the lie-integrator and the hénon-heiles system

8/12/2019 The Lie-integrator and the Hnon-Heiles system

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Iharka SZCS-CSILLIK 250

constant of motion, it is nonintegrable, therefore chaotic. The level of chaoticityvaries with the energy level (Hnon and Heiles 1964; see also Abraham andMarsden 1978; Arnold 1978; Zhong and Marsden 1988; Marsden and Raiu 1994).

Lots of authors dealt with the Hnon-Heiles model and with the associatedproblems, from the most various standpoints. As a very short digression, here welimit ourselves to quoting arbitrarily some contributions of the Romanian authors:Anisiu and Pl (1999); Mioc and Brbosu (2003a, b); Mioc (2004); Pricopi et al.(2006); Mioc and Pricopi (2007); Mioc et al. (2008, 2009).

Resuming, in this paper the Lie-integrator is compared with the Runge-Kuttamethod and a symplectic integrator (Dormand and Prince 1978; Sanz-Serna andCalvo 1984; Channel and Scovel 1990; Forest and Ruth 1990; Forest et al. 1990;

Kinoshita et al. 1990; Yoshida 1990, 1993; Feng and Meng-Zhao 1991; Mei-Gingand Meng-Zhao 1991; Wisdom and Holman 1991; Cartwright and Piro 1992;McLachlan and Atela 1992; Sanz-Serna 1998; Guzzo 2001; Laskar and Robutel2001; Stuchi 2002; Butcher 2003; Csillik 2003).

Section 2 presents the Lie-integrator and details the corresponding algorithmof fourth order (LIE4).

In Section 3 we deal briefly with the Runge-Kutta method, as well as withsymplectic integrators. The Runge-Kutta scheme of fourth order (RK4) is

presented, followed by the symplectic integration scheme for the Lie-integrator(SI4).

In Section 4, the central section of the paper, we present the Hnon-Heiles

system, then we derive the corresponding Lie operator. We write the approximatesolution of the system in both terms of the Lie operator and in explicit form. Tocompare the effectiveness of the methods considered, we resort to numericalapproaches.

Section 5 surveys the main results of our analysis. The advantages andshortcomings of the Lie-integrator are pointed out.

2. THE LIE-INTEGRATOR

Grbner (1967) defined the Lie-operator D as follows:

nn

zz

zz

zzD

++

+

= )()()(

22

11 , (1)

D being a linear differential operator. The point ),,,( 21 nzzzz = lies in the

n-dimensional z-space, the functions )(zi , ni ,1= are holomorphic within acertain domain G (they can be expanded in a convergent power series).


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3 The Lie-Integrator and the Hnon-Heiles System 51

Let the function )(zf be holomorphic in the same region as )(zi , ni ,1= .ThenDcan be applied to )(zf , and if we proceed applying D to f we get

.)(

,)(

,)()()(

,

1

2

22

11

1

0

fDDfD

DfDfD

z

fz

z

fz

z

fzfD

ffD

nn

nn

=

=

++

+

=

=

(2)

TheLie serieswill be defined in the following way:

+++=

=

=

)(!2

)()()(!

),( 22

0

10 zfDt

zftDzfDzfDt

tzL (3)

Because we can write the Taylor-expansion of the exponential function as

20 1 2e ( ...)

2!tD t

f D tD D f= + + + , (4)

),( tzL can be written in the symbolic form

( , ) e ( )tDL z t f z= . (5)

One of the most useful properties of Lie-series is

THEOREM 1 (Exchange Theorem). Let )(zF be a holomorphic function in

the neighborhood of ),,,( 21 nzzzz = , where the corresponding power series

expansion converges at the point ),,,( 21 nZZZZ = . Then we have:

=

=0

)(!

