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Tutorials and Reviews

International Journal of Bifurcation and Chaos, Vol. 20, No. 5 (2010) 1293–1319c© World Scientific Publishing CompanyDOI: 10.1142/S0218127410026496

BIFURCATIONS AND CHAOSIN HAMILTONIAN SYSTEMS

R. BARRIO∗,†, F. BLESA‡ and S. SERRANO∗∗GME, Departamento Matematica Aplicada,

†IUMA,‡GME, Departamento Fısica Aplicada,

Universidad de Zaragoza, E-50009 Zaragoza, Spain

Received July 27, 2009; Revised July 29, 2009

This paper deals with the use of recent computational techniques in the numerical study of quali-tative properties of two degrees of freedom of Hamiltonian systems. These numerical methods arebased on the computation of the OFLI2 Chaos Indicator, the Crash Test and exit basins and theskeleton of symmetric periodic orbits. As paradigmatic examples, three classical problems arestudied: the Copenhagen and the (n+1)-body ring problems and the Henon–Heiles Hamiltonian.All the numerical integrations have been done by using the state-of-the-art numerical libraryTIDES based on the extended Taylor series method.

Keywords : Henon–Heiles Hamiltonian; Copenhagen problem; (n+1)-body ring problem; periodicorbits; chaos indicators; Taylor series method; 2DOF Hamiltonian systems.

1. Introduction

Many real problems coming from Physics or Chemi-stry are formulated as Hamiltonian systems andmost of them admit a particular case that hasjust two degrees of freedom (2DOF). In this situ-ation, the numerical and analytical studies providevery detailed information about the behavior of thesystem.

Our goal in this review paper is to intro-duce some new techniques that permit to studywith much more detail 2DOF Hamiltonian sys-tems. We use the OFLI2 Chaos Indicator [Barrio,2005b, 2006b] to study where the systems seemto be chaotic. Also, we use another technique, theCrash Test [Nagler, 2005], to classify the orbits asbounded, escape or collision. In the cases of severalexits we may compute, instead of the Crash Testthe exit basins that provide a similar information.Finally, we present a study of the families of sym-metric periodic orbits for these problems. Theseorbits have been obtained by using a systematicsearch method [Barrio & Blesa, 2009]. With this

method, we have been able to obtain the “skeleton”of these dynamical systems. The combined use ofall the techniques gives us a complete idea of thedynamics of the problems. A key tool in all of thetechniques is the use of a good numerical ODEintegrator. Obviously, in the last few years thenumerical ODE community has presented manyhigh quality geometric numerical integrators quitesuitable for these problems. We use a recently devel-oped freeware library called TIDES (Taylor Inte-grator for Differential EquationS) that provides uswith an extremely adaptive and useful software forthis class of numerical simulation (see [Abad et al.,2009a, 2009b]).

In Celestial Mechanics the most importantproblem is the n-body problem, but a study of thegeneral case is not easily affordable. In the litera-ture, there is a large number of particular cases andsimplifications. In this paper, we focus our attentionon the Copenhagen and the (n+1)-body ring prob-lems. In Galactic studies a paradigmatic problem isthe Henon–Heiles model, which was posed to study

1293

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1294 R. Barrio et al.

the galactic movement, but it was also one of thefirst models of Hamiltonian chaos. Moreover, thismodel has several applications in Theoretical Chem-istry and is a good example of open Hamiltonians.These three paradigmatic models are two degrees offreedom (2DOF) Hamiltonian systems, a situationthat permits the use of a large number of analyticaland numerical techniques commented above. There-fore, the three problems studied here are just threeparticular examples that illustrate which informa-tion can be obtained by the combination of differentnumerical techniques.

The present paper is organized as follows. Sec-tion 2 presents the different numerical techniques,Sec. 3 gives the numerical results for the n-bodyproblems and Sec. 4 for open Hamiltonians. Finally,we present some conclusions.

2. Numerical Techniques

In this paper, we focus our attention on the use ofstate-of-the-art numerical techniques for the anal-ysis of the qualitative properties of two degrees offreedom Hamiltonian problems. In this section, wedescribe briefly the basic numerical techniques usedin the analysis of the chaotic regions, the collisionsand the exit basins, the computation of periodicorbits and on the numerical integration of ODEs.

2.1. OFLI2 chaos indicator

The first technique that we present here is OFLI2[Barrio, 2005b, 2006b; Barrio et al., 2009c], whichis a fast chaos indicator, that is, a fast numericaltechnique to detect chaos. These past years, a largenumber of numerical techniques to detect chaoshave appeared (FLI, SALI [Skokos et al., 2004],MEGNO [Cincotta et al., 2003], . . .). One of them,based on variational methods, is the Fast LyapunovIndicator (FLI). The OFLI2 indicator extends thedefinition of the FLI and OFLI indicators. The FastLyapunov Indicator (FLI) [Froeschle & Lega, 2000;Fouchard et al., 2003] was introduced as the initialpart (up to a stopping time tf ) of the computa-tion of the Maximum Lyapunov Exponent (MLE[Skokos, 2010]):

FLI(y(t0), δy(t0), tf ) = supt0<t<tf

log ‖δy(t)‖,

where y(t) and δy(t) are the solutions of the systemand the first order variational equations

dydt

= f(t,y),dδydt

=∂f(t,y)

∂yδy.

In order to detect easily periodic orbits, a vari-ation was introduced, called OFLI [Fouchard et al.,2003] (Orthogonal Fast Lyapunov Indicator)

OFLI(y(t0), δy(t0), tf ) = supt0<t<tf

log ‖δy⊥(t)‖

where δy⊥ is the component of δy orthogonal tothe flow at that point. The problem is that theseindicators (and most of the methods that are onthe literature) exhibit spurious structures [Barrioet al., 2009c]. To minimize the appearance of thesespurious artifacts, we developed the OFLI2 method.The method is based on the use of the secondorder variational equations. We use as numeri-cal ODE integrator, a specially developed Taylormethod [Barrio et al., 2005; Barrio, 2006a] thatgives a fast and accurate numerical integration aswe will explain below in Sec. 2.3. The OFLI2 willbe detecting the set of initial conditions where wemay expect sensitive dependence on initial condi-tions. The OFLI2 indicator at the final time tf isgiven by

OFLI2 := supt0<t<tf

log∥∥∥∥{ξ(t) +

12δξ(t)}⊥

∥∥∥∥ , (1)

where ξ and δξ are the first and second order sensi-tivities with respect to carefully chosen initial vec-tors and x⊥ stands for the orthogonal componentto the flow of the vector x. In this case the varia-tional equations up to second order and the initialconditions are

dρ

dt= f(t,ρ), ρ(t0) = ρ0,

dξ

dt=

∂f(t,ρ)∂ρ

ξ, ξ(t0) =f(t0,ρ0)‖f(t0,ρ0)‖

,

dδξj

dt=

∂fj

∂ρδξ + ξ�

∂2fj

∂ρ2ξ, δξ(t0) = 0.

(2)

Note that the last line of Eqs. (2) is written for asingle jth component δξj to simplify the notation.

