Controlling Effect of Geometrically Defined Local Structural Changes on Chaotic Hamiltonian Systems

9.1 MB·English

by  Yossi Ben Zion

Controlling Effect of Geometrically Defined Local Structural Changes on Chaotic
Hamiltonian Systems
Yossi Ben Zion1 and Lawrence Horwitz1,2,3
1Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
2Department of Physics, Ariel University Center of Samaria, Ariel 40700, Israel
3School of Physics, Tel Aviv University, Ramat Aviv 69978, Israel
AneffectivecharacterizationofchaoticconservativeHamiltoniansystemsintermsofthecurvature
associatedwithaRiemannianmetrictensorderivedfromthestructureoftheHamiltonianhasbeen
extended to a wide class of potential models of standard form through definition of a conformal
metric. The geodesic equations reproduce the Hamilton equations of the original potential model
through an inverse map in the tangent space. The second covariant derivative of the geodesic
0 deviationinthisspacegeneratesadynamicalcurvature,resultingin(energydependent)criteriafor
1 unstablebehavior different from theusual Lyapunov criteria. We show here that this criterion can
0 beconstructivelyusedtomodifylocallythepotentialofachaoticHamiltonianmodelinsuchaway
2
that stable motion is achieved. Since our criterion for instability is local in coordinate space, these

n results providea new and minimal method for achieving control of a chaotic system.
a
J PACSnumbers: 45.20.Jj,05.45.Pq,05.45.Gg

1
3
I. INTRODUCTION followmoreregularmotion. Additionalefficientmethods

] have been proposed in [12–16]; some of these methods
D
require tracking the trajectories while others involve the
The issue of controlling chaos has attracted great in-
C addition of interacting terms for which the criteria are
terest in the last two decades; much work has been done
. somewhat sensitive.
n inthecaseofdissipativesystems. Ott,GrebogiandYork
Geometric approaches for the analysis of the stability
i (OGY)[1]developedamethodbywhichchaoscanbesup-
l of a given Hamiltonian have been widely discussed [17–
n pressed by making small time-dependent perturbations
[ 29], for which curvature of a manifold is associated with
in order to shadow one of the infinitely many periodic stability.
1 orbits embedded in the chaotic attractor. This method
The pioneering work of Oloumi and Teychenn´e [30]
v has been extended by many following studies [2–4] and
proposesastabilizationmethodbyconsideringtheGaus-
2 successfully applied to experimental systems [5].
4 sian curvature of the potential energy as a source of
1 As a resultofthe absence ofattractorsinconservative chaos. Evenif the condition of negative curvature of the
0 systemsandthe complex natureofthe phasespacecom- potential and instability of the dynamics are not com-
2. bining regular and irregular regions, constructing an ef- pletely equivalent they were able to successfully control
0 fective method has remained a challenging question. Lai theinstabilitybyavoidanceofnegativecurvatureregions
0 Ding and Grebogi (LDG) extended the OGY method to of the potential energy (ANCRP).
1 Hamiltonian systems by incorporating the notion of sta- In this work we make use of a recently developed ge-
: ble and unstable directions at each periodic point.
v ometrical criterion [25] which is highly sensitive to in-
i In relation to the above question, Chandre et al. [6– stability in Hamiltonian systems, and has been shown
X
8] have developed a method for constructing barriers in to be in complete agreement with the results of the nu-
r phase space to subdue chaotic behavior for large times. merical technique of surface of section (Poincar´e plot)
a
Thismethodisbasedontheintroductionofaspecifically and more effective than other geometric methods [27].
