Table Of ContentReducing or enhancing chaos using periodic orbits
R. Bachelard1, C. Chandre1, X. Leoncini1,2
1 Centre de Physique Th´eorique∗, CNRS Luminy, Case 907, F-13288 Marseille cedex 09, France
2 PIIM, Universit´e de Provence-CNRS, Centre Universitaire de Saint-J´erˆome, F-13397 Marseille, France
(Dated: February 8, 2008)
A method to reduce or enhance chaos in Hamiltonian flows with two degrees of freedom is dis-
cussed. Thismethodisbasedonfindingasuitableperturbationofthesystemsuchthatthestability
of a set of periodic orbits changes (local bifurcations). Depending on the values of the residues,
reflectingtheirlinearstabilityproperties,asetofinvarianttoriisdestroyedorcreatedintheneigh-
borhoodofthechosenperiodicorbits. Anapplicationonaparadigmaticsystem,aforcedpendulum,
illustrates themethod.
6
0 PACSnumbers: 05.45.-a
0
2
Changing the dynamical properties of a sys- I. INTRODUCTION
n
tem is central to the design and performance of
a
J advanced devices based on many interacting par-
A very fruitful information on the dynamics can be
9 ticles. For instance, in particle accelerators, the gained from the study of periodic orbits [1]. First, be-
1 aim is to find the appropriate magnetic elements
cause these particular orbits are generically almost ev-
to obtainan optimalaperture inordertoincrease
erywhere in phase space, and second because they can
] the luminosity of the beam, thus requiring the
D becomputedeasily,i.e.withsomeshortintegrationtime.
decrease of the size of chaotic regions. In plasma
C These periodic orbits together with their stability orga-
physics, the situation is slightly more complex :
nize locally the dynamics. It is then natural to consider
n. Inside a fusion device (like a tokamak or a stel- themasacornerstoneofcontrolstrategies. Forinstance,
i larator), one needs magnetic surfaces in order to
l in order to create invariant tori of Hamiltonian systems,
n increase confinement. These surfaces are invari-
Cary andHanson [2, 3] proposeda method basedon the
[ ant tori of some fictitious time dynamics. A con-
computation of an indicator of the linear stability of a
trol strategy would be to recreate such magnetic
1 set of periodic orbits, namely Greene’s residue [4]. It
surfaces byanappropriate modificationoftheap-
v provides an algorithm to find the appropriate values of
3 paratus (magneticperturbationcausedbyasetof some pre-defined parameters in order to reconstruct in-
4 external coils). On the opposite, in order to col-
varianttoribyvanishingsomeselectedresidues. Firstde-
0 lect energy and to protect the wall components,
velopedfortwo-dimensionalsymplecticmaps,ithasbeen
1 an external modification of the magnetic equilib-
0 extended to four dimensional symplectic maps, and has
rium has to be performed such that there is a
6 beenappliedtostellarators[5](whereperiodicorbitsare
highly chaotic layer at the border (like an ergodic
0 closed magnetic field lines) and particle accelerators [6].
/ divertor). Therefore these devices require a spe-
n cific monitoring of the volume of bounded mag- In this article, we review and extend this residue
i method. The aim is to tune appropriately the param-
l netic field lines. Another example is afforded by
n eters of the system such that appropriate bifurcations
chaotic advection in hydrodynamics : In the long
: occur. It is well-known in the literature that local bifur-
v run to achieve high mixing in microfluidics and
cations occur when the tangentmap associatedwith the
i microchannel devices in particular, the presence
X Poincar´e map obtained by a transversal intersection of
ofregularregionprevents suchmixing, andhence
r theflow,hasaneigenvaluewhichisarootoftheunity. In
a apossiblewaytoenhancemixingistoperturbex-
particular, periodic orbits can lose their stability in case
ternally the system according to some theoretical
of multiple eigenvalueson the unit circle,i.e. when these
prescriptions, in order to destroy invariant sur-
eigenvaluesareequalto1or 1fortwo-dimensionalsym-
faces. −
plecticmaps. ThereforeitisnaturaltoconsiderGreene’s
residuesasawaytolocatethosebifurcations. Inthiscon-
text,vanishingresiduesindicatethespecificvaluesofthe
parameterswheresignificantchangeoccursinthesystem
∗Unit´e Mixte de Recherche (UMR 6207) du CNRS, et des uni-
andhencewillbethebasisforthereductionofchaos(by
versit´es Aix-Marseille I, Aix-Marseille II et du Sud Toulon-Var.