The evolution of the FLI and OFLI indicators isexplained in [Guzzo et al., 2002] for quasi-integrablesystems:

Proposition. Given the Hamiltonian function

Hε(I, φ) = h(I) + ε f(I, φ), (3)

with action-angle variables I1, . . . , In ∈ R andφ1, . . . , φn ∈ S and ε a small parameter, and sup-pose that the functions h and f satisfy the hypothe-ses of both KAM and Nekhoroshev theorems then

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Bifurcations and Chaos in Hamiltonian Systems 1295

the following estimates were given:

(1) If the initial conditions are on the KAM torus

‖(δIε(t), δφε(t))‖ =∥∥∥∥∂2h

∂I2(I(t0))δI(t0)

∥∥∥∥ t

+O(εαt) + O(1),

for some α > 0.(2) If the initial conditions are on a regular reso-

nant motion

‖(δIε(t), δφε(t))‖

= ‖CΛΠΛortδI(t0)‖t + O(εβt) + tO(ρ2)

+O(√

εt) + O(

1√ε

),

with some β > 0,Λ ⊆ Zn a d-dimensional lat-

tice that defines a resonance through the rela-tion ΠΛ(∂h/∂I) = 0 where ΠΛ denotes theEuclidean projection of a vector onto the lin-ear space spanned by Λ and CΛ a linear opera-tor depending on the resonant lattice Λ and theinitial action I(t0).

For the OFLI2, it is possible to obtain similarestimates [Barrio, 2005b]. Then, we expect the FLIand OFLI2 to behave as log t [Barrio, 2005b] forinitial conditions on a KAM tori and on a regularresonant motion but with different rate of growing(and so they grow linearly in a logarithmic timescale as in Fig. 1), tend to a constant value for theperiodic orbits and grow exponentially (in a loga-rithmic time scale) for chaotic orbits.

In Fig. 1, we show the evolution of the OFLI2values in the time interval [1, 10,000] for four partic-ular orbits of the Henon–Heiles problem. The orbitsare indicated with the letters “a” (a periodic orbit),“b” and “c” (orbits on a KAM tori) and “d” (achaotic orbit close to the hyperbolic point).

100 10

210

40

2

4

6

8d c

b

a

time

OF

LI2

Fig. 1. Evolution of the OFLI2 values in the time interval[1, 10 000] for four particular orbits: “a” (a periodic orbit),“b” and “c” (orbits on a KAM tori) and “d” (a chaotic orbitclose to the hyperbolic point).

From now on, all the figures with the OFLI2results use the red color to point the chaotic regionsand blue for the most regular ones, the intermedi-ate colors being the transition from one to anothersituation.

2.2. Crash test and exit basins

In the numerical study of a dynamical system, acommon tool is the Poincare Surfaces of Section(from now on PSS). The basic idea is to select a two-dimensional manifold transverse to most of the tra-jectories of the system and to study their cuts. Anextension of this technique which is useful for sys-tems with several bodies has been developed quiterecently. This extension is denoted as the Crash Test[Nagler, 2004] (in the following CT). CT is based onclassifying the orbits in a PSS depending on severalparameters. For instance, we may study if an orbitis direct or retrograde, bounded or an escape orbit,if the orbit crashes to another body or not, andso on. To each type of orbits, a different color isassigned and so the picture gives much more infor-mation than just the classical PSS. Another exten-sion [Barrio et al., 2009a; Nagler, 2004, 2005] is tomake an analysis of a mesh of initial conditions andto assign them a color depending on their behavior.The difference of this second extension is that nowwe do not look for cuts with the manifold, beingjust a manifold of initial conditions. Now, we maydistinguish among some basic behaviors (describedbelow for the Copenhagen problem and illustratedin Fig. 2 on the left):

• Collision with the central body. We consider thata collision with a body occurs when the distanceof the orbit to the body is lower than rc = 10−4.We give it a blue or cyan color depending on thecollision body.

• Escape orbits. Since we are dealing with a non-integrable problem, it is not easy to prove if themotion is bounded or not. After several numeri-cal tests we have decided to perform a numericalintegration up to a final time tfinal = 20000 and tocheck if, along the computed trajectory, the orbitreaches a value of R = ‖(x, y)‖ > Rmax = 10,then we suppose the orbit is far enough for thetwo bodies, and we stop the integration. Weassign it a yellow (tescape small) or red color(tescape large).

• Bounded motion. In another case, we classify theorbit as bounded orbit and we give it a greencolor.

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1296 R. Barrio et al.

Fig. 2. Left: Schematic color code for the Crash Test: Blue or cyan color — collision, from yellow to red color — escape andgreen color — bounded motion. Right: Color codes for the different escape routes for the Henon–Heiles problem.

Another related technique is the study of theescape basins on open Hamiltonians. As it is wellknown, the exit basin of a particular exit is theset of initial conditions that escape through suchan exit. As expected, usually the geometry of theexit basins is fractal, in fact, they are fat-fractals.Therefore, as in some implementations of the Crash-Test, we analyze a mesh of initial conditions andwe assign them a color depending on their behav-ior. We show in Fig. 2 (right) the different colors foreach exit basins for the Henon–Heiles Hamiltonianand the equipotential lines. The straight lines corre-spond to the escape energy Ee = 1/6. In Sec. 4, westudy the exit basins of this Hamiltonian with somedetail.

2.3. Systematic search ofsymmetric periodic orbits

In Hamiltonian dynamics, the study of the invari-ants provides the key tools in the analysis of the sys-tems. One of the more important sets of invariantsare the periodic orbits of the system. In the numer-ical computation of the skeleton of symmetric peri-odic orbits, we use the classical systematic searchmethod. The basis of this method was alreadyknown by Birkhoff, DeVogelaere and Stromgren andused numerically firstly by Henon [1965], but itscombined use with efficient numerical ODE inte-grators and guaranteed root solvers give rise to

highly powerful methods to locate symmetric peri-odic orbits (for more details see [Barrio & Blesa,2009] and references therein). This method is veryuseful for 2DOF Hamiltonian systems with somesymmetries. In our case, the systems present sev-eral symmetry axes. In the systematic search proce-dures, the first step is to compute the Poincare mapin a particular well-chosen plane. Besides, takinginto account the remarks of [Dullin & Wittek, 1995],the manifold (in our case plane) has to be chosen inorder to be transverse to all orbits so it will describethe complete system. That is, we look for completePoincare sections, but at the same time, we need tohave the symmetry conditions at the sections thatare needed for the systematic search. Without los-ing generality, let us suppose that we choose for thePoincare map the x–x plane imposing y = 0 and yobtained from the Hamiltonian constant. Also, wesuppose that these Poincare sections are symmetricor reverse symmetric with respect to the x-axis, thatare the fixed points of the symmetries. This impliesthat if {x(t), y(t)} is a solution, then {x(t),−y(t)}or {x(−t),−y(−t)} is also a solution. Therefore, inthe reverse symmetry case, if an orbit starts at thex-axis perpendicular to it

(x(0), y(0), x(0), y(0)) = (x0, 0, 0, y0) (4)

and crosses the x-axis again perpendicular, thenthe orbit is closed and symmetric and so it is asymmetric periodic orbit. Thus, we check for this

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Bifurcations and Chaos in Hamiltonian Systems 1297

condition: a new cross perpendicular to the x-axis,which implies

y

(x0, 0, 0, y0;

T

2

)= x

(x0, 0, 0, y0;

T

2

)= 0. (5)

Note that the second cross is done exactly at halfperiod T of the orbit. For other kind of symmetriesthe conditions are similar and therefore the same ora similar method may be used. In order to proceedin the search of periodic orbits, the first step is topresent a mesh in the parameter and variable space.In our systems, we consider autonomous 2DOFHamiltonian systems and therefore the Hamilto-nian is an integral of movement. Thus, we use Hor any other parameter (if any) of the problem, asthe parameter P. Therefore, we perform a mesh inthe plane (x,P). In our case, we use a regular meshand we move on the x variable (or P), which meansthat P (or x) is fixed and we perform numericalintegrations of the system for the different values ofx (or P) in the mesh. Each value of x and P deter-mines a complete set of initial conditions for oursearch (note that x0 = y0 = 0 and y0 is obtainedfrom P). For each set of initial conditions, we inte-grate numerically to compute the Poincare map fora given multiplicity m (number of crossings of thex-axis with y > 0). That is, if we just look for thefirst cross we will only obtain the periodic orbitsof multiplicity one, going further, we also searchfor higher order multiplicities. At the values of thePoincare map we check the sign of x, and by conti-nuity, when a change of sign is detected, this impliesthat the condition (5) is satisfied inside the intervalof the variable x. Thus, another rootfinding processis necessary but now it has to be used in combina-tion with a numerical integrator because we haveto integrate the differential system for each itera-tion of the process. Once the convergence is reachedwe have a set of initial conditions that satisfy thesymmetric periodic conditions (5) in the absence ofnumerical errors.