designed small control term which changes the dynam- Thiscriterionisbasedonanequivalencebetweenmotions
ics from chaotic to regular behavior. Zhang et al. [9] generated by Hamiltonian in the standard form with
have introduced a method for controlling chaos in two- quadratickinematictermsandadditivepotentials(which
dimensionalHamiltoniansystemwhichtheycalledadap- weshallcalltheHamiltondescription)andaHamiltonian
tive integrablemode couplingbasedonthe separationof inwhichthedynamicsisdescribedbyametric-typefunc-
the system into two coupled subsystems one of which is tion of coordinates multiplying the momenta in bilinear
stable and the other unstable; when the unstable system form(whichweshallcalltheGutzwillerdescription)[31–
comes into the vicinity of the integrable system the con- 35]. The mapping between these equivalent descriptions
ditions are reset resulting in an effective adaptive con- was first introduced by Appel [36–39]. The criterion for
trol. Cartwright et al. [10] have constructed a method stability developed in [25] establishes an inverse map-
which permits stabilization of KAM islands throughfor- pingofthemotiondescribedcompletelygeometricallyin
ward iteration of the orbits, and transforming them into the Gutzwiller space back to the Hamilton description,
global attractors of the embedded system. Ciraolo et carrying with it the covariance under diffeomorphisms
al. [11] have used a method of adding a small pertur- thatisapropertyoftheGutwillerdynamics. Theresult-
bation to modify systems, such as nonlinear plasmas, to ing orbits in the Hamilton description, with the special
2
choice of coordinates serving as the basis for the Appel II. THEORY
type relation, reduce to the standard description of the
Hamilton orbits under the Hamilton equations. In gen- It has been shown [25, 36] that a Hamiltoniansystem
eralgeometricform,asgeodesicequations,however,they of the form (we use the summation convention)
aresubjecttoanalysisintermsofgeodesicdeviation,and
theresultingformulacanbereducedtoacomputationin pi2
the specialcoordinatesof a new (symmetric) matrix val- H = +V(x), (1)
2M
ued criterion for stability [25] which has been shown to
be veryeffectiveina widerangeofexamples[17,27,40]. whereV isafunctionofspacevariablesalone,canbeput
into the equivalent form
1
Inthe analysisofthese systems,itwasshownthatthe H = g pipj, (2)
G ij
2M
presence of negative eigenvalues for the stability matrix
in the admissible physical region (for which E > V) re- whereg isconformalandisafunctionofthecoordinates
ij
sultsinachaotictypePoincar´eplot. Sincethe condition alone. Onecaneasilyseethattheorbitsdescribedbythe
islocalincoordinatespace,the resultnecessarilyimplies Hamiltonequationsfor(2)coincidewiththegeodesicson
some degree of ergodicity. There has, however, been no aRiemannianspaceassociatedwiththemetricg ,i.e.,it
ij
proof that the existence of negative eigenvalues is not follows directly from the Hamilton equations associated
strongly associated with the structure of the dynamics with (1) that [31]
more globally, as is often the case, for example, for an-
alytic function theory, where the presence of a complex x¨ℓ =−Γmℓ nx˙mx˙n, (3)
pole implies a distortion of a relatively large region of
where the connection form Γmn is given by
the complex plane. The results of our application to the ℓ
controlofdynamicalsystems,however,indicatesthatthe
1 ∂gkm ∂gkn ∂gnm
regionsofnegativeeigenvaluecorrespondtoaverylocal- Γmn = g + − , (4)
ized phenomena. The observed instability they induce ℓ 2 ℓk(cid:8) ∂xn ∂xm ∂xk (cid:9)
appears to be due to the passageof orbits throughthese
and gij is the inverse of g .
regions, and they are not strongly correlated with the ij
For a metric of conformal form
more global structure of the dynamics.
g =ϕδ , (5)
ij ij
What we have done here is to identify the regions of with inversegij =ϕ−1δ ,on the hypersurfacedefinedby
ij
negative eigenvalue in the coordinate space, and mod- H =H =E =constant,andassumingthattheobserv-
G
ify the Hamiltonian in these regions locally, removing able momenta are the same at every t, the requirement
the source of instability either by removing the insta- of equivalence implies that [25]
bility inducing terms or varying the coupling to these
E
terms to bring the Hamiltonian closer to an integrable ϕ= . (6)
form in these regions. The remaining part of the space E−V(x)
is left to develop according to the full, nonlinear, sym-
To see that the Hamilton equations obtained from (2)
metry breaking, evolution. The effect on the Poincar´e
can,beputintocorrespondencewiththoseobtainedfrom
plots, as we show here, is dramatic. This result provides
theHamiltonianofthepotentialmodel(1),wefirstnote,
strongevidence thatchaotic Hamiltoniansystems derive
from the Hamilton equations for (2), that
their properties from sources that may be thought of as
highly localizedinthe regionsofnegativeeigenvaluesfor
∂H 1
the stability matrix. In addition to giving an interest- x˙i = ∂pGi = Mgijpj. (7)
ing insight into the nature of Hamiltonian chaos, it also
providesaneffectivecontrolmethodinwhichthe Hamil- We then define the velocity field
tonian dynamics is left to exercise its full function in a
largedomainofconfigurations,butisonlyconstrainedin x˙j ≡gjix˙ = 1 pj, (8)
prescribed local regions of the configuration space. i M
coinciding formally with one of the Hamilton equations
implied by (1), for which we label the coordinates xj.
