbounded space (in this exampleK = 2). The solid line is related to a trajectory withα = 1.9 and itsslope is 0.1. The dashed line corresponds to a trajectory with α = 1.5and its slope is 0.5; b). Deviation of momenta from the asymptoticformula for two ASDTs withxn ≈ 2πn in the unbounded space,α =1.9, andK = 2. The dashed line hasp0 = 7 and the solid onep0 = 6;c). Relative deviation of the momenta for the trajectories in Fig 3bfrom the asymptotic formula; d). Deviation of thex-coordinates forthe trajectories in Fig 3b from the asymptotic formula.

where M is an integer, constant step inx in the un-bounded space. Then Eq. (9) withb = 0 gives

xn+1 − xn =1Γ(α)

n∑

k=1

(pk+1 − pk)Vα(n − k + 1) (31)

+p1

Γ(α)Vα(n + 1),

For largen the last term is small (∼ nα−2) and the fol-lowing holds

n∑

k=1

(pk+1 − pk)Vα(n − k + 1) = 2πMΓ(α). (32)

With the assumptionpn ∼ n2−α it can be shown thatfor values ofα > 1.8 considered the terms in the lastsum with largek are small and in the series representa-tion of Vα(n − k + 1) it is possible to keep only terms ofthe highest order ink/n. Thus, (32) leads to the approx-imations

pn ≈ p0 +2πMΓ(α)n2−α

α − 1, (33)

xn ≈ −2πM(2− α)Γ(α)

K(α − 1)nα−1. (34)

In the caseK = 2,α = 1.9 Figs. 3b-3d show two trajec-tories withM = 1 (initial momentap0 = 6 andp0 = 7)

approaching an ASDT: the deviation from the asymp-totic (33) and (34) and the relative difference with re-spect to (33).

5. Period 2 stable trajectory

The SM has two stable points of the periodT = 2trajectory for 4< K < 2π with the property

pn+1 = −pn, xn+1 = −xn. (35)

The same points persist in the numerical experimentsfor the FSM (Fig 4a). These points are attracting mostof the trajectories with smallp0. Assuming the exis-tence of aT = 2 attracting trajectory, it is possible tocalculate the coordinates of its attracting points (xl, pl)and (−xl,−pl). In this case from (8) and (9)

pl =K2

sin(xl), (36)

xl =K

2Γ(α)sin(xl)

∞∑

k=1

(−1)k+1Vα(k) (37)

Finally, the equation forxl takes the form

xl =K

2Γ(α)Vαl sin(xl), (38)

where

Vαl =

∞∑

k=1

(−1)k+1Vα(k) (39)

and can be easily calculated numerically. From (38) thecondition of the existence ofT = 2 trajectory

K > Kc(α) =2Γ(α)

Vαl, (40)

is exactly opposite to (24). It is satisfied above the curveK = Kc(α) on the Fig. 1a. Forα = 2 (40) produces thewell-known conditionK > 4 for the SM. The resultsof calculations of thexl and pl for the casesK = 4.5,1 < α < 2 presented in Fig. 4b-d perfectly coincidewith the results of the direct computations of (8) and (9)with b = 0. After 1000 iterations presented in Figs. 4e,fthe values of deviations|pn − pl| and |xn − xl| are lessthan 10−7.

5

−2 −1 0 1 2−3

−1

1

3

p

x 0.5 1 1.5 2 2.51.5

2

2.5

pl

xl

a b

1 1.4 1.81.5

1.9

2.3

pl

α 1 1.4 1.80.5

1

1.5

2

xl

α

c d

0 400 800−1

0

1x 10−4

p−2.

1055

82

n 0 400 800−1

0

1x 10−4

x−1.

2105

62

n

e f

Figure 4: Period 2 stable trajectory: a). An example ofT = 2 attractorfor K = 4.5, α = 1.9. One trajectory withx0 = 0, p0 = 0.513; b). pl

of xl for the case ofK = 4.5; c). pl of α for the case ofK = 4.5; d). xl

of α for the case ofK = 4.5; e). pn − pl for the trajectory in Fig. 4a.After 1000 iterations|pn − pl | < 10−7; f). xn − xl for the trajectory inFig. 4a. After 1000 iterations|xn − xl | < 10−7.

Figure 5: Cascade of bifurcation type trajectories: a). 120000 itera-tions on a single trajectory withK = 4.5, α = 1.65, p0 = 0.3. Thetrajectory occasionally sticks to a CBTT but then always recovers intothe chaotic sea; b). 100000 iterations on a trajectory withK = 3.5,α = 1.1, p0 = 20. The trajectory very fast turns into a CBTT whichslowly converges to a fractal type area.