Once a symmetric periodic orbit has been cal-culated, we shall study the behavior of the systemin the vicinity of the periodic solution. Therefore,we may analyze the stability of the orbit. The linearstability is studied by means of the eigenvalues ofthe monodromy matrix, that is, the solution at timeT (one period of the periodic orbit) of the matrixlinear differential system (for Hamiltonians of theform H(x, y, x, y) = 1/2(x2 + y2)+U(x, y), in othersituations, we just need to compute the variational

equations for such a case)

δX =

0 0 1 00 0 0 1

−∂2U

∂x2− ∂2U

∂x∂y0 0

− ∂2U

∂x∂y−∂2U

∂2y0 0

δX, (6)

with

X(0) =

1 0 0 00 1 0 00 0 1 00 0 0 1

.

As the system is Hamiltonian the monodromymatrix is a real symplectic matrix and we have thatits eigenvalues {λi, i = 1, . . . , 4} are in reciprocalpairs

λ1λ2 = 1, λ3λ4 = 1,

and the complex eigenvalues are also in complexconjugate pairs. For a periodic orbit, we alwayshave an eigenvalue equal to one and therefore itsreciprocal is also one. Thus, we just have to studythe remaining two eigenvalues λ3 and λ4. As theyare complex conjugate and reciprocal they are onthe unit circle in the complex plane or on the realaxis. In order to have a stable periodic orbit thetwo eigenvalues have to be on the unit circle. If λ3

and λ4 are real, the orbit is unstable. The specialcases λ3 = λ4 = 1 or λ3 = λ4 = −1 are called crit-ical (at these points some bifurcations may occur).Note that in fact we do not need to compute explic-itly the eigenvalues λ3 and λ4. We just need to thetrace the monodromy matrix which is the sum ofthe diagonal elements and it is equal to the sum ofthe eigenvalues. Therefore, we use instead the traceminus two k := trace(X(T )) − 2. Thus a periodicorbit is stable iff |k| < 2, unstable iff |k| > 2 andcritical iff |k| = 2.

When the differential system depends on aparameter P, one wants to know if a periodic solu-tion may be continued, that is, if the p.o. is iso-lated or not when the parameter is changed. Forautonomous Hamiltonian systems, if the periodicorbit is elementary [Meyer, 1970] the orbit can becontinued and so the periodic orbits appear in fam-ilies. Note that if we have a nonelementary periodicsolution (that is, with κ = 2) the periodic orbit mayappear or disappear. A family of periodic orbits

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1298 R. Barrio et al.

is represented by a continuous curve (the charac-teristic curve) in the plane of initial conditions orparameters.

When we have a family of periodic orbits, weare interested to know when it appears, disappearsor when it bifurcates. From the comments above,we know that if κ �= 2 then a periodic orbit is amember of a smooth one-parameter family of peri-odic orbits. Moreover, the converse gives a quiteimportant result: periodic orbits can only appearor disappear when κ = 2.

Above we have defined multiplicity as the num-ber of crossings in the Poincare map we use (itdepends on the symmetries). In order to study whenand how a periodic orbit bifurcates, we need todefine local and global multiplicities:

Definition. We define global multiplicity m of aperiodic orbit with respect to a given Poincare mapas the total number of crossings, once the orienta-tion is fixed, and we define local multiplicity m of abifurcated periodic orbit in the vicinity of the mainperiodic orbit with respect to a given Poincare mapas the number of crossings, once the orientation isfixed, in a neighborhood of the main orbit.

The local multiplicity is defined for bifurcatedorbits (in the main orbit, if it is not a bifurcationof another main orbit, it coincides with the globalmultiplicity).

Once a bifurcation occurs with a local multi-plicity m around a main orbit of global multiplicitym, the global multiplicity of the bifurcated orbit ism · m. We denote by mn : mn−1 : · · · : m1 : m0

an orbit obtained after n bifurcations of local mul-tiplicities m1, . . . , mn from a main orbit of globalmultiplicity m0. The global multiplicity of such anorbit is just m = mn · mn−1 · · · m1 · m0. From nowon, in order to simplify the notation, we will usejust the notation m for both multiplicities if it isclear which is the one in each case.

Now, taking a p.o. of period T and monodromymatrix Π1 = Π(T ) we define ∀m ∈ N, Πm =Π(mT ). Therefore, a p.o. of local multiplicity mcan appear or disappear at points such that κm :=κ(Πm) = 2. These points are called m-bifurcationpoints. So, we look for points in a p.o. where κm = 2or instead points where κ is twice the real part of amth root of the unity, that is

κ = λ3 + λ4 = 2Re(λ3,4) = 2 cos(

2πk

m

)(7)

with k and m coprime natural numbers. The ratiok/m is called the rotation or winding number. Atthis point, the main p.o. bifurcates in p.o. of localmultiplicity m.

Once we know where a periodic orbit bifurcates[Eq. (7)] we are interested to know how. In thispaper, we just describe the typical [Meyer, 1970]local bifurcations of the families of periodic orbits.With typical we mean that other things can hap-pen, but they would be exceptional. In particular,we review the generic bifurcations for 2DOF Hamil-tonians [Meyer, 1970] and we comment on just onebifurcation in the presence of some symmetries. InFig. 3, we show a schematic Poincare section of allthe typical bifurcations for such systems. We sup-pose that the bifurcation occurs at parameter PB

and we write P = PB + ε. Note that we only showone direction, from ε < 0 to ε > 0, but it can bethe opposite.

The first case is a nonelementary periodic solu-tion (κ = 2) and corresponds to the case of asaddle-node bifurcation where two periodic orbitsare created (or destroyed), one stable and anotherone unstable (the unstable orbits are marked with ared cross). This is the only way of creating new fam-ilies of periodic orbits, apart from the boundaries ofthe domain in the definition of the Poincare map.We remark that although from the local multiplicityanalysis, these orbits are of m = 1, the global mul-tiplicity may be anyone, that is, saddle-node bifur-cation periodic orbits of any global multiplicity mthat appear or disappear. The last case is the sym-metric pitchfork bifurcation: from a s.p.o., two newp.o. are created but with less symmetries. Both areisochronous with the main s.p.o. (local multiplicity1 with respect to the main orbit). The main s.p.o.changes its stability character. The Poincare sectionof the period doubling bifurcation is quite similar tothe pitchfork case but now the two crossings defineone s.p.o. of local multiplicity 2 (there are two suchorbits). The generic bifurcations for local multiplic-ities m = 3 and 4 are the touch-and-go bifurca-tions. The cases of local multiplicity m (m > 2 or3 depending on the case) are the creation of reso-nance islands around the main orbit that remainsstable.