)(

ZFDt

ZF , (6)

or

(e ) e ( )tD tDF z F z= . (7)

Making use of it we can demonstrate how Lie-series solve differentialequations. Let us consider the system of differential equations:


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nizt

zi

i ,1),(dd == , (8)

with ),,,( 21 nzzzz = . The solution of (8) can be written as

etDi iz = , (9)

where the i are the initial conditions )0( =tzi and D is the Lie-operator asdefined in formula (1). In order to prove (9), we differentiate it with respect to thetime t:

d

e edtD tDi

i i

z

D Dt = = . (10)

Because of the fact that

)( iiiD = , (11)

we obtain the following result, which turns out to be the original ODE (8):

de ( ) (e ) ( )

dtD tDi

i i i i i i

zz

t= = = . (12)

Now we present the fourth order Lie-integrator (LIE4) for a ),,,( yxyxH

Hamiltonian.The system of differential equations is written as:

==

==

==

==

,),,,(d

d

,),,,(d

d

,),,,(d

d

,),,,(d

d

4

3

2

1

vuyxt

yv

vuyxt

xu

vuyxt

yy

vuyxt

xx

(13)

(where xu = , yv = ) with the initial conditions at time 0=t :

.)0(,)0(

,)0(,)0(

==

==

vu

yx (14)

The corresponding Lie-operator is


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+

+

+

= 4321D . (15)

For a time )( tt + , we obtain the solution under the following form (up to

the fourth order):

)(!4

)(!3

)(!2

)()()( 44

33

22

10

+

+

++=+ DDDDDttx , (16)

)(!4

)(!3

)(!2

)()()( 44

33

22

10

+

+

++=+ DDDDDtty , (17)

)(!4

)(!3

)(!2

)()()( 44

33

22

10

+

+

++=+ DDDDDttu , (18)

)(!4

)(!3

)(!2

)()()( 44

33

22

10

+

+

++=+ DDDDDttv , (19)

where

;))(()(

,))(()(

)())(()(

,)(,)(

314

2131

1112

110

=

=

==

==

DDD

DDD

DDDD

DD

(20)

;))(()(

,))(()(

,)())(()(

,)(,)(

314

213

21112

210

=

=

==

==

DDD

DDD

DDDD

DD

(21)

;)())(()(

,)())(()(

,)()())(()(

,)(,)(

5314

4213

33

1112

310

==

==

===

==

DDDD

DDDD

DDDDD

DD

(22)


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.)())(()(

,)())(()(

,)()())(()(

,)(,)(

5314

4213

34

1112

410

==

==

===

==

DDDD

DDDD

DDDDD

DD

(23)

3. RUNGE-KUTTA AND SYMPLECTIC INTEGRATORS

In numerical analysis, the Runge-Kutta methods constitute an importantfamily of implicit and explicit iterative methods for the approximation of solutionsof ordinary differential equations (Butcher 2003). One member of the family ofRunge-Kutta methods is so commonly used that it is often referred to as RK4.

The family of explicit Runge-Kutta methods is given by

)( 111 ssii kbkbhyy +++=+ , (24)

where),(1 ii ytfk = , (25)

),( 12122 khayhctfk ii ++= , (26)

))(,( 23213133 kakahyhctfk ii +++= , (27)

))(,( 1,11 ssssisis kakahyhctfk ++++= (28)

and s is an integer number (stage number); 1,13121 ,,,,, sss aaaa , sbb ,,1 ,

scc ,,1 are coefficients. In general, the coefficients scc ,,1 , which characterize

the partition of the step h, satisfy the following condition:

.

,

,

1,1

32313

212

++=

+=

=

ssss aac

aac

ac

(29)

They are determined by the condition that the approximate solution 1+ix is equal to

the exact solution corresponding to the first s terms of the Taylor series. TheRunge-Kutta method is consistentif the following relations are satisfied:


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1

;,2,

1

1

1

=

==

=

=

s

j

j

i

j

iij

b

sica

(30)

The Runge-Kutta method of fourth order (RK4) is a numerical techniqueused to solve ordinary differential equation of the form

.)0(

,),(d

d

0yy

yxfx

y

=

= (31)

The RK4 method is based on the following algorithm:

,,0,

,)22(6

1

1

43211

nihtt

kkkkhyy

ii

ii

=+=

++++=

+

+ (32)

where 1+iy is the RK4 approximation of )( 1+ity , and

),(1 ii ytfk = , (33)

++= hkyhtfk ii 12 21

,2

1, (34)

++= hkyhtfk ii 23 21

,2

1, (35)

),( 34 hkyhtfk ii ++= . (36)

Thefourth orderof the RK4 method means that the error per step is of order5h , while the total accumulated error is of order 4h .

