Controlling Escape in the Standard Map
Gabriel I. Díaz, Matheus S. Palmero, Iberê Luiz Caldas, Edson D. Leonel
CControlling Escape in the Standard Map
Gabriel I. D´ıaz, Matheus S. Palmero, Iberˆe Luiz Caldas, and Edson D. Leonel Instituto de F´ısica, IFUSP - Universidade de S˜ao Paulo, Rua do Mat˜ao,Tr.R 187, Cidade Universit´aria, 05314-970, S˜ao Paulo, SP, Brazil Departamento de F´ısica, UNESP - Univ Estadual Paulista,Av. 24A, 1515, Bela Vista, 13506-900, Rio Claro, SP, Brazil (Dated: September 24, 2020)We investigate how the diffusion exponent is affected by controlling small domains in the phasespace.The main Kolomogorov-Arnold-Moser - KAM island of the Standard Map is considered tovalidate the investigation. The bifurcation scenario where the periodic island emits smaller resonanceregions is considered and we show how closing paths escape from the island shore by controllingpoints and hence making the diffusion exponent smaller. We notice the bigger controlled area thesmaller the diffusion exponent. We show that controlling around the hyperbolic points associatedto the bifurcation is better than a random control to reduce the diffusion exponent. The recurrenceplot shows us channels of escape and a control applied there reduces the diffusion exponent.
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I. INTRODUCTION
It is known that stickiness plays an important role re-garding transport properties in several areas of physics,as fluids [1, 2], plasma dynamics [3, 4] and celestial me-chanics [5]. For the generic Kolmogorov-Arnold-Moser(KAM) scenario [6–9], strong fluctuations are observeddue to the presence of Cantori [10] acting as a partialbarrier to the transport of particles. A way to charac-terize the stickiness phenomena is by using the diffusionexponent [11, 12], whose value allows us to distinguishbetween anomalous diffusion, when the exponent is dif-ferent from 1 /
2, and normal diffusion, when the exponentis 1 / II. DESCRIPTION OF THE MODEL AND THEMETHOD
The model under study is the Standard Map [16, 17],which describes the motion of a particle constrained tomove in a ring. The particle is kicked periodically by anexternal field. The dynamics of the Standard Map (SM)is described by a mapping T SM ( p n , q n ) = ( p n +1 , q n +1 )that gives the position and momentum just before the( n + 1) th kick T SM : (cid:26) p n +1 = [ p n + k sin( q n )] mod (2 π ) q n +1 = [ q n + p n +1 ] mod (2 π ) , (1)where the parameter k controls the intensity of the non-linearity of the mapping. Since the determinant of theJacobian matrix is the unity, the mapping preserves thearea on the phase space. A plot of the phase space of SMis shown in Fig. 1. FIG. 1: Plot of the phase space for the Standard Map con-sidering the control parameter k = 1. One sees from Fig. 1 a coexistence of chaotic domainsaround regular ones, the regions of regular motion are a r X i v : . [ n li n . C D ] S e p generally formed by invariant curves arranged in com-plex structures called Kolmogorov-Arnold-Moser (KAM)islands [6–9].We follow the procedure presented in [12] to measurethe diffusion exponent around KAM islands. The proce-dure is based in the studies of Scafetta and Grigolini [11]to determinate the diffusion exponent by using entropymeasurements. A Brief description of the method is thefollowing, (for more details see [12]):1. Choose a two dimensional I (horizontal divisions) × I (vertical divisions) grid with help of the relation10 × diffusion rangefluctuation size ∼ grid divisions .2. Set an ensemble of initial conditions around theKAM island to apply the mapping.3. Apply the mapping 1 to the ensemble of orbits. Ateach iteration construct an histogram [ h ij ] countinghow many points are inside each grid box. Thenmeasure the entropy by means of the equation S = − (cid:80) Ii =1 (cid:80) Ij =1 h ij ln ( h ij ).4. After some iterations search of a interval in timewhere the entropy grows linearly[21] with ln ( n ).5. Use the equation S = A + δ ln ( n ) to calculate thediffusion exponent δ .A variant of the method where we control an small areain phase space has to consider a conditional statement if in the SM. We consider additional steps in the previousmethod:2 (cid:48) Choose a control area in phase space [22].3 (cid:48)
At every iteration for each element of the ensembleof orbits ask if the orbit has entered in the controlarea. If it does it, re-initiate the orbit choosing anew initial condition randomly from our ensembleof orbits at n = 0. III. PREVIOUS RESULTS
Similarly to [12] we measure the area of the main islandin the SM and compare it to the diffusion exponent. Toget the area of the main island separating chaos fromregular behavior in the phase space we see what regionschaotic orbits visit and what regions are not visited. Weconsider values of the parameter k where a set of chaoticorbits can visit all the chaotic sea provided enough timeis allowed. We divide the phase space into grid of boxesand ask if any of the chaotic orbits visited a given gridelement. If so we mark it as yellow indicating a boxof chaotic behavior. If none orbit has landed in a boxwe paint is as a black characterizing a box of regularbehavior.The main islands center is an elliptic fixed point ofcoordinates q = π , p = π . It is located inside a box of regular domain, hence black. Selecting all the boxes ofblack color that are connected, i.e., that are first nearestneighbors, starting from that one, we are able to find anapproximation of the main island area by a black regionof simple connected boxes. We call to the total area ofthis boxes divided by total area, (2 π ) , the normalizedarea. FIG. 2: Plot of the diffusion exponent (red line) and the nor-malized area of the main island (blue line) vs. the parameter k . The region between vertical lines corresponds to the inter-val k ∈ [1 . , .