Laboratoireaffili´ea`laFRUMAM(FR2291) creation of invariant tori) as in Refs. [2, 3] but also for
the destruction of regular structures.
In Sec. II, we review some basic notions on periodic
orbitsofHamiltoniansystemsandtheirstability,andwe
explain the details of the residue method. We give the
condition on the residues of a pair of Birkhoff periodic
orbitstocreateaninvarianttorusintheir vicinity,anda
2
similar condition which leads to a destruction of nearby dynamics will change with variations of the parameters.
invariant tori. In Sec. III, we apply this method to the Generically, periodic orbits and their linear stability are
destruction and creationof librational and rotationalin- robustto smallchanges of parameters,except at specific
variant tori of a particular Hamiltonian system, a forced values where bifurcations occur. The proposed residue
pendulumwithtwointeractingprimaryresonances,used methodtocontrolchaosdetectstheserareeventstoyield
as a paradigm for the transition to Hamiltonian chaos. the appropriate values of the parameters leading to the
prescribed behavior on the dynamics.
The residue method which leads to a reduction or an
II. THE RESIDUE METHOD enhancement of the chaotic properties of the system is
based on the change of stability of periodic orbits upon
WeconsideranautonomousHamiltonianflowwithtwo a change of the parameters of the system. For α = 0,
degreesoffreedomwhichdependsonasetofparameters let us consider two associated Birkhoff periodic orbits
denoted α=(α1,α2,...,αm) Rm : (i.e. periodic orbits having the same action but different
∈
anglesintheintegrablecaseandhavingthesamerotation
z˙ =J H(z;α), number on a selected Poincar´e section), one elliptic
∇ Oe
and one hyperbolic . Let us call R and R their
where z = (p,q) ∈ R4 and J = (cid:18)I02 −0I2 (cid:19), and I2 raensdidRues(.0)W<e 0h.avWe eROeshl(i0g)ht>ly 0mo(adnifdy stmheealplearratmhaehnterosneα)
h
being the two-dimensional identity matrix. In order to
untiltheellipticperiodicorbitsbecomesparabolic. Some
determine the periodic orbits of this flow and their lin-
particular situations arise at some critical value of the
earstabilityproperties,wealsoconsiderthetangentflow parameters α=α :
c
written as (i) : R (α )=R (α )=0.
e c h c
d
dtJt(z)=J∇2H(z;α)Jt, (ii) : Re(αc)=0 while Rh(αc)<0.
whereJ0 =I4 and∇2H istheHessianmatrix(composed (iii) : Re(αc)=1 while Rh(αc)<0.
by second derivatives of H with respect to its canonical
variables). For a given periodic orbit with period T, the
The first case is associated with the creation of an in-
spectrum of the monodromy matrix JT gives its linear
variant torus. The two latter cases might be associated
stability property. As the flow is volume preserving, the
with the destruction of invariant tori (the ones around
determinant of such a matrix is equal to 1. Moreover, if
the elliptic periodic orbit). The third one is associated
∗ ∗
Λisaneigenvalue,soare1/Λ,Λ and1/Λ . Astheorbit
with a period doubling bifurcation. In this latter case,
is periodic, Λ = 1 is an eigenvalue with an eigenvector
the change of stability of the new elliptic periodic or-
in the direction of the flow. Its associated eigenspace is
bit has to be considered. Other interesting cases occur
at leastofdimension 2 since there is another eigenvector
depending on the set of selected periodic orbits. The
with eigenvalue 1 coming from the conserved quantity
situation (i) resembles the integrable situation where all
H = E. Therefore, according to the remark above, the
theresiduesofperiodicorbitsofconstantactionarezero.
orbit is elliptic if the spectrum of JT is (1,1,eiω,e−iω)
It is expected that an invariant torus is reconstructed in
(and stable, except at some particular values), or hyper-
this case. It can be associated with a transcritical bifur-
bolic if the spectrum is (1,1,λ,1/λ) with λ R∗ (un-
∈ cation (an exchange of stability), a fold, or another type
stable). The intermediate case is when the spectrum is
of bifurcation. In the situation (ii), a change of stabil-
restricted to 1 or 1 and the orbit is called parabolic.
