Table Of ContentA mechanism for stickness, dealing with extreme events
Taline Suellen Kruger1, Paulo Paneque Galuzio1, Thiago de Lima Prado1,
Sergio Roberto Lopes1,∗ Jos´e Danilo Szezech Jr2, and Ricardo Luiz Viana1
1Departamento de F´ısica, Universidade Federal do Parana´, Curitiba, PR, Brazil
2 Departamento de Matema´tica e Estat´ıstica, Univ. Est. de Ponta Grossa, Ponta Grossa, Parana´, Brazil
(Dated: January 13, 2015)
In this letter we study how hyperbolic and non hyperbolic regions in the neighborhood of a
resonant island perform a important role allowing or forbidding stickiness phenomenon around
islands in conservative systems. The vicinity of the island is composed by non hyperbolic areas
that almost prevent the trajectory to visit the island edge. For some specific parameters there are
tiny channels embedded in the non hyperbolic area that are associated to hyperbolic fixed points
5
presentintheneighborhood oftheislands. Suchchannelsallow thetrajectory tobeinjected inthe
1
innerportionofthevicinity. Whenthetrajectorycrossesthebarrierimposedbythenonhyperbolic
0
regions,itspendsalongtimetoabandonthesurroundingoftheisland,sincethebarrieralsoprevents
2
the trajectory to scape from the neighborhood of the island. In this scenario the non hyperbolic
n structures are the responsible for the stickiness phenomena, and more than that, the strength of
a the sticky effect. We reveal that those properties of the phase space allow us to manipulate the
J
existenceofextremeevents(andthetransport associated toit)responsible for thenonequilibrium
2 fluctuationof thesystem. Infact wedemonstrate thatmonitoring verysmall portions of thephase
1 space(namely≈4×10−4 %ofit)itispossibletogenerateacompletelydiffusivesystemeliminating
long time recurrences that result from thestickiness phenomenon.
]
D
PACSnumbers: 05.45.Pq,05.60.Cd,05.45.Ac
C
.
n
Understanding the transport properties of Hamilto- plasmafusiondevices[2,16],celestialmechanics[17]and
i
l niansystemisoneofthemajorobjectiveinthestatistical manyothersfoundapplicationsinstickinessoccurrences.
n
analysis of dynamical systems [1, 2]. Recent works have
[ Previous works have studied properties of the bound-
shown that for a large class of systems, including those
ariesbetweenregularandchaoticregionsfocusingmainly
1
presenting mixed phase space, the transport can not be
in the role of the stickiness in the dynamics [6, 18] and
v
treated considering just ergodic theory or random phase
8 alsothe cantoristructuresderivedfrom the breakof tori
1 approximation [4]. One class of such systems is the low [19]. When the system presents stickiness, the dynamics
7 dimensional (3/2 or 2 degrees of freedom) Hamiltonian of orbits in chaotic sea is observed to be intermittent,
2 system (1/2 degree of freedom corresponds to a periodi-
where after periods of chaotic motion away from the in-
0
caldisturbance). LowdimensionalHamiltoniansystems,
fluence of a sticky island, the system presents periods of
.
1 commonlypresentnonuniformphasespacescomposedby
almost regular motion. However a mechanism based on
0 regular (islands) and chaotic regions. The interface be-
thetopologyoftheislandvicinityandhowthattopology
5
tween these regions is far from being a smooth surface
1 affects an injected and subsequently ejected trajectory
and the dynamics near the edge between chaotic and
: from the sticky area is not completely known.
v regular regions is very complex and have been not well
i Here we study how characteristics of the topology of
X understood so far. The complexity comes mainly due
the system, namely hyperbolic and non hyperbolic re-
to the presence of stickiness in the boundaries of islands
r
gions on the neighborhood of a resonant island perform
a [5]. The sticky effect forcesa trajectoryinjected into the
an important role in order to establish the presence and
boundaryareatostayneartheboundaryforlongperiods
morethan that, thestrengthof thesticky effect. Weshow
oftime. Oneofthemainconsequencesofthisphenomena
thatthestickyeffectisassociatedwiththepresenceofin-
is the existence of power law tails in the Poincar´e recur-
jection channels related to the crossingof stable and un-
rence times making the systemto presentdistributionof
stablemanifoldsofhyperbolicfixedpointsinthevicinity
recurrence times displaying algebraically decay for long
of the island, allowing the trajectories to shift between
times rather than a exponential decay as expected for a
sticky and nonsticky areas of the phase space. We show
normaltransportsystem[3,4,7–11]. Insuch a situation
that the effectiveness of such channels to capture trajec-
these systems are characterized as out of equilibrium.
toriestothestickyareaiscloselyrelatedtothedegreeof
Some features of the kinetics of Hamiltonian systems hyperbolicity of the close surrounding areaof hyperbolic
are important to understand anomalous transport and fixedpointslocatedintheneighborhoodofanisland. Fi-
super diffusion. The phase space topology of theses sys- nallywemakeuseofthepresenceofthehyperbolicchan-
tems playsa crucialrolein the anomaloustransportand nelstoavoid(control)extremerecurrenceeventsandthe
in sticky phenomena [12, 13]. Many problems of science anomalous transport associated to it, resulting from a
such as particle advection in fluids [14, 15], transport in trajectory injection into the sticky area.