The above stability analysis is done for pla-nar symmetric periodic orbits with respect to per-turbations of the initial conditions in the plane.Note that a planar problem may also be under-stood in 3D and thus it is interesting to see thestability with respect to perturbations of the initial

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Bifurcations and Chaos in Hamiltonian Systems 1299

Fig. 3. Typical bifurcations for 2DOF Hamiltonians.

conditions perpendicular to the plane of motion. Toreach that goal the planar formulation is extendedto 3D [Hadjifotinou & Kalvouridis, 2005] just byadding

z = −∂U

∂z.

Note that to have planar motion, we need thatz = z f(x, y, z) and so, if z(0) = z(0) = 0the movement will be planar. Following Henon[1973], we just compute the part of the monodromymatrix Z(T ) of z and z. That is (taking intoaccount that we study planar periodic orbits, and

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1300 R. Barrio et al.

so z(t) = z(t) = 0)

δZ =

0 1

−∂2U

∂z20

δZ

with

Z(0) =(

1 00 1

).

Now, the orbit is vertically stable iff both eigen-values λ5 and λ6 of Z(T ) lie on the complex unitcircle, unstable if both are real (note that the mon-odromy matrix Z(T ) is a symplectic matrix and soλ5λ6 = 1) and critical iff λ5 = λ6 = ±1 (in thiscase, vertical bifurcations are possible).

2.4. Adaptive ODE integrator:Taylor series method

In the process of determination of periodic orbitswe obviously have to integrate the differential sys-tem, normally for a short time, with very high preci-sion, especially for highly unstable periodic orbits.Moreover, in the study of bifurcations and stabil-ity of periodic orbits we also have to integrate thefirst order variational equations using as initial con-ditions the identity matrix. To reach this goal, wemay, obviously, use any numerical ODE method,like for example, a Runge–Kutta. These last fewyears, in the computational dynamics community[Guckenheimer & Meloon, 2000], one of the pre-ferred methods is the Taylor method.

The Taylor method is one of the oldest numer-ical methods for solving ordinary differential equa-tions but it is scarcely used in the numerical analysiscommunity. Its formulation is quite simple [Corliss& Chang, 1982]. Let us consider the initial valueproblem y = f(t,y). Now, the value of the solutionat ti (that is, y(ti)) is approximated by yi from thenth degree Taylor series of y(t) at t = ti (the func-tion f has to be a smooth function). So, denotinghi = ti − ti−1,

y(t0) =: y0,

y(ti) � yi−1 + f(ti−1,yi−1)hi

+12!

df(ti−1,yi−1)dt

h2i

+ · · · + 1n!

dn−1f(ti−1,yi−1)dtn−1

hni

=: yi.

Therefore, the problem is reduced to thedetermination of the Taylor coefficients {1/(j +1)! djf/dtj}. This may be done quite efficiently by

automatic differentiation (AD) techniques (for moredetails see [Barrio et al., 2005]). Note that theTaylor method has several good features; it givesdirectly a dense output in the form of a power series,therefore being quite useful when a G-stop criteriamay be used (as in the computation of Poincare sec-tions), it can be formulated as an interval methodgiving guaranteed integration methods (used, byinstance, in the computer proof of chaos [Galias &Zgliczynski, 1998]), Taylor methods handle directlyhigh order differential equations just by taking intoaccount that the Taylor coefficients for the solu-tion and its derivatives are evidently related, Taylormethods of degree n are also of order n and so Tay-lor methods of high degree provide numerical meth-ods of high order (therefore, they are very usefulfor high-precision solution of ODEs, as needed, forexample, in some fine studies of dynamical systems[Simo, 2003] and in the computation of unstablep.o.).

Taking a brief look at the practical implemen-tation of the Taylor series method, we remark thatin the literature there are efficient variable-stepsizevariable-order (VSVO) formulations. For example,in [Barrio et al., 2005] the variable-stepsize formu-lation is based on the error estimator using thelast two coefficients, giving the following stepsizeprediction

hi+1 = fac · min

Tol∥∥∥∥{

1(n − 1)!

f (n−2)(ti)}∥∥∥∥

∞

1n−1

,

Tol∥∥∥∥ 1

n!f (n−1)(ti)

∥∥∥∥∞

1n

where fac is a safety factor and Tol the usererror tolerance. A very simple order selection thatonly depends on the user error tolerance is given[Jorba & Zou, 2005] by the formula n(Tol) =−(1/2) ln Tol. See [Barrio, 2005a; Barrio et al.,2005] for a more extensive analysis and compara-tives with variable-stepsize variable-order formula-tions of the Taylor method. In Table 1, we presentsome comparisons on the Henon–Heiles problemwith initial conditions (x0, y0,X0, Y0) = (0, 0.52,0.371956090598519, 0) and E = 0.157494996 inthe time interval [0, 200] using the Taylor methodand the well established code dop853 developed

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Bifurcations and Chaos in Hamiltonian Systems 1301

Table 1. CPU time and final error using dop853 and a Taylor method with VSVO formu-lation for the Henon–Heiles problem using different compiler options (CO): double precision(dp) for tolerance level 10−10, quadruple precision (qp) for tolerance level 10−20 and multipleprecision (mpf90) for tolerance level 10−64.

Taylor dop853

CO Tol CPU RelErr CPU RelErr

dp 10−10 0.53E−02 0.201E−10 0.34E−02 0.205E−06

qp 10−20 0.30E+00 0.300E−20 0.30E+01 0.102E−17

mpf90 10−64 0.89E+01 0.144E−65

by Hairer and Wanner [Hairer et al., 1993]. Thiscode is based on an explicit Runge–Kutta of order8(5, 3) given by Dormand and Prince with step-size control and dense output. Both methods areonly compared in double and quadruple precisionusing the Lahey LF 95 compiler because the dop853cannot be directly used in multiple precision. Themultiple-precision tests are done using Fortran90 and the MPFUN90 multiple precision pack-age [Bailey, 1995]. Note that for low precision thedop853 code is a bit faster but when the precisiondemands are increased the Taylor method is by farthe fastest, being the only reliable method for veryhigh precision.

As we are interested not only in the differ-ential equations but also in the variational equa-tions and to avoid its explicit generation we havedevised [Barrio, 2006a] an alternative that permitsus to obtain the solution of the variational equa-tions without computing them explicitly. Therefore,we have to obtain a numerical solution of y(t) andLδy(t0)y(t), Lδy(t0)y(t) being the Lie derivative ofthe solution y(t) with respect to the vector δy(t0)(that is, in this case the directional derivative). Notethat

Π = (Le1y(t) | Le2y(t) | Le3y(t) | Le4y(t))

with (e1, e2, e3, e4) the canonical base of R4.

The Taylor series method computes the Taylorseries of the solution of the differential equation andthe Taylor series of the partial derivatives of thesolution

δy(ti) =∂y(ti)∂y(t0)

· δy(t0) = Lδy(t0)y(ti)

� Lδy(t0)y(ti−1) + Lδy(t0)f(ti−1)hi

+12!Lδy(t0)f

(2)(ti−1)h2i

+ · · · + 1n!Lδy(t0)f

(n−1)(ti−1)hni .