As regards the symplectic integrators, it is very well known that they havethe following properties (e.g., Yoshida 1990): area preserving, time reversibility,and constant step-size (this guarantees that there is no secular change in the error ofthe total energy).

It is known that for Hamiltonian systems of the form

)()( qp VTH += (37)


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explicit symplectic schemes exist. In this formula the q variables are generalizedcoordinates, the p variables are conjugate generalized momenta, whereas theHamiltonian H corresponds to the total mechanical energy ( T and V stand forthe kinetic energy and potential energy of the system, respectively).

Here we dwell on Neris (1987) explicit symplectic integrator in terms of Liealgebraic language. To do this, we begin by writing the Hamiltonian equations inthe following form (using Poisson bracket):

qp

z

pq

zzz

z

==HH

Ht

)}(,{d

d. (38)

Introducing the differential operator },{)( HDH zz = , (38) can be written as

)(d

dz

zHD

t= . (39)

Therefore, the exact time-evolution of the solution of equation (39), )(tz ,

from 0=t to =t , is given by

)0(]exp[)( zz HD= . (40)

For the Hamiltonian of the form (37), where VTH DDD += , we have theformal solution

( ) exp[ ( )] (0)A B = +z z , (41)

where TDA =: and VDB =: are two operators which do not commute. Further, wesuppose that the set of the real numbers ),( ii dc , ni ,1= , satisfies the equality

=

++=+n

i

nii OBdAcBA

1

1)()exp()exp()](exp[ , (42)

where n is the integrators order. Let a mapping from )0(zz= to )(= zz begiven by

zz

=

=

n

i

ii BdAc

1

)exp()exp( . (43)

The above application is symplectic because it is a product of elementarysymplectic mappings. We can write the explicit equation (43) in the followingform:


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niV

dpp

Tcqq

i

i

q

iii

p

iii

,1,

1

1

1=

+=

+=

=

=

q

p

q

p, (44)

where ),( 00 pq=z and ),( nn pq=z . This system of equations is an nth order

symplectic integration scheme. The numerical coefficients ),( ii dc , ni ,1= , are notuniquely determined from the requirement that the local truncation error is of order

n . If one requires the time reversibility of the numerical solution, one can

determine it uniquely.In this context, the fourth order Lie-integrator schema is given by the system

of equations (44) for 4,1=i , where the values of the coefficients ),( ii dc , 4,1=i ,are:

.0,22

2

,22

1

,)22(2

21

,)22(2

1

43

3

2

331

3

3

32

341

=

=

==

==

==

dd

dd

cc

cc

(45)

4. HNON-HEILES SYSTEM AND APPLICATION

In 1964, Michel Hnon and Carl Heiles modelled the motion of a star withina galaxy, representing the gravitational attraction of the galaxy by a potentialhaving cylindrical symmetry (Hnon and Heiles 1964). The chosen potentialenergy is given by that of a planar oscillator to which two cubic terms were added:

3222

3

1)(

2

1),( yyxyxyxV ++= . (46)

Those terms made the problem nonlinear and nonintegrable. Thistwo-dimensional system has only one constant of motion, hence nonintegrable, infact chaotic. Hnon and Heiles found that, for very low values of the total energy


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( 12/1=E ), the system is very close to being integrable, since the majority ofintersection points are placed along the curves corresponding to the intersectionswith the tori. By increasing the energy, the regular curves progressively disappear,through a phase where they are reduced to tiny isolated zones persists. For 6/1=E the whole area is filled by points generated by a single trajectory. So, at lowenergies, the system is practically integrable. Numerical methods could then

provide much help in guessing if a system is very close to the condition ofintegrability or not. In the Hnon-Heiles problem, the system, as the energy grows,

becomes a chaotic system, the intersections of the trajectories with the surface ofsection thickly fill the area determined by energy conservation. The exponentialdivergence of the trajectories is determined by the initial conditions according tothe equations of the system. This fact does not help in predicting the systems

behaviour, because in practical cases one has to deal with experimental errors.We introduce the Hnon-Heiles Hamiltonian in the most general form:

322222

3

1)(

2

1CyyDxByAxvuH ++++= . (47)

The corresponding equations of motion are

+=

=

=

=

.