2
Here we characterize a hyperbolic region of the phase
space S as an ensemble for which the tangent phase
space splits continuously into stable (SM) and an un-
stable (UM) manifolds. SM and UM are invariant under
the system dynamics: infinitesimal displacements in the
stable (unstable) direction suffer exponentially decay as
time goes forward (backward) [20]. In addition, it is re-
quired that the angles between the stable and unstable
directions to be uniformly bounded away from zero. In
thisway,inordertoquantifythedegreeofnonhyperbol-
icity relatedto the phenomenawe describe in this letter,
let us consider an initial condition (p ,x ) and an unit
0 0
vector v, whose temporal evolution is given by
vn+1 =J(pn,xn)vn/||J(pn,xn)vn||, (1)
whereJ(pn,xn)istheJacobianmatrixofthemap. Forn
largeenough,v isparalleltotheLyapunovvectoru(p,x)
associatedtothemaximumLyapunovexponentλuofthe
FIG. 1. (color online) (a) Phase space for the kicked rotor
map orbit starting by (p ,x ). A backward iteration of
0 0 map,darkerregionsaroundthemainislandreflectstheeffect
the same orbit gives us a new vector vn that is parallel of the stickiness. (b) Phase space distribution of angle be-
to the direction s(p,x), the Lyapunov vector associated
tween unstable and stable manifold, dark (blue) tones mean
to the minimum Lyapunovexponentλs [21]. Forregions strong non hyperbolicity. (c) An example of the edge detec-
where λs < 0 < λu the vectors u(p,x) and s(p,x) are tionalgorithmusedheretocomputethevicinityoftheisland.
tangenttotheUMandSM,respectively,ofapoint(p,x). Intheinsetwedisplayamagnification showing detailsofthe
The (non)hyperbolic degreeof aregionS canbe stud- edge. (d) Probability distribution function of the angle be-
tween stable and unstable manifolds, ρ(θ).
iedcomputingthelocalanglesbetweenthetwomanifolds
θ(p,x)=cos−1(|u·s|), (2)
theresonantisland. Itisclearthatthemajorpartofthe
for (p,x) ∈ S [22]. So, θ(p,x) ∼ 0 denotes tangency vicinities of the island is composed by strong nonhyper-
betweenUMandSMat(p,x). Thegeneralmethodused bolic areas, represented by dark blue areas in Fig. 1-b
to calculate the θ angles follows the reference [22]. (tangencies between UM and SM). A small fraction of
Chaotic orbits of two dimensional mappings are often the vicinity is observedin red–yellow– greentones and
non hyperbolic since the SM and UM are tangent in in- corresponds to angles greater than 300 and responsible
finitely many points. As an illustration of this effect we for the weak hyperbolic part of the vicinities. The role
consider a periodically kicked rotor subjected to a har- of both areas will be clear later on in the text. In Fig
monic potential function - the Chirikov-Taylor map [2], 1-b we also define the angle φ = arctan(p/x), defined in
whose dynamics is two dimensional. The dynamics of a the interval [−π/2,+π/2], so each point in the vicinity
periodically kicked rotor can be described in a periodic of the island can be identified by a single number. The
phase space [−π,π)×[−π,π), whose discrete-time vari- 9+9(symmetric) hyperbolic fixed points present in the
ables pn and xn are respectively the momentum and the neighborhood of the main island are identified as black
angularpositionoftherotorjustafterthe nthkick,with bullets in Fig. 1-b. In order to make clear what we call
the dynamics given by the following equations: the vicinity of the island, Fig 1-c displays the result of
ouralgorithmforedgeislanddetection,detailsmagnified
pn+1 =pn+Ksin(xn),mod2π (3)
in the inset. Finally in Fig 1-d we plot the probability
xn+1 =xn+pn+1,mod2π (4) distribution function of the angles between SM and UM
where K is related to the kick strength. ρ(θ), so ρ(θ)dθ represents the probability to find a angle
In order to exemplify the dynamics of the Chirikov- between θ and θ +dθ in the phase space ensemble dis-
Taylor map and its sticky phenomena, Fig. 1-a presents played in Fig. 1-b. The large plateau for small θ angles
a portion of the phase space for a typical trajectory of reflectsthe strongnonhyperboliccharacteroftheregion.