In the next section, we reviewthe basic ideas underly-
Tocompleteourcorrespondencewiththedynamicsin-
ingthegeometricalcriterion,andinthefollowingsection ducedby (1), considerthe Hamiltonequationfor p˙i gen-
we give numerical results for the application of our con-
erated by H ,
G
trol procedure for the cases of an oscillator with broken
symmetryandapotentialwhichmaybederivedfromthe ∂H 1 ∂g
p˙ℓ =− G =− ijpipj. (9)
Toda form. ∂x 2M ∂x
ℓ ℓ
3
0.6 0.8 0.8
0.6 0.6
0.4
0.4 0.4
0.2 0.2 0.2
y 0 Py 0 Py 0
−0.2 −0.2
−0.2
−0.4 −0.4
−0.4
−0.6 −0.6
−0.6 −0.8 −0.8
(aI) −0.6 −0.4 −0.2 x0 0.2 0.4 0.6 (aII) −1 −0.5 y0 0.5 1 (aIII) −1 −0.5 y0 0.5 1
0.8 0.8
0.6
0.6 0.6
0.4
0.4 0.4
0.2 0.2 0.2
y 0 Py 0 Py 0
−0.2 −0.2 −0.2
−0.4 −0.4 −0.4
−0.6 −0.6
−0.6
−0.8 −0.8
(bI) −0.6 −0.4 −0.2 x0 0.2 0.4 0.6 (bII) −1 −0.5 y0 0.5 1 (bIII) −1 −0.5 y0 0.5 1
1 1
0.5
0.5 0.5
y 0 Py 0 Py 0
−0.5 −0.5
−0.5
−1 −1
(cI) −0.5 x0 0.5 (cII) −1 −0.5 y0 0.5 1 (cIII) −1 −0.5 y0 0.5 1
1 1
0.5 0.5 0.5
y 0 Py 0 Py 0
−0.5 −0.5 −0.5
−1 −1
(dI) −0.5 x0 0.5 (dII) −1 −0.5 y0 0.5 1 (dIII) −1 −0.5 y0 0.5 1
FIG.1: EffectofcontrolonthemodelofEq. (19). Thefirstcolumnshowsthephysicalregion(closedblackcurves)corresponding
to the sequence of energies, 1,1,1,1 corresponding to a,b,c,d. The regions of negative eigenvalues (instability) are shown in
5 4 3 2
gray. ThesecondcolumnshowsthePoincar´eplotsfortheuncontrolledsystem. ThethirdcolumnshowsthePoincar´eplotsfor
the controlled system (Eq. (21) with r = 0.4 for all cases). The (red) dashed line in column one shows the boundary of the
control modification.
With (8) this becomes sor index to reach the Hamilton manifold, is exactly the
“truncated” connection (11).
x¨ℓ =−Mmℓnx˙mx˙n, (10) Substituting (5) and (6) into (10) and (11), the Kro-
necker deltas identify the indices of x˙m and x˙n; the re-
where
sulting square of the velocity cancels a factor of (E −
1 ∂g V)−1, leaving the Hamilton-Newton law derived from
Mmℓn ≡ 2gℓk ∂xnkm. (11) the Hamilton equations directly from (1). Eq. (10) is
thereforeageometricallycovariantformoftheHamilton-
Eq. (10) has the form of a geodesic equation, with Newtonlaw,exhibitingwhatcanbeconsideredanunder-
a truncated connection form [25]. Note that perform- lying geometry of standard Hamiltonian motion.