46] (Colors on line).
It is possible to see in Fig. 2 that whenever the areadecreases abruptly the diffusion exponent increases. Fur-thermore the area grows, while the exponent decreases,until a critical value when the area abruptly decreasesonce more, with its corresponding increase in the diffu-sion exponent. We mark two values of k , region betweenthe vertical lines in Fig. 2, at this interval happens thelargest decrease in the area, for the values of k consid-ered.We see in Fig. 3 the transition marked in Fig. 2. Whenpassing from k = 1 .
45 to k = 1 .
46 the main island ejects aresonance of smaller islands, therefore reducing the areaof the main island. However, each ejected island has anelliptic periodic point in the middle, and by the Poincar´eBirkhoff theorem [18, 19] exists their corresponding hy-perbolic fixed points pair. According to the conjecturepresented in Ref. [12] the action of the stable and un-stable manifolds of such hyperbolic points is responsiblefor the changes in diffusion behavior since they providelarge channels to escape from the main island [13–15].
IV. NUMERICAL RESULTS
For a given value of k from the SM. Eq. 1, we searchthe hyperbolic points, associated to the smaller islandsejected, and control a circular area around them [23]when we calculate the diffusion exponent.In table I we see the change in diffusion exponent whenthe area around the hyperbolic points is controlled. Fordifferent values of the parameter k we have different peri- k = 1 . k = 1 . k Period δ (with control) δ (no control)1 .
46 6 0 . ± .
001 0 . ± . .
47 6 0 . ± . . ± . .
48 6 0 . ± .
003 0 . ± . .
85 16 0 . ± .
002 0 . ± . .
86 16 0 . ± .
005 0 . ± . .
87 16 0 . ± .
002 0 . ± . .
96 10 0 . ± .
009 0 . ± . .
97 10 0 . ± .
01 0 . ± . .
98 10 0 . ± .
007 0 . ± . r = 0 . ods in the resonance islands, but in every case the diffu-sion exponent become smaller than without control. Thevalues of k were chosen near three decays in the normal-ized area from Fig. 2.We examine now the effect of the control area, wechange the radius of the circular ball around the hy- perbolic points and see how does the diffusion exponentchanges, see Fig. 4 FIG. 4: Plot of the diffusion exponent against the radius ofthe controlling ball for two values of the control parameter,namely k = 1 .
46 and k = 1 . It is possible to see in Fig. 4 that a bigger value radiusof control area translates into a smaller value of the diffu-sion exponent, meaning that more orbits are re-initiatedsince it is more likely than a orbit enters in the controlarea. If we change the position of the control area, thediffusion exponent also changes, as can be seen in Figs.5 and 6. We we notice the diffusion exponent seems tobe more affected when the control area is around an hy-perbolic point (red point) than when is around anotherrandom point (blue points).
FIG. 5: Plot of diffusion exponent for k = 1 .
46, the positionof the control point is indicated with blue points with itsdiffusion exponent next to them. The red point indicatesthe hyperbolic fixed point. The green points are the initialensemble of conditions. The gray dots in the backgroundshows the Standard Map portrait for that value of k (Colorson line). Finally we search if there is a connection between thediffusion exponent and the recurrence in phase space. For
FIG. 6: Plot of the diffusion exponent for k = 1 .