,2

,

,

22CyDxByv

DxyAxu

vy

ux

(48)

For the values 1==== DCBA of the coefficients, we obtain the nonintegrableHnon-Heiles Hamiltonian:

322222

3

1)(

2

1),,,( yyxyxvuvuyxH ++++= . (49)

The equations of motion for the coordinates and momenta of the Hnon-Heilessystem are given by:

+=

==

=

==

===

=

==

.d

d

,2d

d

,dd

,d

d

22 yxyy

H

t

vv

xyxx

H

t

uu

vv

Ht

yy

uu

H

t

xx

(50)


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The initial conditions for a time are denoted as follows:.)(,)(,)(,)( ==== vuyx (51)

To derive the corresponding Lie operator, we use the following system ofequations:

,,2,, 422

321 =+======= (52)

and the Lie operator is:

+

+

+

= 4321D , (53)

which, substituting (52), can be written as:

++

+

+

= )()2( 22D . (54)

So, for a time )( + the solution is:

( ) e , ( ) e ,

( ) e , ( ) e .

D D

D D

x y

u v

+ = + =

+ = + = (55)

We can write the solution in the following form:

2 11 2 1

( ) e ( ) ( ) ( ) ( ) ,2! ( 1)!

nD n

D D D O nn

+ = = + + + + +

(56)

2 11 2 1( ) e ( ) ( ) ( ) ( ) ,

2! ( 1)!

nD n

y D D D O nn

+ = = + + + + +

(57)

2 11 2 1( ) e ( ) ( ) ( ) ( ) ,

2! ( 1)!

nD n

u D D D O nn

+ = = + + + + +

(58)

2 11 2 1( ) e ( ) ( ) ( ) ( ) ,

2! ( 1)!

nD n

v D D D O nn

+ = = + + + + +

(59)

where the expressions of )(iD , ni ,1= , are:

,)(2)()(

,)()(,)(

113

121

=

==

DDD

DDD

(60)


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whereas the expressions of )(iD , ni ,1= , are:

,)()()()(

,)()(,)(

212113

121

+=

==

DDDD

DDD

(61)

We explicitly write for ni ,0= :

,)()(2)()(

...

,114)(20)323(8

)(56)15(8)(42)(

,8)35(4)2(16

)(202022)(

,12)5(2)(10)(

,)(246)(,)(2)(

,2)(,)(,)(

2

0

22

2

2222

22227

2222

226

225

2243

210

=

=

++

++=

+++

++=

++++=

+++=+=

===

n

k

knkkn

nnDDCDD

D

D

D

DD

DDD

(62)

and

,))()()()(()()(

...

,122112)58(16)(20)15(8)(42)(

,836102)53(4

)(10)(20)(11)(

,12)5(2)(10)(

,)(2)(2)(3)(

,)(2)(

,)(,)(,)(

2

0

222

2

222

22227

224422

2222226

225

2222224

3

22210

=

=

+++=

+++++

+++=

++=

+++=

=

+===

n

k

knkknkkn

nn DDDDCDD

D

D

D

D

D

DDD

(63)


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The expressions for )(iD and )(iD , where ni ,1= , can be obtained easily:

.)()(

,)()(

1

1

=

=

+

+

ii

ii

DD

DD (64)

Now the equations of motion of the Hnon-Heiles system can be integratednumerically with the Lie-integrator method (up to order n; see Pl and Sli 2007).

LIE4: Using (62)(64), the numerical solutions (56)(59) of the Hnon-Heiles system (50) with the initial conditions (51) are:

,24

))(246(

6))(2(

2)2()(

422

32

++++

+++++=+x

,24

))(2)(2)(3(

6))(2(

2)()(

4222222

3222

++++

+

+

+++=+y

,24

)12)5(2)(10(6

))(2

46(2

))(2()2()(

422

322

2

+++++

++

++++++=+u

.24

)12)5(2)(10(6

))(2)(2

)(3(2

))(2()()(

422

32222

222

22

+++

++

+++

+++=+v

(65)