the system for K = 3.0. The denser areas near the is- The topological properties of the phase space in the
land result from the sticky effect due to the time the vicinitiesofanislandplayanimportantroleinthesticky
trajectory remains near the edge of the island. In or- mechanism. In order to explore the relation between
der to characterize the hyperbolicity of the surrounding topologicalcharacteristicsofthephasespaceandtheway
areas of the island Fig. 1-b displays the local angle be- trajectoriesvisitanislandvicinity andstickto it, wede-
tween stable and unstable manifolds (θ), Eq. (2), near fine the probability density function F(1)(φ) for trajec-
in
3
FIG.2. (a) Probability distribution function of incomeangle
in the vicinity of a island of the Chirikov-Taylor map. (b)
Average time spent in the sticky area as a function of the
injectedangle.(c) Magnification ofpanel(a)and(b)nearthe
maxima of F(1)(φ)
in
tories injected into the vicinities of the island consider-
ing the vicinity computed by our algorithm (Fig. 1-c).
F(1)(φ)dφ is the probability that a typical chaotic tra- FIG.3. (a) Probability distribution function of incomeangle
in into the vicinity of the main island of the Chirikov map. (b)
jectory to visit the vicinity of the island through a angle
Averagespenttimeinthestickyarea asafunction ofthein-
(1)
between φ and φ+dφ. In Fig. 2 we plot Fin (φ), panel jectedangle. (c)Probabilitydistributionfunctionofoutcome
(a) as well as the average time the trajectory remains anglefromthevicinityofthemainislandoftheChirikovmap.
in the vicinity of the island when injected by a particu- (d)Averagespenttimeinthestickyareaasafunctionofthe
ejected angle. (e) Probability distribution of injected angle,
lar angle, panel (b). In panel (c) we magnify the gray
subjected to the condition that the time of stickiness to be
region of panel (a) and (b). Observe that although the
greater than 1000 iterations
majorpartofthetrajectoriesareinjectedbyjustfewan-
gles inside the red-yellow-greentones regions aroundthe
island in Fig. 1-a (weak hiperbolic regions ), these spe-
cifictrajectoriesspend,inaverage,ashorttime mapping the stable manifold of the hyperbolic fixed point do not
the sticky area. Trajectories are easily injected into the cross the tiny hyperbolic channel produced by the cross-
stickyareasbyweakhyperbolicareassurroundingtheis- ing between stable and unstable manifolds of the fixed
landbutalmostallofthemarealsoeasilyejectedfromit. point and can not be injected into the stick area.
Those trajectories do not contribute for the phenomena Considering those trajectories injected into the stick-
of stickiness and do not make any substantial changesin iness remaining mapping the edge for at least 100 iter-
the Poincar´erecurrence time for the dynamics. ations, we compute in Fig. 3-b the average time they
In order to distinguish sticky trajectories from those spend near the island (sticky trajectories) as a function
that just reach the island edge and leave it quickly, we of the injected angle. In Fig. 3-c we graph the prob-
compute the probability density function F(100)(φ), of ability distribution function of sticky trajectories as a
in
trajectoriesinjected into the stickyareabya specific an- function of the ejected angle Fo1u0t0(φ). It is clear the al-
gleconsidering that once a trajectory reaches the vicinity most discrete nature of the distribution. All ejected tra-
of the island, it remains mapping the same set of points jectories follow the unstable manifold of the hyperbolic
as computed by our algorithm of island edges detection fixed points moving along a narrow channel departing
for at least 100 iterations. We identify such trajectories from the fixed point. We plot in Fig. 3-d the average
as sticky ones. The result is plotted in Fig. 3-a. Almost time the trajectories stay in the sticky region (at least
all sticky trajectories are injected in very specific inter- 100 iterations inside the sticky area)as a function of the
valsofangles. Eachangleintervalsaredirectlyrelatedto outcome angle. From Fig. 3-d we can conclude that
theangularlocationofthechainofperiodicpoints(setas stickytrajectoriesareejected onlyby few angleintervals
black bullets in Fig. 1-c). It is possible to conclude that φout = φ(max(τout)). Therefore, we are able to calcu-
the trajectories are injected into the sticky area when late the probability F(φ)dφ that a given trajectory will
th
they are tangent to the stable manifold of the 18 or- enter the sticky region considering only trajectories that
der fixed points chain located in the vicinity of the main leave these region through the angle φout. The result is