96, the posi-tion of the control point is indicated with blue points with itsdiffusion exponent next to them. The red point indicates thehyperbolic fixed point. The green points are the initial ensem-ble of conditions. The gray dots in the background shows theStandard Map portrait for that value of k (Colors on line). an initial ensemble of conditions we calculate the numberof times any of them enters a small box of a grid. Ac-cording to the number of times any point entered, a boxis assigned with a given color, see Fig. 7. FIG. 7: Recurrence plot for the initial ensemble of orbits(green points). Every box in the grid has a color assigneddepending on the number of points visited them, yellow indi-cates a larger concentration of points, black a small concen-tration (Colors on line).
In Fig. 8 wee see a color grid with the value of thediffusion exponent when the control area is in each gridbox. We notice the diffusion exponent near the hyper-bolic point it is smaller than in other regions. Even more,this figure has similar structure as shown Fig. 7 confirm-ing small channels were the control is better than awayfrom those channels.
FIG. 8: Diffusion exponent for the initial ensemble of orbits(green points). Every box in the grid has a color assigneddepending on the value of the diffusion exponent when thecontrol area is in the grid box (Colors on line).
V. CONCLUSION
We calculated the diffusion exponent around the mainisland in the Standard map. We focused in the bifurca-tion scenario were the main KAM island emits smallerresonance islands. For different values of the nonlinearparameter k we showed that when considering an smallcontrol area, around hyperbolic points associated to theresonances, it is possible to change the diffusion expo-nent making it smaller. This is mainly due to the factthat the control action closes paths to escape from themain island. We also showed that the bigger the controlarea the smaller the diffusion exponent, since it is moreprobable that an orbit will enter the control area.We showed that changing the position of the controlpoint the diffusion exponent changes, for random controlpoints this diffusion exponent is not much affected aswhen we consider the hyperbolic point.Finally, when comparing Fig. 7 and Fig. 8, we sawthat a recurrence plot shows channels of escape, placesof high recurrence, where a controlled point made thediffusion exponent smaller. ACKNOWLEDGMENTS
G.I.D. thanks the fellowship from National Councilfor Scientific and Technological Development (CNPq).M.S.P., I.L.C., and E.D.L. acknowledge Sao Paulo Re-search Foundation (FAPESP) Grants No. 2018/03000-5, No. 2018/03211-6, and No. 2019/14038-6. I.L.C.and E.D.L. acknowledge National Council for Scien-tific and Technological Development (CNPq) Grants No.300632/2010-0 and No. 301318/2019-0. [1] P. J. Morrison, Physics of Plasmas vol. 7, 2279, 2000.[2] D. del-Castillo-Negrete, Phys. Plasmas vol. 7, 1702, 2000.[3] M. Roberto, E. C. Silva, I. L. Caldas, R. L. Viana, Phys.Plasmas vol. 11, 214, 2004.[4] C. G. L Martins, M. Roberto, I. L. Caldas, IEEETRANSACTIONS ON PLASMA SCIENCE vol. 42,2764, 2014.[5] G. Contopoulos, M. Harsoula, N. Voglis and R.Dvorak,J. Phys. A: Math. Gen. vol. 32, 5213, 1999.[6] A. N. Kolmogorov “On the Conservation of Condition-ally Periodic Motions under Small Perturbation of theHamiltonian”. Dokl. Akad. Nauk. SSR., vol. 98, pp 2,1954.[7] J. Moser, “On invariant curves of area-preserving map-pings of an annulus”. Nachr. Akad. Wiss. II, pp. 1, 1962.[8] V. I. Arnold, “Proof of a theorem of A. N. Kolmogorov onthe preservation of conditionally periodic motions undera small perturbation of the Hamiltonian”. Uspekhi Mat.Nauk ,vol. 18, 1963 (English transl.¿ Russ. Math. Surv,vol. 18)[9] I. C. Percival, “A variational principle for invariant toriof fixed frequency”. Journal of Physics A: Mathematicaland General, vol. 12, 1979.[10] J. D. Meiss, Rev. Modern Physics vol. 64, 795, 1992.[11] N. Scafetta and P. Grigolini, “Scaling detection in timeseries: diffusion entropy analysis”. Physical Review E,vol. 66, pp. 036130, 2002.[12] G. I. D´ıaz, M. S. Palmero, I. L. Caldas, and E. D. Leonel,“Diffusion entropy analysis in billiard systems”. PhysicalReview E, vol. 100, pp. 042207, 2019.[13] Y. S. Sun, L. Zhou and J. L. Zhou, “The role of hyper- bolic invariant sets in stickiness effects”, in