ing parallel transport on the local flat tangent space of Since the coefficients Mℓ constitute a connection
mn
theGutzwillermanifold(forwhichΓmn andg arecom- form, they can be used to construct a covariant deriva-
ℓ ij
patible), the resulting connection, after raising the ten- tive. Itisthiscovariantderivativewhichmustbeusedto
4
1 1
0.8
0.6
0.5 0.5
0.4
0.2
y Py 0 Py 0
0
−0.2
−0.5 −0.5
−0.4
−0.6
−1 −1
(aI) −0.6 −0.4 −0.2 x0 0.2 0.4 0.6 (aII) −1 −0.5 y0 0.5 1 (aIII) −1 −0.5 y0 0.5 1
1.5 1.5
0.8
0.6 1 1
0.4 0.5 0.5
0.2
y Py 0 Py 0
0
−0.2 −0.5 −0.5
−0.4
−1 −1
−0.6
−1.5 −1.5
(bI) −0.6 −0.4 −0.2 x0 0.2 0.4 0.6 (bII) −1 −0.5 y0 0.5 1 (bIII) −1.5 −1 −0.5 y 0 0.5 1
1 1.5 1.5
1 1
0.5
0.5 0.5
y 0 Py 0 Py 0
−0.5 −0.5
−0.5 −1 −1
−1.5 −1.5
(cI) −0.5 x0 0.5 (cII) −1 −0.5 0 y 0.5 1 1.5 (cIII) −1.5 −1 −0.5 y0 0.5 1 1.5
1.5 1.5
1
1 1
0.5
0.5 0.5
y 0 Py 0 Py 0
−0.5 −0.5
−0.5
−1 −1
−1.5 −1.5
(dI) −0.5 x0 0.5 (dII) −1 −0.5 0 y 0.5 1 1.5 (dIII) −1.5 −1 −0.5 y0 0.5 1 1.5
3
2
1 1.5 2
1
0.5 1
0.5
y 0 Py 0 Py 0
−0.5 −1
−0.5 −1
−2
−1.5
−1 −2 −3
(eI) −0.5 x0 0.5 (eII) −1.5 −1 −0.5 y0 0.5 1 1.5 (eIII) −2 −1 y0 1 2
FIG.2: EffectofcontrolonthemodelofEq. (20). Thefirstcolumnshowsthephysicalregion(closedblackcurves)corresponding
tothesequenceofenergies,0.5,0.6,0.8,1.0,1.5correspondingtoa,b,c,d,e. Theregionsofnegativeeigenvalues(instability)are
showningray. ThesecondcolumnshowsthePoincar´eplotsfortheuncontrolledsystem. ThethirdcolumnshowsthePoincar´e
plots for thecontrolled system (Eq. (22) with α=0.5 for all cases). The (red) dashed linein column one shows theboundary
of the control modification.
5
compute the rate of transport of the geodesic deviation orbits are viewed geometrically as geodesic motion, is a
ξℓ =x′ℓ−xℓ along the (approximately common) motion new condition for instability [25], based on the underly-
of neighboring orbits in the Hamilton manifold, since it ing geometry, for a Hamiltonian system of the form (1),
follows the geometrical structure of the geodesic curves providing new insight into the structure of the unstable
on {xℓ}. andchaoticbehaviorofHamiltoniandynamicalsystems.
The relation The method has been successfully applied to many po-
tentialmodels[17,25,27,40]. Wenowapplythemethod,
∂Mℓ
ξ¨ℓ =−2Mℓ x˙mξ˙n− mnx˙mx˙nξq, (12) whichis simple andstraightforward,since the instability
mn ∂xq criterionisdefinedlocallyincoordinatespace,toextract
from a chaotic Hamiltonian the part of the dynamics in
obtained from (10), can be factorized in terms of the
the regions for which V has negative eigenvalues which
covariant derivative
lead to unstable motion . We show by direct simulation
∂ξℓ that this procedure is remarkably effective.
ξℓ = +Mℓ ξm. (13)
;n ∂xn nm Theresultsconfirmthenotionofthelocalityofourcri-
terion, since the (local) removal of parts of the Hamilto-
One obtains
nianinducinginstabilitybythiscriterionstronglyaffects
D 2 the global stability of the motion.
M ξℓ =R ℓ x˙qx˙nξm, (14)
D t2 M qmn
M
III. RESULTS AND DISCUSSION
where the index M refers to the connection (11), and
R ℓ = ∂Mqℓm − ∂Mqℓn +Mk Mℓ −Mk Mℓ InthefollowingwegivesomeexamplesofHamiltonian
M qmn ∂xn ∂xm qm nk qn mk motion which are unstable and exhibit chaotic behavior.
(15) In two dimensions, the terms in the potential that break
correspondstothecurvatureassociatedwiththeconnec- rotationalsymmetryhaveaform,intheseexamples,that
tion form Mmℓn This expression does not coincide with induces chaotic behavior. The matrix V computed over
the curvature of the Gutzwiller manifold (given by this the physically accessible region exhibits certain regions
formula with Γℓqm in place of Mqℓm), but is a dynami- with negative eigenvalues. We have arranged our simu-
cal curvaturewhichisappropriateforgeodesicmotionin lation to replace the Hamiltonian in the regions of nega-
{xℓ} tiveeigenvaluesbyaHamiltonianinwhichthesymmetry
Withtheconformalmetricinnoncovariantform(5),(6) breaking terms are either removed or for which the cou-
(inthecoordinatesysteminwhich(6)isdefined),thedy- pling coefficient is decreased, but in regions of positive
namicalcurvature(15)canbewrittenintermsofderiva- eigenvalues,theHamiltonianremainsinitsoriginalform.
tivesofthepotentialV,andthegeodesicdeviationequa- The changesare carriedinrelatively smallregionsof the
tion (14) becomes configurationspaceintheneighborhoodoftheregionsof
negative eigenvalues.