6

−2 −1 00

1

2

3 x 104

p

x −2 0 2−1000

−500

0

500

1000

1500

p

x

a b

Figure 6: Examples of phase space forK > 2π: a). An attractingballistic trajectory withK = 6.908745,α = 1.999, p0 = 0.7; b). Achaotic trajectory forK = 6.908745,α = 1.9.

6. Cascade of bifurcations type trajectories (CBTT)

Period 2 stable trajectories have limited basins ofattraction. Trajectories that don’t fall into those ar-eas reveal a diverse variety of properties, from periodtwo slow attracting trajectories to fractal type attractorsand cascade of bifurcations type trajectories (CBTT).Fig. 5a presents a single chaotic trajectory which sticksto the areas similar to the cascade of bifurcations whichare well-known for the logistic map. In Fig. 5b a sin-gle trajectory falls very rapidly into one of the attractingCBTTs. Because the bifurcation diagram of the logisticmap has fractal properties (see for example Chapter 2 in[28]), it is expected that the structure to which this tra-jectory slowly converges also possesses fractal features.

The properties of this type of attractors, as well as theproperties different types of observed during computersimulations chaotic, attracting, and ballistic trajectoriesfor K > 2π (see Fig 6) will be considered in the subse-quent article.

7. Fractional attractors and their stability

The problems of existence and stability of the frac-tional attractors for the systems described by the FDEswere addressed in a few recent papers. It was noticed in[22] that the properties of the fractional chaotic attrac-tors are different from the properties of the “regular”chaotic attractors and may have some pseudochaoticfeatures. The problem of existence of multi-scroll frac-tional chaotic attractors was considered in [29]. Theproblem of stability of the stationary solutions (fixedpoints for ODEs) of systems described by the fractionalODEs and PDEs was considered in [30, 31, 32]. Inthe above mentioned articles the equations contained theCaputo fractional derivatives, whereas in the present ar-ticle the Riemann-Liouville fractional derivative is used.This fact does not allow a direct comparison of the re-sults. The results [22, 29, 30, 31, 32] were supported

by a relatively small number of computations and thisis understandable, taking into account all the difficultiesof performing numerical simulations for the equationswith fractional derivatives.

The use of the FSM, which is equivalent to the origi-nal FDE, allows performing thousands of runs of simu-lations of the kicked fractional system with two param-eters:K andα. The FSM also allows making some an-alytic deductions and revealing some properties of thefractional attractors which were not reported before:

a). The stability of the fixed point (0,0) of the FSMis different not only from the stability of the fixed pointin the domain of the regular motion (zero Lyapunov ex-ponent) of the SM, but also from the stability of fixedattracting points of the regular (not fractional) dissipa-tive systems like, for example, the dissipative standardmap (Zaslavsky map) [33]. The difference is in the wayin which trajectories approach the attracting point. Inthe FSM this way depends on the initial conditions. Forexample, in Fig. 1f there are two trajectories approach-ing the same fixed point: one is fast spiraling into theattractor and the other is slowly converging.

b). Stable period 2 attracting trajectories exist onlyin the asymptotic sense - they do not represent any realperiodic solutions. If the initial condition is chosen ina period two stable attracting point, this trajectory willimmediately jump out of this point and where it will enddepends on the values ofK andα.

c). All the FSM attractors exist in the sense thatthere are trajectories which converge into those attrac-tors. But if an initial condition is taken on any of theattracting trajectories (except for the fixed point), theywill most likely not evolve along the same trajectory.

8. Conclusion

In this article properties of the phase space of theFSM were investigated. It was shown that islands ofregular motion of the SM in the FSM turn into at-tractors (points, attracting trajectories, and fractal-likestructures). Properties of the attracting fixed points, pe-riod two trajectories, ASCTs, and ASDTs were consid-ered. This consideration allows the description of theevolution of the dynamical variablex of the originalfractional dynamical system, a system described by theFDE reducible to the FSM. Physical interpretation ofthe momentum, defined through a fractional derivativefrom the variablex, is unclear.

The explanation of the CBTTs, which are interest-ing phenomena, requires further detailed investigation.Chaotic trajectories that spend some time near CBTTs,

7

which can be called sticky attractors in analogy to stickyislands of the SM, are good candidates for the investi-gation of anomalous diffusion. Transport was not con-sidered in this article. How general the properties ofthe phase space of the FSM are will become clear afterfurther investigations of different fractional maps, mapswith memory which can be derived from the FDEs, andparticular those suggested in [15], will be conducted.The fact that so many physical systems can be reducedto studying of the SM gives a hope that those physicalsystems which can be reduced to studying the FSM willbe found.

Acknowledgments

We express our gratitude to H. Weitzner for manycomments and helpful discussions. The authors thankA. Kheyfits for suggesting the use of generating func-tions to solve the FSM fixed point stability problem.This work was supported by the Office of Naval Re-search, Grant No. N00014-08-1-0121.

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