We may now compute the coefficients 1/(j + 1)!Lδy(t0)f (j)(ti−1) by rules of automatic differen-tiation of the elementary functions (±, ×, /,ln, sin, . . .) obtained in [Barrio, 2006a]. Automaticdifferentiation gives a recursive procedure to obtainthe numerical value of the reiterated derivativesof the elementary functions at a given point. Wepresent here the rules for the sum, product bya constant, product, division and real power offunctions adapted to our problems (in fact validfor any Taylor approximation of any Hamiltoniansystem):

Proposition. If f(t, y(t)), g(t, y(t)): (t, y) ∈R

s+1 �→ R are functions of class Cn and given avector v ∈ R

s, we denote

f [j,0] :=1j!

djf(t)dtj

, f [j,1] :=1j!Lvf (j),

that is, the jth Taylor coefficient of the functionf(t,y(t)) and of its Lie derivative with respect tov, respectively. Then, we have

(i) If h(t) = f(t) ± g(t) then h[n,i] = f [n,i] ± g[n,i].(ii) If h(t) = αf(t) with α ∈ R then h[n,i] =

α f [n,i].(iii) If h(t) = f(t) · g(t) then

h[n,0] =n∑

j=0

f [n−j,0] · g[j,0],

h[n,1] =n∑

j=0

(f [n−j,0] · g[j,1] + f [n−j,1] · g[j,0]).

(iv) If h(t) = f(t)/g(t) then

h[n,0] =1

g[0,0]

f [n,0] −

n−1∑j=0

h[j,0] · f [n−j,0]

,

June 1, 2010 15:41 WSPC/S0218-1274 02649

1302 R. Barrio et al.

h[n,1] =1

g[0,0]

f [n,1] − h[n,0] · f [0,1] −

n−1∑j=0

(h[j,0]

· f [n−j,1] + h[j,1] · f [n−j,0])

.

(v) If h(t) = f(t)α with α ∈ R and f [0,0] �= 0,then

h[0,0] = (f [0,0](t))α,

h[n,0] =1

nf [0,0]

n−1∑j=0

(n α − j(α + 1))h[j,0]

· f [n−j,0],

h[0,1] =1

f [0,0]αh[0,0] · f [0,1],

h[n,1] =1

nf [0,0]

−nh[n,0] · f [0,1]

+n−1∑j=0

(n α − j(α + 1)) (h[j,0]

· f [n−j,1] + h[j,1] · f [n−j,0])

.

The use of high-precision numerical integratorsin the determination of periodic orbits is justified,for instance, by the search of highly unstable peri-odic orbits.

To perform a short comparison of the Taylorseries method for solving ODEs and, in a directway, variational equations, we show in Fig. 4 sev-eral numerical tests with the Kepler problem. Wecompare again with the well established softwaredop853 [Hairer et al., 1993].

The Kepler problem, that describes the planartwo body motion with eccentricity e, is given by:

x = − x

(x2 + y2)3/2, y = − y

(x2 + y2)3/2,

and initial conditions x(0) = 1− e, y(0) = x′(0) = 0and y′(0) =

√(1 + e)/(1 − e). The integration time

is [0, 200 · 2π] (that is, 200 periods). We have con-sidered two values of the eccentricity, e = 0.7and e = 0.99. Note that this problem is usuallyemployed as a test for variable-stepsize strategiesfor high values of the eccentricity (near 1).

In all the numerical tests, we observe as forhigh-precision that the Taylor method presents withdifference the best behavior. Moreover, we canappreciate the different slope of the variable ordermethod (Taylor method) and the fixed order one(dop853), being clear that for high precision thevariable order schemes become more competitivebecause they are more versatile.

10-12

10-1010-810-610-410-3

0.1

0.2

0.30.4

Kepler (e=0.7): ∂/∂x0

com

pute

r tim

e

taylordop853

10-7

10-6

10-4

10-2

100

0.2

0.4

0.60.8

1Kepler (e=0.99): ∂/∂x

0

com

pute

r tim

e

relative error

10-810-610-410-2100

0.2

0.4

0.60.8

1Kepler (e=0.7): ∂2/∂x

0∂y

0

com

pute

r tim

e

10-8

10-6

10-4

10-2

100

0.5

1

1.52

2.5Kepler (e=0.99): ∂2/∂x

0∂y

0

com

pute

r tim

e

relative error

k1 k2

k3 k4

Fig. 4. Precision-computer time diagrams for the dop853 and Taylor method for the Kepler problem for two values of theeccentricity in the first order sensitivities with respect to the initial condition x0 and second order sensitivities with respectto the initial conditions x0 and y0.

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Bifurcations and Chaos in Hamiltonian Systems 1303

To end this section, we remark that the Tay-lor series method can be currently used along withthe new state-of-the-art numerical library TIDES(Taylor Integrator of Differential EquationS) thathas just been developed by Abad, Barrio, Blesa andRodrıguez [Abad et al., 2009a, 2009b]. (Contact theauthors to obtain the software.)

3. Simplifications of the n-BodyProblem

In Celestial Mechanics the most important prob-lem is the n-body problem, but a study of the gen-eral case is not easily affordable. In the literature,there are many specific cases and simplifications.In this review paper, we focus our attention on theCopenhagen [Barrio et al., 2009a] and the (n + 1)-body [Barrio et al., 2008a] ring problems. These twosimplifications are two degrees of freedom (2DOF)Hamiltonian systems, a situation that permits theuse of a large number of analytical and numeri-cal techniques. Therefore, the two problems studiedhere are just two specific examples that illustratewhich information can be obtained by the combi-nation of different numerical techniques.

3.1. The Copenhagen problem

The three-body problem is one of the oldest prob-lems in dynamical systems. The Restricted Three-Body Problem (RTBP) supposes that the mass ofone of the three bodies is negligible. It was con-sidered by Euler (1772) and Jacobi (1836), and itcan serve as an example of classical chaos. For theremaining two bodies, the case of equal masses wasfirst investigated by Stromgren and his colleaguesof the Copenhagen group. Its name is derived fromthe series of papers published by them starting in1913.

Defining the distances to the respective pri-maries as:

r21 = (x + µ)2 + y2,

r22 = (x − (1 − µ))2 + y2,

where µ = m1/(m1 + m2) with m1 and m2 themasses of the two bodies, the equations of motionof the Restricted Three-Body Problem are

x − 2y = x − (1 − µ)x + µ

r31

− µx − 1 + µ

r32

,

y + 2x = y − (1 − µ)y

r31

− µy

r32

.

The only known integral of motion of this problemis the Jacobian constant

C = −(x2 + y2) + 21 − µ

r1+ 2

µ

r2+ x2 + y2

that can be used to define the effective energy asEJ = −C/2. Since the Copenhagen problem is theparticular case of m1 = m2, we have to take thevalue µ = 1/2. In Fig. 5, we show on the left, aschematic drawing of the problem.

In Figs. 6–8 we show, for the Copenhagen prob-lem, the PSS, the CT (Ci) and the OFLI2 (Oi)pictures for several values of the energy EJ on the(x, y) and (x, x) planes. The initial conditions arex(0) and y(0) given by the point on the (x, y) plane,x(0) = 0 and y(0) determined by the energy EJ ,or x(0) and x(0) given by the point on the (x, x)plane, x(0) = 0 and y(0) obtained by the energyEJ . Each figure consists of a regular grid of 600×600points. The cases PS1, PS2, C1, C2, O1 and O2 areobtained for EJ = −1.73, cases PS3, PS4, C3, C4,O3 and O4 for EJ = −0.75 and cases PS5, PS6, C5,C6, O5 and O6 for EJ = 0.0. Note how the corre-spondence between the CT and the OFLI2 plots ishigh and how the classical PSS plots give less infor-mation. On white, we plot the forbidden regions.

r

r1

2

x1x2

b1

b2b3r(7)

^ ^

Copenhagenproblem

(7+1)-bodyring problem

Fig. 5. Left: Spatial configuration of the Copenhagen problem. Right: Spatial configuration of the (7+1)-body ring problem.