− ity occurs : The elliptic periodic orbit turns hyperbolic
Whether or not the parabolic periodic orbit is stable de-
while the hyperbolic one stays hyperbolic. It is generi-
pendsonhigherorderterms. Inamoreconciseform,the
cally characterized by a stationary bifurcation. In this
abovecasescanbe summarizedusingGreene’sdefinition
case, the destruction of invariant curves is expected in
of a residue which led to a criterion on the existence of
general whether there are librational ones (representing
invariant tori [4, 7] :
the linear stability of an elliptic periodic orbit) or the
4 trJT neighboring rotational ones.
R= − . An extra caution has to be formulated since this
4
method only provides an indicator of the linear stability
We notice that the 4 (instead of 2 for 2D maps) in the of periodic orbits. The nonlinear stability (or instabil-
numerator comes from the two additional eigenvalues 1 ity) has to be checked a posteriori by a Poincar´e section
comingfromautonomousHamiltonianflows. IfR ]0,1[, for instance. This method only states that a bifurcation
∈
the periodic orbit is elliptic; if R < 0 or R > 1 it is has occurred in the system, whether it is a stationary,
hyperbolic; and if R = 0 and R = 1, it is parabolic transcritical, period doubling or other types of bifurca-
and higher order expansions give the stability of such tions. A more rigorous and safer control method would
periodic orbits. Since the periodic orbit and its stability require to consider the global bifurcations, like the ones
depend on the set of parameters α, the features of the obtained by the intersections of the stable and unstable
3
manifolds of two hyperbolic periodic orbits in the spirit 1.5
of Ref. [8]. However such a control method would be
computer-time consuming (determination of the stable 1
and unstable manifolds) and hence not practical if some
short time delay feedback is involved in the control pro- p 0.5
cess.
0
III. APPLICATION TO A PARADIGMATIC −0.5
MODEL 0 1 2 3 4 5 6
x
2
We consider the following forced pendulum system 0.2
1
with 1.5 degrees of freedom 0.1
3
0
p2 −0.1
H(p,x,t)= +ε(cosx+cos(x t)). (1) p
2 − −0.2
−0.3
A Poincar´e section of Hamiltonian (1) is depicted on −0.4
Fig.1 for ε=0.065andonFig.2 for ε=0.034. In order −0.5
to modify the dynamics of Hamiltonian (1), we add an
0 1 2 3 4 5 6
additional (control) parameter k : We consider a family x
of Hamiltonians of the form
FIG. 1: Poincar´e sections of Hamiltonian (1) with ε=0.065.
2
p k 2 The arrows indicate the elliptic periodic orbits for the three
H (p,x,t)= +ε(cosx+cos(x t))+ ε cos(2x t),
c 2 − 2 − cases considered here.
(2)
wherekisnottoolargeinordertoconsiderasmallmod-
ification of the original system, and minimizing the en-
ergy cost needed to modify the dynamics. Other choices 1.5
offamiliesofcontroltermsarepossible(notrestrictedto
kcos(2x t)). Inparticular,moresuitablechoicesofcon-
1
−
trol terms would include more Fourier modes. We have
selected a one-parameter family which originates from
p 0.5
another controlstrategywhichhas been provedto be ef-
fective [9]. The goal here is to determine the particular
valuesoftheparameterksuchthatsuitablemodifications 0
of the dynamics (which will be specified later) occur.
The algorithm is as follows : First, we determine −0.5
0 1 2 3 4 5 6
two periodic orbits of Hamiltonian (1), an elliptic and x
0.5
a hyperbolic one with the same rotation number on
the Poincar´e section, using a multi-shooting Newton- 0.45
Raphson method for flows [1]. Then we modify contin-
0.4
uously the control parameter k and follow these two pe-
riodic orbits. We compute their residues as function of p0.35
k.
