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A review study of the 3-particle Toda lattice and higher order truncations: The even-order cases (Part II)

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Tutorials and Reviews
International Journal of Bifurcation and Chaos, Vol. 20, No. 11 (2010) 3391–3441c
World Scientiﬁc Publishing CompanyDOI:10.1142/S0218127410027799
A REVIEW STUDY OF THE 3-PARTICLE TODALATTICE AND HIGHER ORDER TRUNCATIONS:THE EVEN-ORDER CASES (PART II)
LOUKAS ZACHILAS
Lecturer in Applied Mathematics,Department of Economics, University of Thessaly,43 Korai str., GR-38333, Volos, Greecezachilas@uth.gr
Received February 13, 2010
We complete the study of the numerical behavior of the truncated 3-particle Toda lattice (3pTL)with even truncations at orders
n
= 2
k
,
k
= 2
,...,
10. We use (as in Part I): (a) the methodof Poincar´e surface of section, (b) the maximum Lyapunov characteristic number and (c) theratio of the families of ordered periodic orbits. We derived some similarities and quite manydiﬀerences between the odd and even order expansions.
Keywords
: Toda lattice; Hamiltonian dynamics; Poincar´e surface of section; periodic orbits.
1. Introduction
The special case of the periodic 3-particle Todalattice (3pTL) was studied by Contopoulos andPolymilis (henceforth CP) [1987]:
H
=12(˙
q
21
+ ˙
q
22
+ ˙
q
23
) + exp(
q
1
−
q
2
)+ exp(
q
2
−
q
3
) + exp(
q
3
−
q
1
)
−
3 (1)where
q
i
,
˙
q
i
(
i
= 1
,
2
,
3) are the positions andmomenta of the three particles, respectively.In the present paper (Part II), we study thetransformed and truncated Toda lattice systems of order
n
= 2
k
,
k
= 2
,...,
10. The choice of truncat-ing the approximation at the 20th order is due tothe complexity and the heavy computer consump-tion of the 117 terms of
H
20
.The general form of the truncated Hamiltoni-ans is:
H
i
=12(˙
x
2
+ ˙
y
2
) + Φ
i
(
x,y
) =
E
(2)where Φ
i
(
x,y
) is the potential function of eachapproximation (
i
= 2
k
,
k
= 2
,...,
10) and
E
isthe energy integral.Similarly to Part I [Zachilas, 2010], we havechosen the plane
x
= 0 as surface of section. Weintegrate numerically the equations of motion for-ward in time and ﬁnd the successive intersectionsof the orbit by the surface (Poincar´e consequents).Equation (2) deﬁnes the curve of zero velocity (i.e.“limiting curve”), if we consider
x
= ˙
x
= 0:
E
=12˙
y
2
+ Φ
i
(0
,y
) (3)for given value of the energy
E
.In Sec. 2 we present the even-order truncations.Section 2.1 provides some general remarks aboutthe nature of those potentials, and Sec. 2.2 describesthe ﬁrst even-order potential, the fourth orderone (
H
4
). As it is a quite well studied potential[Udry & Martinet, 1990], we just underline themain characteristic features and we present thecrucial diﬀerences with respect to the third order
3391
3392
L. Zachilas
Hamiltonian [Part I]. We follow the same schemeof presentation for the truncations of sixth order(Sec. 2.3), eighth order (Sec. 2.4) and tenth order(Sec. 2.5). Section 2.6 describes in a groupingscheme the ﬁve similar potentials of 12th order, 14thorder, 16th order, 18th order and 20th order.In Sec. 3, we summarize the conclusions of thiswork and underline the most important remarks.The Appendix includes (as in Paper I) all the18 Hamiltonians (
H
3
,...,H
20
), odd and even-order truncations, in terms of their power seriesexpansion.
2. The Even-Order Systems
The even-order Hamiltonians are found analytically(exactly as we have found the odd-order ones, inPart I) by expanding the exponential terms andare given in the Appendix. These are
H
4
,
H
6
,
H
8
,
H
10
,
H
12
,
H
14
,
H
16
,
H
18
and
H
20
[see Appendix,Eqs. (A.2), (A.4), (A.6), (A.8), (A.10), (A.12),(A.14), (A.16) and (A.18)], where the subscriptsshow the order of the approximation. Let us recallthat, if we set ˙
x
= ˙
y
= 0 in Eq.(2) and for every
Fig. 1. The proﬁles of the nine even-order potentials along with the proﬁle of the full exponential Toda, which is the innermostfor
y <
0, and the outermost for
y >
0 (in purple color).
value of the energy
E
, the equation Φ
i
(
x,y
) =
E
deﬁnes contours of isopotential curves, called zerovelocity curves (ZVC). On the other hand, if wetake
x
= 0, we get curves on the (
E,y
) plane of theform Φ
i
(0
,y
) =
E
.
2.1.
Common characteristics of even-order systems
Figure 1 presents the proﬁles of the nine potentialsalong with the proﬁle of the full exponential Toda(the outermost in purple color). The
y
-coordinateis on the horizontal axis and Energy (
E
) is on thevertical axis. All potentials look similar in theirproﬁle: a double barrier (on both sides), whichbounds the motion along the
y
-axis. There is nolocal maximum on the right-hand side of the poten-tial (like in odd-order truncations) and thus wehave no value of escape energy. The successiveeven-order truncations (as we add on higher even-order terms) approach the full exponential Todapotential. The right-hand branch approaches thefull exponential from inside, while the left-handbranch approaches from outside. The diﬀerence of
A Review Study of the 3-Particle Toda Lattice and Higher Order Truncations
3393
width of the potential well, as we move to highereven-order truncations, is rather small (especially atenergy levels less than
E
= 15). The only peculiar-ity, in all proﬁles, is the proﬁle of
H
4
, which inter-sects all higher order proﬁles.