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1304 R. Barrio et al.

Fig. 6. PSS plots on the (coordinate x versus coordinate y)-plane (on the left) and (coordinate x versus velocity x)-plane(on the right) for several values of the energy EJ of the Copenhagen problem.

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Bifurcations and Chaos in Hamiltonian Systems 1305

Fig. 7. CT plots on the (coordinate x versus coordinate y)-plane (on the left) and (coordinate x versus velocity x)-plane (onthe right) for several values of the energy EJ of the Copenhagen problem.

June 1, 2010 15:41 WSPC/S0218-1274 02649

1306 R. Barrio et al.

Fig. 8. OFLI2 plots on the (coordinate x versus coordinate y)-plane (on the left) and (coordinate x versus velocity x)-plane(on the right) for several values of the energy EJ of the Copenhagen problem.

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Bifurcations and Chaos in Hamiltonian Systems 1307

The escape and bounded regions have a more reg-ular behavior and the collision regions have a morechaotic behavior. The position of the two primariesare marked with pink dots or pink vertical lines onthe (x, y) and (x, x) planes respectively. For highvalues of the energy, there are many more escapeorbits, and the bounded motion regions decrease insize. There is a correspondence between escape andbounded motion for the CT and nonchaotic regionsfor the OFLI2, since they are regular motions. Onthe PSS plots of the escape region we have, obvi-ously, just a few points. On the other side, collisionorbits are next to chaotic zones.

In Fig. 9, we show the evolution with the energyof the OFLI2 and the skeleton of symmetric peri-odic orbits, studying the variation in the x-axisby fixing y = x = 0, depending on the energyEJ . We note that these pictures give us a clearidea of the evolution of the system. When EJ islow the system has large forbidden regions and themotion is highly regular. Increasing EJ , the systembecomes more and more complex. The OFLI2 plotshows that the chaotic behavior appears mainly inthe range EJ ∈ [−1.75, 0]. When EJ is very high,the behavior is again more regular. The positionof the two primaries are marked by discontinuous

OFLI2

−4−3

−2−1

01

2−2

.5−2

−1.51

0.50

0.51

1.5

coor

dina

te x

POlimit

m= 1m=2m=3m=4

Energy EJEnergy EJ

2−4−3

−2−1

02

1−2

.5−21.5−1

−0.50

0.51

1.52

Fig. 9. OFLI2 plot and the skeleton of symmetric periodic orbits (PO) up to multiplicity m = 4 showing the evolution withenergy (coordinate x versus energy EJ plots) for the Copenhagen problem.

June 1, 2010 15:41 WSPC/S0218-1274 02649

1308 R. Barrio et al.

vertical lines. The skeleton of symmetric periodicorbits is produced to global multiplicity m = 4.In the figure, we have used a color code for dif-ferent multiplicities. Regions with a great num-ber of periodic orbits denote also the regions withchaotic behavior (see the OFLI2 plot). Note thateach point on the curves stands for the initial con-ditions of a symmetric periodic orbit and eachcurve is a family of periodic orbits. We remarkthat our results are in agreement with the skele-tons of symmetric periodic orbits computed byPapadakis and Goudas [Goudas & Papadakis, 2006;Papadakis & Goudas, 2006] for this problem.

3.2. The (n + 1)-body ring problem

If we now add more bodies in our problem, passingfrom the three-body problem to the more genericn-body problem (n > 3), but maintaining a 2DOFproblem, we may use other simplifications. The(n + 1)-body ring problem describes the motion ofan infinitesimal particle attracted by the gravita-tional field of (n + 1) primary bodies. These bodiesare distributed in a planar ring configuration, thatis, a central primary of mass βm and the rest n pri-maries of equal mass m are located at the vertices ofa regular polygon that is rotating on its own planeabout the center with a constant angular velocity.In Fig. 5, we show on the right, a schematic drawingof the problem for the case n = 7. The stars denotethe equilibrium points, on just two axes of sym-metry. The equilibrium points in green are presentonly for some values of β and the equilibrium pointsin red are always present. The stars (equilibriumpoints) with half part solid are stable for some val-ues of β and n. This problem was initially posedby Maxwell in order to study the stability of themotion of Saturn’s rings. After a suitable choice ofunits (for details, see [Arribas & Elipe, 2004; Bar-rio et al., 2008a, 2009a]) we obtain the equations ofmovement

x − 2y = −∂U

∂x, y + 2x = −∂U

∂y,

where

U(x, y) = −12(x2 + y2) − 1

∆

(β

r0+

n∑i=1

1ri

)

is the effective potential,

r0 =√

x2 + y2, ri =√

(x − xi)2 + (y − yi)2,

i = 1, . . . , n,

are the distances of the particle from each primaryand ∆ = M (Λ + βM2) with M = 2 sin θ and

Λ = sin2 θ

n∑i=2

1sin(i − 1)θ

.

The peripheral’s coordinates in this system are

xi =1M

cos 2(i − 1)θ, yi =1M

sin 2(i − 1)θ.

Under this formulation the system posses a Jaco-bian type integral of motion

C = 2U + (x2 + y2).

This problem, setting X = x−y and Y = y+x,accepts a Hamiltonian formulation

H(x, y,X, Y ) =12(X2+Y 2)−(xY −yX)+U∗(x, y),

with U∗(x, y) = U(x, y)+(1/2)(x2 +y2). Along thispaper we fix n = 7.

The linear stability of this configuration undersmall perturbations has been studied by severalauthors [Salo & Yoder, 1988, 1991]:

Proposition. The (n + 1) ring problem configura-tion is unstable if n < 7. If n ≥ 7 there is a βs(n)such that for β ≥ βs(n) the configuration is stable.

Therefore it is interesting to study the dynam-ics of this system for n ≥ 7.

The dynamical behavior of the (n + 1)-bodyring problem system is in some sense similar tothe Copenhagen problem but, obviously, much morecomplicated. First we remark that the number ofprimaries does not affect the dynamical behavior ofthe system too much (obviously, for more bodies thechaotic regions grow).

In Fig. 10, we show on the top plot, for the(7+1)-body ring problem, in a region on the (x,C)plane, the skeleton of symmetric periodic orbits upto global multiplicity m = 4. We have used dou-ble precision with an error tolerance Tol = 10−14.In the figure, we use a color code for the differentmultiplicities (the vertical lines correspond to theposition of the two primaries that are on the x-axis;the cental primary and one ring primary placed atdistance r(7), the radius of the ring). In the pic-tures (middle and bottom) we show the horizontalstability of the different families of periodic orbits.In (B) we show in green the stable orbits, in bluethe critical ones and in red the unstable orbits. Wenote that there are three zones with stable periodicorbits: on the top left, bottom left and near the

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Bifurcations and Chaos in Hamiltonian Systems 1309

Fig. 10. Top: Symmetric periodic orbits (y(0) = x(0) = 0) for the (7 + 1)-body ring problem versus the Jacobiconstant C. Middle and bottom: Horizontal stability (B (normal scale) and C (logarithmic scale)) of the symmetric periodicorbits.

primaries. The unstable families of periodic orbitsare all around the picture adopting “muscle-like”figures. In plot (C) we use a decimal logarithmicscale of the absolute value of the horizontal stabil-ity index in order to show “how” unstable are the

orbits. We observe as there are orbits with a sta-bility index greater than 108. Due to the existenceof periodic orbits with a very high value of stabilityindex, their computation is quite challenging at thenumerical precision.