0.3
A first analysis is done on librational invariant tori
(around the primary resonance located around p 0). 0.25
≈
We point out in Fig. 1 three particular elliptic periodic
0.2
orbits(and their associatedhyperbolic ones),labeled1,2 0 1 2 3x 4 5 6
and 3, with, respectively, Q = 4, Q = 9 and Q = 13
intersections with the Poincar´e section (t = 0 mod 2π). FIG. 2: Poincar´e section of Hamiltonian (1) for ε=0.034.
These orbits will be used for two purposes : First we
follow the idea of Cary and Hanson on the construction
of invariant tori. Then we extend the residue method to
the destruction of these tori. A similar analysis is done
on rotational invariant tori (the example of the golden- Brieflywedeterminethecontrolparameterksuchthat
mean invariant torus is treated). This case allows us to there is a creation of an invariant torus if the original
compare the residue method with another approach on system does not have one, and the destruction of an in-
the control of Hamiltonian systems. variant torus if the system does have one.
4
0.15
0.1
0.1
0.05
0
R0.05
p−0.05
−0.1
0 −0.15
−0.2
−5 0 5 10 −0.25
k 2.5 3 3.5 4
x
FIG. 3: Residues R of the elliptic and hyperbolic (bold line)
FIG.4: Poincar´esectionofperiodicorbitsandsometrajecto-
periodicorbitsofCase1(Q=4)asfunctionsoftheparameter
ries for Case 1 (Q=4) and Hamiltonian (2) with ε=0.065.
k for Hamiltonian (2) with ε=0.065.
The trajectories in gray are for k = 0 and the ones in black
are for k = 5. At k = kc, an invariant torus of the system
is represented (bold line). The arrows indicate the change of
A. An exchange of stability associated with the locations of theperiodic pointsas k increases.
creation of librational invariant tori
1.5
For the case Q = 4, Fig. 3 represents the values of 1
the residues of the elliptic and hyperbolic periodic or-
bits asfunctions ofthe controlparameterk [see Eq.(2)]. 0.5
At k = kc 2.747, both residues vanish which means R 0
≈
that they become parabolic periodic orbits as in the in-
−0.5
tegrablecase. Byincreasingk,wenoticethatbothorbits
exchange their stability which is the manifestation of a −1
transcriticalbifurcation while eachof the periodic orbits
−1.5
undergo individually a tangent bifurcation. This type −2 0 2 4 6
k
of bifurcation has been observed in Refs. [10, 11]. At
k = k , an invariant torus is reconstructed. In order to
c FIG. 5: Residues R of the elliptic and hyperbolic periodic
checktherobustnessofthemethod,onecouldarguethat orbits of Case 2 (Q= 9) as functions of the parameter k for
since this invariant torus is composed of periodic orbits, Hamiltonian (2) with ε=0.065.
it is not expected to be robust. However, by continuity
in phase space, an infinite set of invariant tori is present
intheneighborhoodofthe createdinvarianttorus. Most k 2.79. However, even if these residues do not vanish
≈
of them have a frequency which satisfy a Diophantine (andthereforenoexchangeofstabilitybythecreationof
conditionandhencewhichwillpersistundersuitablehy- an invariant torus), there is a significant regularization
pothesis on the type of perturbations. of the dynamics at this specific value of the parameter
The locations of the different periodic points on the (not shown here). For Q = 13, the residues vanish for
Poincar´e section as k varies are indicated by arrows on k 2.76 (see Fig. 6) and there is a transcritical bifurca-
≈
Fig. 4. The change of stability of these periodic points tionassociatedwiththecreationofasetofinvarianttori
isassociatedwiththe creationofaninvarianttorus(also like in Figs. 3 and 4. The associated phase space shows
representedinFig.4bytheplotoftheseparatrices). We a significant increase of the size of the resonant island
noticethatapartfromtheexchangeofstability,thephase
space inthe neighborhoodof these periodic orbits is still
regular (the chaotic region around the hyperbolic peri- B. Enhancing chaos near a resonant island
odic orbits is not well developed), and hence the regular
nature of phase space has not been changed locally (or In this section, we address the destruction of a reso-
one needs to consider higher values of the parameters). nantislandby breakinguplibrationalinvarianttori. We
The same analysis can be carried out on a set of pe- notice that onFig.5, a bifurcationoccurs atk 1.254
≈−