2.2.
The fourth order truncation
The fourth order truncation [given by Eq. (A.2)] isa very well known potential and studied in [Udry &Martinet, 1990]. The ZVC are plotted in Fig. 2along with a 3D mesh grid. The isopotentials (onthe bottom of the box) are closed curves and theyremain closed even for big energy values (since wehave no escape energy).We remind that the invariant curves on thesurface of section (
x
= 0 and ˙
x >
0) have beencalculated by solving numerically the equations of motion, Eqs.(3).
2.2.1.
Invariant curves and periodic orbits
In Fig. 3, we give the diagram of characteristics of the fourth order truncation. Although the diagramstops at
E
= 2, we have calculated invariant curvesfor
E >
2, in order to prove the statement of CP[1987]: “
The system of fourth order seems to be inte-grable. No stochasticity appears even for large val-ues of energy
”, is totally false. The truth is thatsince there is no escape energy, we have calculatedinvariant curves up to the huge value of
E
= 150000and the results are very impressive.Family I (marked in red dots) is stable all theway up to high energies. Family II, which is a fam-ily that did not exist in the odd-order systems, isstable (marked in light green dots) from the bot-tom of the well up to
E
= 1
.
515. At this point,it becomes unstable (marked further on in greendots) and it bifurcates inversely to two new fami-lies (IIa and IIb) of the same period. Both families,according to H´enon’s stability index, are unstableand they disappear at energy
E
= 0
.
4422
,
just onthe curve of the surrounding ZVC. Family IV (andits symmetric, Family V) is marked in light bluedots. It is stable up to
E
= 1
.
515 (like Family II)and then it becomes unstable, bifurcating inverselyto a new unstable family, IVa. Near this value of energy, the closed invariants of Families IV and V —as we will describe below — disappear, leaving justan unstable invariant point. Finally, Family VI (andits symmetric, Family VII), which begins as unsta-ble (marked in orange dots at
E
= 0), becomesstable at
E
= 0
.
4422 and bifurcates an unstablefamily, called VIa. Family VI remains stable up tohigh energies (i.e.
E
= 200). A very interesting phe-nomenon (which will be described below with the
Fig. 2. Contour plot of the ZVC (Zero Velocity Curves) of the fourth order truncation, along with a 3D mesh grid.
3394
L. Zachilas
Fig. 3. Diagram of characteristics of the main Families I, II, IV/V and VI/VII. We can also see the bifurcating Families, IIaand IIb, IVa and VIa.
help of the diagrams of Poincar´e surface of section)is the behavior of Families I, IIa, VIa and IVa nearthe value
E
= 0
.
984 (
y
= 0
.
611), where it seemsthat we have a collision of the four branches. Afterthis value, Families I and IIa continue normally, butIVa and VIa join onto a branch.Figure 4(a) depicts the Poincar´e surface of section at an energy value which was used in HH(
H
3
), namely
E
= 0
.
0833333 (=1/12). By choos-ing this value, we wished to see if there are anysimilarities and/or diﬀerences with the well-knownHamiltonian of the third order. In [Udry & Mar-tinet, 1990], the authors have done a step by steptransition from
H
3
to
H
4
, at a speciﬁc energy level,with the use of an auxiliary variable. In Fig. 4(a), wecan see ﬁve well-separated regions of closed invari-ant curves. The red invariants (at the right) belongto Family I, the green ones (in the middle) belongto Family II, the blue ones (at the left) belong toFamily III, the magenta ones (at the top) belong toFamily IV and ﬁnally the dark blue (at the bottom)to Family V. We can, also, distinguish two unsta-ble invariant points on the
y
= 0 axis, marked asVI (˙
y
= 0
.
2) and VII (˙
y
=
−
0
.
2), which will playa crucial role later on. The two important energiesin
H
3
were:
E
= 0
.
125 (=1/8) and
E
= 0
.
16666(=1/6, the escape energy). In
H
4
they are not pre-sented, since we have no important change, and thepicture of the invariants remains the same [as inFig. 4(a)].In the next two plots [Figs. 4(b) and 4(c)]energy takes the values 0.4 and 0.5 correspondingly.The changes are very crucial, since at
E
= 0
.
4422,Family VI becomes stable; while near the ZVC twonew families are born (Families IIa and IIb, whichare inverse bifurcations of Family II, see Fig. 3). InFig. 4(b), Family I is in red, Family II in green,Family III in blue, Family IV in magenta and Fam-ily V in dark blue. The region of closed invari-ants of Family IV (and V) increased, while at thesame time, the regions of Families I and III havedecreased. The scene is quite diﬀerent in Fig. 4(c).Two new regions of closed invariant curves haveappeared (the one around the stable Family VI andthe second around its symmetric, VII). The regionsof Families I and III became smaller and the regions
A Review Study of the 3-Particle Toda Lattice and Higher Order Truncations
3395Fig. 4(a). Poincar´e surface of section of the fourth order truncation at
E
= 0
.
083333. We can see the ﬁve main families of closed invariants. Family I is in red color, Family II in green color, Family III in blue color, Family IV in magenta color andFamily V in dark blue color. We, also, denote the two unstable invariant points, VI and VII, which play crucial roles.Fig. 4(b). Poincar´e surface of section at
E
= 0
.
4. We can see the ﬁve main families of closed invariants. Family I is in redcolor, Family II in green color, Family III in blue color, Family IV in magenta color and Family V in dark blue color. The twounstable invariant points are VI and VII.

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