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1310 R. Barrio et al.

r(7)coordinate x

Jaco

bi c

onst

ant C

F1d

F1r

F1

0.5 1 1.5 2−1

−0.5

0

0.5

1

coordinate x

coor

dina

te y

0.8 1 1.2 1.4

−0.2

−0.1

0

0.1

0.2

0.3

coordinate x

coor

dina

te y

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

coordinate xco

ordi

nate

y

F1

F1d

F1r

Fig. 11. Magnification of the skeleton of symmetric periodic orbits for the (7 + 1)-body ring problem and some selectedperiodic orbits of families of multiplicity m = 1.

Finally, in Fig. 11 we present a magnificationof the global skeleton of symmetric periodic orbits,but now up to global multiplicity m = 12. Again, inthe figure, we have used a color code for the differ-ent multiplicities (the vertical line corresponds tothe position of one ring primary at distance r(7)).Besides, the thin lines correspond to unstable peri-odic orbits, whereas the thick lines correspond tostable ones. In the figure, the profusion of unsta-ble periodic orbits is clear inside the more chaoticregion (C ∈ [−7,−5]), but there are two main fam-ilies of stable periodic orbits of multiplicity one (onred) that generate several bifurcations. By instance,in the small figures on the right, we present severalorbits of three families of symmetric periodic orbitsof multiplicity m = 1 extracted from the magnifica-tion plot. Orbits of F1d and F1r are direct and ret-rograde, respectively, around a primary body andorbits of F1 evolve around the ring.

4. Open Hamiltonians:The Henon–Heiles Hamiltonian

The classical Henon–Heiles Hamiltonian [Henon &Heiles, 1964] (HH), introduced in the study of galac-tic dynamics to describe the motion of stars arounda galactic center, is a three-fold symmetric per-turbation of the 1 : 1 harmonic oscillator, and isgiven by

H =12(X2 + Y 2) +

12(x2 + y2)

+(

yx2 − 13y3

). (8)

The HH system presents different symmet-ries: the spatial group is a dihedral group D3,whereas the complete symmetry group is D3 × T(T is a Z2 symmetry, the time reversal symmetry).This Hamiltonian has attracted a large number of

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Bifurcations and Chaos in Hamiltonian Systems 1311

-0.4 -0.2 0 0.2 0.4 0.6 -0.5

0

0.5

coordinate y

velo

city

Y

E: 1/12

-0.4 -0.2 0 0.2 0.4 0.6 -0.5

0

0.5

coordinate y

velo

city

Y

E: 1/8

-0.5 0 0.5 1

-0.4

-0.2

0

0.2

0.4

0.6

velo

city

Y

E: 1/6

coordinate y

-0.6

O3b

O3

O2

P3

P2

O1

P1

Fig. 12. Left: Poincare sections of the Henon–Heiles problem for different values of the energy and a magnification of theOFLI2 for E = 1/6 on the (y, Y )-plane. Right: Evolution of the OFLI2 in the same cases.

June 1, 2010 15:41 WSPC/S0218-1274 02649

1312 R. Barrio et al.

researchers, and it has been studied mainly for val-ues of the energy below the escape energy Ee = 1/6(for values of the energy below escape Ee the levelcurves are closed and all the orbits are bounded,exhibiting chaotic or regular motion, for the escapeenergy the equipotential lines form an equilateraltriangle, and for higher values the vertices of thetriangle open and most of the orbits are unbounded,escape orbits).

In order to show the behavior for low values ofthe Energy, in Fig. 12, we plot the Poincare sec-tions and the OFLI2 pictures at tf = 100 (O1) andtf = 300 (O2 and O3) for the Henon–Heiles problemfor values of energy E below the escape energy onthe surface x = 0. For E = 1/12, most of the orbitsare as regular as in the Poincare sections as OFLI2pictures show. Note that the OFLI2 locates the sep-aratrices and there are no spurious structures. ForE = 1/8, the OFLI2 gives much more informationthan the Poincare sections and locates, without aselection of the orbits, the periodic orbits and thechain of islands inside the chaotic sea. For E = 1/6,most of the orbits are chaotic and so the Poincaresection gives a cloud of points. OFLI2 gives moreinformation about the transition from regular tochaotic orbits as seen in the magnification (O3b),done for tf = 500, where we may appreciate thecomplex behavior among the two regular regions.

For values of energy greater than the escapeenergy Ee and due to the symmetry of the sys-tem, there are three exits [Aguirre et al., 2001;Aguirre & Sanjuan, 2003; Aguirre et al., 2009]:Exit 1 (y → +∞), Exit 2 (y → −∞, x → −∞)

and Exit 3 (y → −∞, x → +∞). During thelast few years, several papers [Aguirre et al.,2001; Aguirre et al., 2009; Barrio et al., 2008b,2009b; de Moura & Letelier, 1999; Papadakiset al., 2005] have focused their attention on thestudy of the escape basins of the HH system.Besides, on the (x,E)-plane, we observe an infi-nite sequence of intervals of stable and unsta-ble periodic orbits that accumulate at the escapeenergy. This behavior was first proved theoreti-cally in [Churchill et al., 1980] and recently Brack[2001] has obtained numerical and analytical esti-mates of such sequences in the case of isochronousbifurcations.

As the exit basins exhibit a fractal geometry,it is interesting to give some estimations abouttheir fractality. Because these fractals are of positivemeasure, their Minkowski–Bouligand dimension (orbox-counting dimension)

dimMB(S) := limε→0

log N(ε)

log(

1ε

) � 2

coincides with the space dimension; therefore, wemay use the fat-fractal analysis of Umberger andFarmer [1985] to classify these fractals. We assumethat the area of the covering set with N(ε) squaresof side ε is µ(ε) = N(ε) ε2, of the form µ(ε) = µ0 +K εγ : K a positive constant, µ0 the limiting measureof the set and γ the fat-fractal exponent. Performinga Levenberg–Marquardt nonlinear fit using 500 <n < 1000 (all plots have resolution 1000× 1000) weobtain γ, the fat-fractal exponent.

E=0.20 E=0.25

E=0.35 E=0.50

x

y

-1.3 1.3

1.3

-1x -1.2 1.2

x-1.3 1.3 x-1.5 1.5

y

1.4

-1

y

1.5

-1.3

y

1.5

-1.5

E1

E3E2

B

Fig. 13. Exit basins on the (x, y)-plane for different energies.

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Bifurcations and Chaos in Hamiltonian Systems 1313

In Fig. 13, we show the exit basins on the (x, y)-plane for different energies E. The color codes forthe exit basins are as follows: green — boundedmotion, blue — exit 1, yellow — exit 2, and red —exit 3. We observe that when the energy grows,the structure becomes simpler with more sharpbounds; therefore, we expect the fat-fractal expo-nent to increase. In Table 2, we present the expo-nents obtained for the three exits.

In Fig. 14, we show the exit basins on the(x,E)-plane by fixing y = X = 0. As before, weobserve a simplification of the geometry when theenergy grows, giving an asymptotic band structureas shown in plot B.

In Fig. 15, we show OFLI2 plots, some rescaledmagnifications of interesting regions (B, C1, C2)

Fig. 14. Exit basins on the (x,E)-plane.

Table 2. Fat-fractal exponent γ of the exitbasins on the (x, y)-plane for different energiesshown in Fig. 13.