riodic orbits which are located in a more chaotic region, fortheCase2(Q=9)whentheresidueoftheellipticpe-
like for instance the two cases Q = 9 and Q = 13 of riodicorbitbecomesequalto1. Theneighborhoodofthis
Fig. 1 outside the regular resonant island. The values periodic orbit becomes a chaotic layer and the qualita-
of the residues as functions of the parameter k are re- tivechangeinthedynamicsisseensincethechaoticlayer
spectively represented on Figs. 5 and 6 for Q = 9 and becomes thicker atthis value ofthe parameter. However
Q = 13. For Q = 9, we notice that the residues do not sincethisperiodicorbitwasinitially(atk =0)alreadyin
vanishintherangeofk weconsideredalthoughthereare theouterchaoticregion(seeFig.1),theregularizationis
small and extremum at the same value of the parameter not drastic. In order to obtain a more significantchange
5
2 1.5
1.5
1 1
0.5
R 0 p 0.5
−0.5
0
−1
−1.5
−0.5
−2 0 1 2 3 4 5 6
−2 0 2 4 x
k
0.2
FIG. 6: Residues R of the elliptic and hyperbolic periodic
orbitsof Case 3 (Q=13) asfunctions oftheparameterk for 0.1
Hamiltonian (2) with ε=0.065. 0
−0.1
p
−0.2
in the dynamics and a large chaotic zone, one needs to −0.3
select a periodic orbit inside a regular region,like for in- −0.4
stance the one with Q = 13. A bifurcation occurs at −0.5
k 2.484 where the residue of the elliptic periodic or- 0 1 2 3 4 5 6
≈− x
bitcrosses1(seeFig.6). APoincar´esectionforthelatter
case is depicted on Fig. 7, and shows that a significantly
FIG.7: Poincar´esectionofHamiltonian(2)forε=0.065and
large neighborhood has been destabilized by the control k=−2.484.
term. We notice that the ratio between the size of the
control term and the one of the perturbation is equal to
kε 0.16. We also notice that this last value of k is For Q = 2, the residues of the elliptic and hyperbolic
≈
largerthanthe onerequiredforQ=9. Asexpected,one periodic orbit vanish at k 4.3, and for Q = 3, at
needs a largeramplitude to destabilize a regioncloser to k 3.4. For Q = 5, both r≈esidues vanish at k = 2.98
aregularone. Amoreeffectivedestabilizationprocedure an≈d also at k 5.07. At these values of the parameter,
can be obtained with the periodic orbit Q = 4 which is the phase spac≈e is locally filled by invarianttori where it
inside the regular region. However, as mentioned, the is also expected that the goldenmean invariant torus is
value necessaryfor this destabilization(k 12.5or for present. We notice that the elliptic periodic orbit with
≈−
k 16.3)is too large;hence we discardit because ofour Q = 8 (the next one in Greene’s residue approach for
≈
restriction on energy cost. the analysis of the golden mean torus) is destabilized at
ε 0.0325. Therefore, there is no elliptic periodic orbit
≈
with Q=8 at ε=0.034 and the analysis using the cou-
C. Creation of the goldenmean rotational invariant pledelliptic/hyperbolicperiodicorbitscannotbe carried
torus out. However,by following the two (initially hyperbolic)
periodic orbits with Q = 8, we see that both residues
In this section, we apply the same approach on rota- vanish at k 2.84.
≈
tional invariant tori. It allows us to compare the results Wecomparedthesevaluesofstabilizationwiththeone
with the ones obtained by a control method proposed in givenbyamethodoflocalcontrolbasedonanappropri-
Refs. [9, 12, 13]. First,the idea is to lookat the creation atemodificationofthepotentialtoreconstructaspecific
of a specific invariant torus. For instance, we select a invariant torus [9, 12]. Such method provides explic-
torus which has been widely discussed in the literature itly the shape (and amplitude) of possible control terms
(see Ref. [14] and references therein), the goldenmean whereastheoneusedinthisarticlehasbeenguessedfrom
one, which has a frequency ω =(3 √5)/2 for Hamilto- thesereferences. By appropriatetruncation(keepingthe
−
nian (2). We choose ε = 0.034, and we first notice that main Fourier mode), this method provides
when k =0, this Hamiltonian does not have such an in-
varianttorus (since its criticalvalue is ε 0.02759[14]). ε2
≈ f(x,t)= cos(2x t),
The purpose here is to find the value of the control pa- 2ω(1 ω) −
−
rameter k needed by the residue method to reconstruct
this invariant torus (such that Hamiltonian (2) has this where ω = (3 √5)/2, as an approximate control term.
