Multifractal analysis of tori destruction in a molecular Hamiltonian system
A. M. Tarquis
Departamento de Matema ´tica Aplicada, Escuela Te ´cnica Superior de Ingenieros Agro ´nomos, Universidad Polite ´cnica de Madrid,
28040 Madrid, Spain
J. C. Losada and R. M. Benito*
Departamento de Fı ´sica y Meca ´nica, Escuela Te ´cnica Superior de Ingenieros Agro ´nomos, Universidad Polite ´cnica de Madrid,
28040 Madrid, Spain
F. Borondo
†
Departamento de Quı ´mica, C– IX, Universidad Auto ´noma de Madrid, Cantoblanco 28049 Madrid, Spain
Received 2 August 2001; published 20 December 2001
In this paper, an analysis of the phase space structure of the isomerizing molecular system LiNC/LiCN,
using Poincare ´ surfaces of section and frequency analysis, is presented. The scaling structure of the frequency
map in the chaotic region next to the regular part corresponding to the stable linear isomer LiNC is studied
using multifractal analysis. This approach is a way to characterize quantitatively the complexity in the mecha-
nism of the tori destruction in a molecular Hamiltonian system that exhibits soft chaos as the vibrational energy
of the system increases.
DOI: 10.1103/PhysRevE.65.016213 PACS numbers: 05.45.Ac, 05.45.Df, 05.45.Tp
I. INTRODUCTION
From the vibrational point of view, molecules can be con-
sidered as Hamiltonian systems formed by a collection of
nonlinear anharmonic coupled oscillators. The corresponding
classical dynamics can be interpreted in terms of phase space
structures that, although envisaged by Poincare
´
at the end of
the nineteenth century, could only be properly studied after
the development of modern digital computers 1. For low
levels of excitations molecular motions take place in the vi-
cinity of the minima of the potential energy surface, defined
within the Born-Oppenheimer approximation. In this har-
monic regime the motion is regular, corresponding to the
well known normal mode picture 2. The combination of
anharmonicities and strong mode couplings, as vibrational
energy increases, makes molecules nonintegrable dynamical
systems, with the possibility of undergoing chaotic motion
3. The celebrated Kolmogorov-Arnold-Moser KAM theo-
rem provides a very powerful framework to understand this
transition to chaos. When some perturbation acts on an inte-
grable system some tori are destroyed, but those with ‘‘irra-
tional enough’’ frequency ratios in the sense of the KAM
condition, called KAM tori, survive 4. In two-degrees-of-
freedom 2dof systems, these structures establish a hierar-
chical organization of phase space. The family of persistent
KAM tori, parametrized by a Cantor set of frequency vectors
in the ‘‘holes’’ of which chaotic behavior takes place, consti-
tutes an impenetrable barrier for the flux of trajectories
across. The destroyed tori turn into periodic orbits PO, ho-
moclinic tangles, and cantori 3. Periodic orbits correspond
to resonant motion rational frequency ratio and are orga-
nized in phase space according to a Farey tree distribution
5. Emanating from each unstable PO fixed point there are
two associated manifolds, one incoming and another outgo-
ing, whose repeated crossings form the homoclinic tangle: a
band of stochasticity that can be tiled, classifying the differ-
ent regions according to their dynamical properties 1. Can-
tori are fractal objects 6, originated by the destruction of
the ‘‘not irrational enough’’ tori in the unperturbed system;
that act as partial barriers in the chaotic regions of phase
space. As perturbation increases, the fractal dimension of
these structures decreases, and the corresponding barrier ef-
fect weakens 7; at the same time more and more KAM tori
enter into this category. Then, the dynamical bottleneck in a
given phase space region corresponds to the most intact can-
tori, i.e., the last broken KAM torus corresponding to that
with the most irrational frequency ratio. According to the
continued fraction theory 8, this corresponds to the golden
mean, defined as
=1 +
1
1 +
1
1 +•••
=
1 + 5
2
. 1
The destruction of tori has been systematically studied in
the standard map 3 by a number of authors 9, and some
fractal structures in the diagrams of the breakup of tori had
been identified by Schmidt and Bialek 10.
There are numerous methods to investigate the structure
of phase space. In systems with a 2dof composite Poincare
´
surface of section SOS, consisting of the intersection of the
trajectories at a given energy with suitable surfaces, there is
more informative than other tools 11, such as Lyapunov
exponents or Kolmogorov entropy. Unfortunately, a SOS is
not feasible for systems with more than 2dof.
An alternative method is that of frequency analysis FA,
which is based on a Fourier representation of trajectories.
The FA method involves monitoring the variations of the
fundamental frequencies of the system with time. In the case
of regular motion, an analytical representation for the solu-
tion of the Hamilton equations of motion is obtained,
*Electronic mail: rbenito@fis.etsia.upm.es
†
Electronic mail: f.borondo@uam.es
PHYSICAL REVIEW E, VOLUME 65, 016213
1063-651X/2001/651/0162139/$20.00 ©2001 The American Physical Society 65 016213-1