E Exit γ

E = 0.20 1 0.4606 (±0.0861)E = 0.20 2 0.4356 (±0.0853)E = 0.20 3 0.4581 (±0.0844)

E = 0.25 1 0.5953 (±0.0674)E = 0.25 2 0.6006 (±0.0670)E = 0.25 3 0.6045 (±0.0671)

E = 0.35 1 0.8164 (±0.0416)E = 0.35 2 0.8278 (±0.0488)E = 0.35 3 0.8242 (±0.0456)

E = 0.50 1 0.8904 (±0.0409)E = 0.50 2 0.8801 (±0.0395)E = 0.50 3 0.9199 (±0.0325)

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1314 R. Barrio et al.

Fig. 15. OFLI2 plots on the (y, E)-plane.

exhibiting fractal geometry. Our first remark is thatthe KAM tori disappear on the y-axis around E ≈0.2113.

In Fig. 16, we show OFLI2 plots and somerescaled magnifications B, C1 and C2 which areobtained from the OFLI2 plots by painting just thepoints of the region of bounded motion (the rest onwhite) and using a new color scale (all the pointshave regular behavior) in order to highlight the linesof the families of periodic orbits and the differenceson the regular region.

Both Figs. 15 and 16 (B, C1, C2) show regionsof bounded motion where the self-similarity repeatseverywhere: not only at the edges of the KAMregion, but also deep inside the escape regions thereare safe zones that show regular behavior.

In the unbounded region and far from theescape energy we have located several regularregions of bounded motion, such as in plots Cand C1 and C2 of Figs. 15 and 16. Moreover,these regions present a self-similar structure (seethe magnification chains C1–C2 in both figures).

The skeleton of periodic orbits in Fig. 17 showssome bifurcations that configure the system. Wenote the presence of families of the normal modesthat determine the behavior for large E. Followingthe skeleton of periodic orbits in Fig. 17, we observethat these regions originate from saddle-node bifur-cations of different families of periodic orbits. Dueto these bifurcations, new families of periodic orbitsappear suddenly, with one stable branch (the onethat creates the region) and one unstable branch.We have detected dozens of such regular regions,but we hypothesize that in fact there are infinitelymany on such a region of energy E (with multi-plicity m growing to infinity). Note that we areinside the escape region [Barrio et al., 2009b] andnot in a bounded ergodic region, where it is wellknown that infinitesimal small stable regions appearcontinuously. From picture B we see that these sta-ble regions are small in size and that they are notpresent for E greater than E ≈ 0.27 since the escaperegions dominate and the saddle-node generationstops.

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Bifurcations and Chaos in Hamiltonian Systems 1315

Fig. 16. OFLI2 plots on the (y, E)-plane.

In Fig. 17, we show a joint OFLI2 and s.p.o.plot on the (y,E)-plane. Note that in such plots,any point corresponds to the initial conditions ofone orbit. We can see how both techniques comple-ment one another. Each figure consists of a regu-lar grid of 1000 × 1000 points (106 orbits) and wehave used double precision with an error toleranceTol = 10−14. In plot A, we have used a color codefor the different multiplicities of the periodic orbits.We show the s.p.o. (x(0) = Y (0) = 0) versus theEnergy constant E up to multiplicity m = 5. Inplot B, we also show the stability of the orbits (ingreen the stable and in red the unstable ones). The

forbidden region is located below, like in the previ-ous figures.

Figure 18 illustrates some bifurcations on theproblem, without doing a complete study of Hamil-tonian bifurcations under symmetries which is outof the scope of the present paper (see Fig. 3 andits explanation in Sec. 2.3 and the extensive litera-ture on this subject [de Aguiar et al., 1987; Dellnitzet al., 1992; Mao & Delos, 1992; Meyer, 1970]). Inthe plot on the left, we show the OFLI2 plot ofjust one safe region on a color scale and some fam-ilies of symmetric periodic orbits on plane (y,E).To illustrate the bifurcations we are interested in,

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1316 R. Barrio et al.

Fig. 17. Skeleton of symmetric periodic orbits, superposed on the OFLI2 plots of regular bounded regions (gray scale on thetop, black lines at the location of the periodic orbits and color scale on the bottom) on the (y, E)-plane ((a) shows differentmultiplicities on the color legend and (b) shows in green the stable orbits and in red the unstable ones).

we plot only some of the families. On the right,we show the stability index κ versus the Energy E(left) and on the bottom figure κ versus the coor-dinate y. The main family of multiplicity m = 1

is that of black color. The other families bifurcatefrom this intersection point (points given by (7)).Note that the bifurcated families always begin witha value of κ = 2 and the main family of periodic

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Bifurcations and Chaos in Hamiltonian Systems 1317

−0.11 −0.105 −0.1 −0.095 −0.09 −0.085 −0.08 −0.075 −0.070.2529

0.253

0.2531

0.2532

0.2533

0.2534

0.2535

Ene

rgy

E

−6 −4 −2 0 2 4 60.2529

0.253

0.2531

0.2532

0.2533

0.2534

0.2535

Stability index k

−0.11 −0.105 −0.1 −0.095 −0.09 −0.085 −0.08 −0.075 −0.07−6

−4

−2

0

2

4

6

Coordinate y

Sta

bilit

y in

dex

k

m=1m=3

m=1

Fig. 18. Generic and y-axis symmetric bifurcations in a regular region above the escape energy. On the left, both OFLI2 andfamilies of periodic orbits. Right and bottom plots show the stability index κ of these families. The multiplicity m = 1 orbit,shown on the bottom-right, is created at the saddle-node bifurcation and has no D3 symmetry.

orbits does not present D3 symmetry. It appearswith a saddle-node bifurcation (SN), which is anonelementary periodic solution (κ = 2) and cor-responds to the case where two periodic orbits arecreated (or destroyed), one stable and another oneunstable. This is the only way of creating new fami-lies of periodic orbits, apart from the boundaries ofthe domain in the definition of the Poincare map.The stable branch changes its stability index until itreaches κ = 2 again, where a isochronous pitchforkbifurcation (P) appears. It is a symmetric pitch-fork bifurcation: from a symmetric periodic orbit,two new isochronous periodic orbits are created butwith less symmetries (the symmetry in this case ison the y-axis, not D3) and the main symmetric peri-odic orbit changes its stability character after thebifurcation. Besides, we plot the generic touch-and-go bifurcation, as an example of a known genericbifurcation [Mao & Delos, 1992] where an unsta-ble periodic orbit of multiplicity m = 3 touches thecenter m = 1 periodic orbit and “bounces” whilethe main orbit remains stable. None of the orbits

disappear. Note how the skeleton of periodic orbitsconfigures the safe region.

5. Conclusions

In this review paper, by using some state-of-the-artnumerical techniques, we have done a brief analy-sis of the paradigmatic classical Copenhagen and(n + 1)-body ring problems and the Henon–HeilesHamiltonian as examples to show how the combi-nation of the different techniques give new sightson these classical problems: the OFLI2 permits tolocate the region of initial conditions where weexpect chaos, the CT to classify the kind of orbits(bounded, unbounded, collision, . . .) and the sys-tematic search method to obtain the different fam-ilies of symmetric periodic orbits. Moreover, all thetechniques complement one another providing muchreassurance on the results. The numerical integra-tion of the ODEs has been done by a highly adap-tive Taylor series method (using the freeware libraryTIDES), quite useful when we need a high precision

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1318 R. Barrio et al.

or we need to compute the solution of the varia-tional equations.

Acknowledgments

The authors R. Barrio and S. Serrano have beensupported during this research by the SpanishResearch Grants MTM2009-10767 and the authorF. Blesa has been supported by the SpanishResearch Grant AYA2008-05572/ESP.

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