island. All trajectories that are not tangent enough to plotted in Fig. 3-e. The great similarity between F and
4
F(100) suggests that the or previous conclusions are con-
in
sistent. Therefore,wecanarguethatthelocalmaximum
ofF(100) representthestickyangles,i. e.,theanglesthat
in
once a trajectoryis injected from one of them, there is a
greatprobability that this trajectory turn to be stuck to
theisland. Thesemaximacorrespondtothesameregion
where are located hyperbolic points, confirming the hy-
potheses that these points provide channel for a typical
trajectory to enter in the sticky region. Figs. 3-a and
3-e, clearly show us that both figures are almost iden-
FIG.4. (coloronline)(a)Degreeofhyperbolicityofthevicin-
tical, a strong suggestion that all trajectories leave the ity (ǫ = 0.0025) of one of the eighteen fixed points present
stickyregionsbythe hyperbolic channelsdepartingfrom around the main island of the map 4. (b) ρ(θ) character-
the hyperbolic fixed points. ized by just on maximum when computed near a fixed point
around themain island
Thepresenceofstickinessaroundanislandispredicted
by some theory [5, 18] but an analysis of Fig. 3-a shows
) 10
thatthe effectivenessofa trajectoryinjectionorejection 0)
byaparticularhyperbolicchannelisnotthe sameforall (10F in 1
fixed points, as can be observed by the different ampli- (
x
tude of maxima of Fi(n100) in Fig. 3-a. In order to make Ma 0.1
clear the role of the hyperbolicity of the close vicinity of 0.1 1
θ(ρ )
the hyperbolic fixed points in the injection and ejection max
phenomenaofstickytrajectories,firstly,wepresentasan
example, in Fig. 4-a the degree of hyperbolicity of one FIG.5. (coloronline)Theeffectivenessoftheinjection chan-
of the 18 fixed points presents around the main island nels measured using the maxFi1n00 as a function of the angle
of the Chirilov-Taylor map. As observed in Fig. 4-b the between UM and SM.
probability density function ρ(θ) presents just a sharp
maximum due to the almost unique angle between sta-
wetrackthepositionofthetrajectoryandonceitmapsa
ble andunstable manifoldcomputedin the close vicinity
smallcircleofradius0.003centeredinoneofthe18fixed
of the fixed point. All other fixed points present simi-
points we perturb the trajectory, so a possible crossing
lar sharp peaks in the probability density function ρ(θ),
of the channel and consequent stick of the trajectory is
nevertheless each hyperbolic point has its own angle for
avoided. Numerically, we perform a restart of the tra-
the maximum of ρ(θ) characterizing its own degree of
jectory outside the injection channel. Results for the
hyperbolicity. Secondly, to demonstrate the relation be-
Poincar´e recurrence time for the system with and with-
tween the degree of hyperbolicity of the close vicinity of
out the control mechanism for two values of K, K =3.0
the fixed points, and the effectiveness of the hyperbolic
and K = 3.565 (a large stickiness case) are plotted in
channelsrelatedtoeachfixedpointtocapturestickytra-
Fig. 6. Black bullets and green up-triangle display the
jectories, we graph in Fig. 5 the maxima of the function
F100(φ)asafunctionofthedegreeofhyperbolicitymea- time distributions for Poincar´e recurrence without any
in
control mechanism. As can be observed for large recur-
sured by the angle for which the function ρ(θ) presents
rence time, a strong fluctuation of the exponential law
a maximum θ(ρ ). The red line is a power law fit-
max
is observed. In fact for large recurrence time the dis-
ting that serves us as eye guide. The result presented in
tribution has a power law decay as a result of the sticky
Fig. 5 clearly shows that the effectiveness of the chan-
phenomenaintherecurrencetime. Thetimedistribution
nels is a function of the degree of hyperbolicity of the
ofthe Poincar´erecurrencetime for the systemsubjected
close region of the fixed points. Small values of θ(ρ )
max
is related to the fact that just a very small portion of
the surrounding area of the fixed point is occupied by
the injection/ejection hyperbolic channel. As a result 106 K=3.0 Control off
K=3.0 Control on
the function Fi(n100)(φ) presents a relatively small maxi- 104 K=3.565 Control off
mum, meaning that just a small fraction of trajectories P K=3.565 Control on
can cross the channel in an injection or ejection process 102
from the sticky area. 100
0 2×104 5×104 8×104 1×105
Such properties of the phase space allow us to manip- τ
ulate the non equilibrium fluctuation of the system. To
show that it is possible to control the non equilibrium FIG.6. (coloronline)RecurrencetimefortheChirikov-Taylor
fluctuations that arise due to the presence of stickiness, map with and without control
5
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