RK4: The numerical solutions of the Hnon-Heiles system (50), with the

notation )1(yx= , )2(yy= , )3(yx= , )4(yy= and initial conditions )(0jy , 4,1=j ,

are:


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,

,,0,4,1,)22(61

1

)(4

)(3

)(2

)(1

)()(1

htt

nijkkkkhyy

ii

jjjjji

ji

+=

==++++=

+

+ (66)

where

),( )()(1j

iij

ytfk = , (67)

++= hkyhtfk jjiij )(

1)()(

2 2

1,

2

1, (68)

++= hkyhtfk jjiij )(2)()(3 2

1,21 , (69)

),( )(3)()(

4 hkyhtfkjj

iij ++= . (70)

SI4 (Csillik 2004): Using the system of equations (50)(51) (notation:

1qx= , 2qy= , 1px= , 2py= ), equations (44) for the Hnon-Heiles system are:

,))()((

,)2(,

,

22

212

122

2111

11

1

2

1

22

11

111

jjjj

jj

jjjj

jj

j

j

jj

jj

jj

qqqdqp

qqqdqp

pcqq

pcqq

+=

+=+=

+=

4,1=j , (71)

where the values of the coefficients ),( jj dc are given by (45).

A surface of section (SOS), also called a Poincar section (see, e.g., Rasband1990), is a way of presenting a trajectory in an n-dimensional phase space in an(n1)-dimensional space. By picking one phase element constant (in our Hnon-Heiles system case: 0=x ) and plotting the values of the other elements, each time

the chosen element has the desired value, an intersection surface is obtained. Thesurfaces of section for the Hnon-Heiles equation (49) with energy 8/1=E ,

plotting )(ty versus )(ty at values where 0)( =tx , are presented in Fig. 1 (using

RK4 integrator), in Fig. 2 (using SI4 integrator), and in Fig. 3 (using LIE4integrator). All three integrators are made with MATLABand MAPLE(Baker 1981;Lynch 2000; Moler 2004).


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Fig. 1 Poincar section of the Hnon-Heiles equation (49) using RK4 integrator.

Fig. 2 Poincar section of the Hnon-Heiles equation (49) using SI4 integrator.

A Poincar map can be interpreted as a discrete dynamical system with astate space of codimension 1 with respect to the original continuous dynamicalsystem. Because it preserves many properties of periodic and quasiperiodic orbitsof the original system and has a lower dimensional state space it is often used foranalyzing the original system. In practice this is not always possible as there is nogeneral method to construct a Poincar map. The Hnon-Heiles system was used tostudy the motion of stars in a galaxy, because the path of a star projected onto a


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plane looks like a tangled mess and the Poincar map shows the structure moreclearly. Comparing Figs. 1, 2 and 3, one can see that Fig. 3 with LIE4 integratorhas more details.

Fig. 3 Poincar section of the Hnon-Heiles equation (49) using LIE4 integrator.

5. CONCLUSIONS

The Lie-integrator method appears to be very effective as compared withother numerical integrators of differential equations. The solution written in the

power series expansions of the independent variable can be slightly different forevery step. The Lie-integrator method (LIE4) is quite precise and fast, as has beenshown in our comparative test computations with the Runge-Kutta (RK4) andsymplectic integrators (SI4). Symplectic integrators are very effective wheneccentricities are small; the Lie-integrator is a better choice in studies such as this

one, where very large eccentricity orbits are explored (Pl and Sli 2007). Theconvergence of the Lie-integrator can be regarded as a kind of a symplecticintegrator, which conserves the integrals of motion. The desired accuracy can befixed and controlled by two different parameters: the time step and the number ofLie-terms. The only disadvantage is that the derivation of recurrence in the Lie-terms may nevertheless lead to very lengthy expressions when the right-hand sidesare complicated and lengthy themselves (Dvorak et al. 2005). The Lie integrator is


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straightforward and comprehensible in its construction; it allows an easy handling.So, the method is very well suited for the calculation of orbits in celestialmechanics. Many practical differential equations evolve on a Lie group; it

preserves the geometry of the configuration space.

Acknowledgments. Valuable comments of Dr. Vasile Mioc are duly acknowledged.

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Received on 15 August 2009

the lie-integrator and the hénon-heiles system

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