−
invariant torus). Therefore the amplitude is k=1/ω(1 ω) 4.24 which
− ≈
The idea of doing this follows Greene’s residue crite- isofthesameorderasthevaluesobtainedbyzeroingthe
rion. By performing an appropriate change of stability residues. However, we point out that smaller values are
on higher and higher order periodic orbits, the ampli- obtained by looking at higher periodic orbits.
tude of the control term should be smaller and smaller. Therefore, an efficient control strategy is to combine
6
theadvantagesofbothmethods: First,thespecificshape 2
of the terms that haveto be added to regularizethe sys-
tem is obtained using the method of Ref. [9]. Then the 1
amplitudes of these terms are lowered using high order
0
periodic orbits. By considering the control term used in
R
this article, we expect that zeroing the residues of high
−1
period will not be feasible with just this term (as it is
the case for instance in Fig. 5). A more suitable form of −2
controltermswouldbeconstructedfromanexactcontrol
term which is −3
−1.5 −1 −0.5 0
k
ε2
f(x,t) = cos(2x t)
2ω(1 ω) − FIG. 8: Residues R of the elliptic and hyperbolic periodic
− orbits with Q=13 and also of the one with Q=26 (dashed
2 2
ε ε
line) born out of a period doubling bifurcation for Hamilto-
cos2x cos2(x t).
−4ω2 − 4(1 ω)2 − nian (2) with ε=0.0275.
−
However, it should be noticed that a control term given
by Ref. [9] is not always experimentally accessible. The changesofdynamicsoccurringastheparameterisvaried.
idea is to use a projection of this control term onto a Wenoticethatthebehaviorsdescribedbelowaregeneric
basisofaccessiblefunctions. Thisprojectedcontrolterm for all the neighboring periodic orbits.
wouldgiveanideaofthetypeofcontroltermstobeused
Theresiduesoftheseperiodicorbitsasfunctionsofthe
for the residue method.
parameter k are shown in Fig. 8. We notice that the el-
Wewouldliketostressthatintheabsenceofellipticis- lipticperiodicorbitchangesitsstability,i.e.becomeshy-
landsaninitialguessforthe Newton-Raphsonmethodis perbolic,atk 1.205(whereitsresiduebecomesequal
not straightforward from the inspection of the Poincar´e to 1). A close≈in−spection of the Poincar´e section shows
section. In particular, it is not easy to select the ap- onFig.9thatitundergoesaperioddoubling bifurcation
propriate hyperbolic periodic orbits which will lead to a intoanellipticperiodicorbitwith26intersectionsonthe
significantchangeinthe dynamics. However,once it has Poincar´e section (and winding ratio 10/26) which has a
been located, the method can follow them by continuity residue zero at the bifurcation. By following the residue
in the same way as the elliptic ones since the Newton- of this elliptic periodic orbit (depicted by a dashed line
Raphson method does not depend on the linear stability in Fig. 8) we see that it vanishes for k = 1.6256. At
of these orbits. This makes the method more difficult this value of the parameter and for higher−value in am-
(althoughpossible)tohandleforjusthyperbolicperiodic plitude, all the periodic orbits considered here (the two
orbits. with Q = 13 and the one with Q = 26) are hyperbolic.
Thereforeitisexpectedthatthereisachaoticzoneinthis
areaanditisavalueatwhichthetorusisexpectedtobe
D. Destruction of the goldenmean rotational broken (confirmed by a close inspection of the Poincar´e
invariant torus section).