D2ξ
=−VPξ, (16) We take for illustration here a simple and important
Dt2
case of coupled harmonic oscillators with perturbation
where the matrix V is given by
1
V(x,y)= (x2+y2)+6x2y2 (19)
3 ∂V ∂V 1 ∂2V 2
V = + . (17)
ℓi M2v2∂xℓ∂xi M ∂xℓ∂xi
(cid:8) (cid:9) and a generalization of the Toda potential
and
1 1 3 1
V(x,y)= (x2+y2)+x2y− y3+ x4+ y4. (20)
vivj 2 3 2 2
Pij =δij − , (18)
v2
The first of these is known to generate chaotic behav-
with vi ≡ x˙i, defining a projection into a direction or- ior;aswehaveshownpreviously,thetransitiontochaotic
thogonal to vi. behavior as a function of energy and of the coupling to
Instability should occur if at least one of the eigenval- the perturbation is well described by our geometric cri-
ues of PVP is negative, in terms of the second covariant terion [25]. In the present work, we set the coupling to
derivatives of the transverse component of the geodesic thechaosinducingperturbationtozerointheregionsfor
deviation. This condition is easily seen to be equivalent whichthegeometriccriterionresultsinnegativeeigenval-
tothesameconditionimposedonthespectrumofV,and ues.
is thus independentofthe directionofthe motiononthe In our previous study of the second example Eq. (20)
orbit. we studied the sensitivity of the dynamical behavior to
The conditionimplied by the geodesic deviationequa- the choice of energy [27]. In the presentwork, at the en-
tion (16), in terms of covariant derivatives, in which the ergies for which the system exhibits chaotic behavior we
6
have removedthe chaos inducing perturbation in the re- whereαstandsforthe limit(ahorizontaldashedline)of
gionsforwhichthegeometriccriterionresultsinnegative the region of positive eigenvalues.
eigenvalues.
Fig. 1 shows the effects on the dynamical behavior of In both cases, for sufficiently high energies the uncon-
the change of the coupling to a stable value in the re- trolledsystemsbecomelessstableandachaoticsignature
gions of negative eigenvalues for the perturbed oscillator appears. We examine the method for different values of
potential (19). The results for different values of energy theenergy,allcorrespondingtochaoticmotionoftheun-
are shown; these values correspond to different sizes of controlled system. One can easily see that the Poincar´e
the physically accessible region. Note that the radius of plotsincontrolledsystemspresentalmostcompletelyreg-
the regionofpositiveeigenvalues doesnot changeinthis ular motion.
example.
The first row shows the physical region and the lo-
cation of negative eigenvalues. The interior of the cir-
cle corresponds to a region in which no negative eigen-
value occur, chosen in this way to simplify the compu-
tation. The second row shows the surface of section
Poincar´e plot(surface of section) of the original uncon-
trolled Hamiltonian indicating chaotic dynamics. The
IV. CONCLUSION
third row shows the Poincar´eplot generated by the con-
trolled potential:
1(x2+y2)+6x2y2 x2+y2 <r2 Weseefromthesecomputationsthatinregionsofpos-
V(x,y)= 2 (21)
(cid:26) 1(x2+y2) x2+y2 ≥r2 itive eigenvalues the Hamiltonian with chaos inducing
2
terms do not generate instability. If the chaos induc-
where r stands for the radius of the region of positive ing terms are removed or decreased in coupling in the
eigenvalues. regionsofnegativeeigenvalues,thechaoticmotionofthe
Fig. 2 shows the effect of control on second system system is reduced or disappears. We conclude that the
(20), using the controlled potential: chaotic behavior of the system is associated with local
properties of the Hamiltonian, and that our local crite-
1(x2+y2)+x2y− 1y3+ 3x4+ 1y4 y >−α rioniseffectiveinidentifyingtheseregions. Furthermore,
V(x,y)= 2 3 2 2
(cid:26) 1(x2+y2) y ≤−α ourprocedureprovidesaneffectivemethodforcontrolof
2
(22) chaotic systems of this type.
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