It is important to notice that a vanishing residue does
In this section, we consider Hamiltonian (2) with ε = notautomaticallyimplythatthereisacreationofanin-
0.0275. We notice that for k = 0, Hamiltonian (2) does variant torus, contrary to the previous cases which were
have the rotational goldenmean torus. The purpose is obtained by using jointly the elliptic and hyperbolic pe-
to find some small values of the parameter k where this riodicorbits(andvanishingresiduesinbothcases). Here
invariant torus is destroyed. We notice that this case is the hyperbolic periodic orbit associatedwith these ellip-
easiertofindthaninthe previoussectionsinceitiswell- ticperiodicorbits(whichistheperiodicorbitfromwhich
known that any additional perturbation will end up by the new elliptic orbit was born out by a period doubling
destroying an invariant torus generically. Here it means bifurcation)stayshyperbolicastheresidueoftheelliptic
that there will be largeintervals ofparametersfor which onevanishes. Thisfeatureis generic: The sameanalysis
the torus is broken (contrary to the case of the creation has been carriedout for higher order elliptic periodic or-
of invariant tori). However we will add an additional bitsclosetothegoldenmeaninvarianttorus,i.e.theones
assumption that the parameters for which this invariant withwindingratio8/21,13/34,21/55,34/89: First,the
torus is destroyed has to be small compared with the values of the control parameter for which the residues
perturbation. We also notice that the destruction of the (which are around 0.25 for k = 0 and increase as k de-
goldenmeaninvarianttorusis firstobtainedfornegative creases) cross 1 are computed and reported in Table I
values of the control parameter (see Fig. 8). (denoted k(R = 1)). At these values of the parameters,
First we illustrate the method by considering specific a period doubling bifurcation occurs for each of them.
ellipticandhyperbolicperiodicorbits(withwindingratio Then we follow the residues of the elliptic periodic or-
5/13) near the goldenmean torus which will show the bits with double period Q = 42, Q = 68, Q = 110 and
7
that the goldenmean invariant torus is destroyed by this
additionalperturbationbutnottheonesintheneighbor-
hood. Ifoneislookingatlargescaletransportproperties,
0.4 these other invariant tori have to be taken into account.
0.39
0.38 0.4017
p0.37 Concluding remarks
p
0.36
0.35 0.4016 In this article, we reviewed and extended a method of
control of Hamiltonian systems based on linear stabil-
0.34 3.05 3.1 x3.15 3.2
ity analysis of periodic orbits. We have shown that by
0.33
0 1 2 3 4 5 6
x varyingtheparameterssuchthatthe residuesofselected
periodic orbits cross 0 or 1, some important bifurcations
happeninthesystem. Thesebifurcationscanleadtothe
FIG.9: Poincar´esectionaroundtheperiodicorbitwithwind- creation or the destruction of invariant tori, depending
ingratio5/13(indicatedwithcrosses)forHamiltonian(2)for on the situation at hand. Therefore we have proposed
ε=0.0275 and k=−1.215. The period orbit with period 26
a possible extension of the residue method to the case
(indicated by circles) results from a period doubling bifurca-
of increasing chaos locally. Moreover,we have compared
tion of theonewith period 13 (represented by crosses on the
two methods of chaos reduction, and by taking advan-
Poincar´e section).
tage of both methods, we have devised a more effective
control strategy. It is worth noticing that the extension
Q 13 21 34 55 89 ofCary-Hanson’smethodtofourdimensionalsymplectic
k(R=1) -1.205 -0.705 -0.435 -0.273 -0.179 mapshasbeendoneinRefs.[6,15]fortheincreaseofdy-
kc -1.626 -0.935 -0.566 -0.350 -0.225 namic aperture in accelerator lattices. The extension to
the destruction of invariantsurface would be to consider
TABLE I: Values of the parameter k at which the residue of the change of linear stability of selected periodic orbits.
the elliptic periodic orbit with period Q crosses 1 (denoted However, it would require to consider new types of bi-
k(R = 1)) and at which the residue of the elliptic periodic
furcations which occurs in the system, like for instance,
orbitwithperiod2Qobtainedbyperioddoublingbifurcation
Krein collisions [16].
at k(R=1) vanishes(denoted kc).
Q = 178. The parameter values at which these residues Acknowledgments
vanish are also reported in Table I. For instance, using
the periodic orbit with winding ratio 8/21, we obtain This work is supported by Euratom/CEA (contract
k = 0.935 as the value at which the residue of the bi- EUR 344-88-1 FUA F). We acknowledge useful discus-
−
furcated elliptic periodic orbit with winding ratio 16/42. sions, comments and remarks from J.R. Cary and the
If we consider higher order periodic orbits, it happens Nonlinear Dynamics group at CPT.
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