Dissipation as a mechanism of energy gain

R. Egydio de Carvalho,1,* C. Vieira Abud,1 and F. Caetano Souza2

1Departamento de Estatística, Matemática Aplicada e Computação-DEMAC, Instituto de Geociências e Ciências Exatas-IGCE,
Universidade Estadual Paulista-UNESP, Av. 24A, No. 1515, Bela Vista, 13506-900 Rio Claro, São Paulo, Brazil

2Instituto de Física de São Carlos, Universidade de São Paulo, 13560-970 São Carlos, São Paulo, Brazil
�Received 29 August 2007; revised manuscript received 17 January 2008; published 7 March 2008�

Some properties of the annular billiard under the presence of weak dissipation are studied. We show, in a
dissipative system, that the average energy of a particle acquires higher values than its average energy of the
conservative case. The creation of attractors, associated with a chaotic dynamics in the conservative regime,
both in appropriated regions of the phase space, constitute a generic mechanism to increase the average energy
of dynamical systems.

DOI: 10.1103/PhysRevE.77.036204 PACS number�s�: 05.45.Pq, 05.90.�m

I. INTRODUCTION

Many works on dissipative systems report results con-
cerning the minimization or the elimination of the dissipation
effects in different fields of science, such as molecular phys-
ics �1�, optimal network �2,3�, Raman lasers �4�, turbulent
fluid �5�, viscous liquid �6�, and Fermi quantum gas �7�. In
an opposite approach, other works take advantage of the
presence of dissipative effects and there are results reported
in fields of science such as mesoscopic device �8�, optics �9�,
relativity �10�, nanoscience �11,12�, and nonlinear encryption
�13�.

In this paper we will follow the later approach by consid-
ering a nonlinear dynamical system where dissipative effects
are present and, for a particular set of parameters, the system
experiences a gain in its average energy. The common sense
establishes that when dissipative effects are present in a sys-
tem, it loses energy, what is correct in principle, but when
chaos is also present in the dynamics, the combination with
dissipation and attractors can produce this scenario. Our es-
sential idea is to generate attractors, through bifurcations of
elliptic fixed points in stable focus �14�, for velocities which
are higher than the typical average velocity of the system in
the nondissipative regime. In the next section we present the
model we are using and in Sec. III we show our results and
the conclusions.

II. MODEL

We will revisit the time-dependent annular billiard
�15,16�, but now under the presence of inelastic collisions
with both boundaries. This work is a natural extension of the
studies developed on Fermi acceleration where the pulsating
annular billiard was used as model �17,18�. We will seek to
understand and describe the behavior of the average velocity
of a particle as a function of the collisions with the bound-
aries. The model consists of a circle circumscribed in another
circle in whose annular region a classical particle moves
freely. The boundaries defined by the circumferences play
the role of infinite potential barriers. The geometry of the

model may assume two configurations: �i� concentric circles
or �ii� eccentric circles where the distance between both cen-
ters is called by eccentricity. For the nondissipative static
case �15,16�, where both boundaries are static, the chaotic
sea grows with the increasing of the eccentricity. For the
limit case where the circles are concentric, the system is
globally integrable because both the energy and the angular
momentum are preserved. On the other hand, for the time-
dependent elastic case, the oscillations of the boundaries
break the invariant tori and even for the concentric configu-
ration it is possible to have a chaotic sea. In previous works
�17,18� we considered the boundaries of the annular billiard
to pulsate periodically in time and now, in this work, we will
take advantage of that approach to introduce inelasticity in
the boundaries.

In the static elastic case, a particle moves in straight lines
between two collisions with the circles and, at each collision,
its energy and both the tangential and radial components of
the linear momentum are preserved. The motion is governed
by two nonlinear maps, one of them describes the dynamics
when the particle does not hit the inner circle, between two
consecutive collisions with the outer circle, and the other one
foresees the collision with the inner circle. For both situa-
tions, the angle of incidence that the particle makes with the
normal vector of any circle is the same as the angle that the
particle exits from the collision, characterizing specular re-
flections. The corresponding phase space is essentially deter-
mined by two angles defined at the instant of each collision
with the external circle. One of them is measured with re-
spect to the normal vector of the outer circle and the other
one is associated with the arc length counted from the ab-
cissa axis. The reader may find out the details of the corre-
sponding calculations in �15,16�.

For the elastic time-dependent case, the radii of the circles
are functions of time given by r�t�=r0+�r cos��t+�n� and
R�t�=1+�R cos�t+�n�, where r0 and 1 correspond to the
static radii of the inner and outer circles, respectively. The
quantities �n and �n are the oscillating phases of each
boundary in the nth iteration of the maps, �r and �R, are the
boundaries amplitudes of oscillation, and � is the ratio be-
tween the two frequencies. The regions �r0−�r ,r0+�r� and
�1−�R ,1+�R� define the collision zones for both the inner
and outer circles. When the particle attains a collision zone it*Corresponding author. regydio@rc.unesp.br

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may suffer a collision. Depending on the velocity of the par-
ticle and on the phase of the moving boundary, at the en-
trance of a collision zone, it may suffer many others succes-
sive collisions before leaving it. It is possible to look at the
problem of successive collisions through two versions, one
called complete which consider all possibilities of collisions
in the numeric calculations and another one, called simpli-
fied, which keeps the boundaries static but the particle
changes its momentum as if the boundaries were oscillating.
The computational calculations are faster in the simplified
version and it is expected that the results are essentially simi-
lar as the ones from the complete version, thus we will con-
sider the simplified version of the time-dependent dissipative
annular billiard model in this paper.

Due to the symmetry of the billiard, the momentum trans-
fer is similar to the central force problem, in such way that
only the radial component of the momentum is changed at
each collision. We will consider an index n in order to iden-
tify the nth collision of the particle with the outer circle so
that the corresponding values of the particle’s velocity are
given by

V�n+1�� = − �− eVn� + �1 + e�U� ,

V�n+1�� = Vn�,

where � and � denote the radial and tangential components,
respectively, U is the velocity of the boundary which may be

Ṙ�t� or ṙ�t� depending on which boundary the collision hap-
pened, and the parameter e� �0,1� is the restitution coeffi-
cient. For e=1 there is no dissipation, see Fig. 1�a�, but for
e�1 the areas on the phase space are not anymore conserved
and attracting fixed points emerge, see Figs. 2�a� and 2�b�.

III. RESULTS AND CONCLUSIONS

The inelastic time-dependent annular billiard has an ex-
pressive quantity of control parameters and after several
simulations we have chosen the following parameter set: r0
=0.4, �=0.5, �R=�r=0.01, and �n=�n. We emphasize that
this choice is representative for the annular billiard and that
we will consider only the concentric case because in this
nondissipative case it is well known that the particle has
limited velocities due to the existence of spanning curves in
the phase space �17,18�, which is convenient for the purpose
of this work. This characteristic allows us to perform a more
precise comparison among the scenarios determined by the
values of e. When the boundaries oscillate, there is another
phase plane defined by the pair of coordinates �� ,−V��
which is more appropriated to observe the behavior of the
particle velocities. Figure 1�a� shows a plot of 20 initial con-
ditions on the Poincaré section �� ,−V�� for the case of null
dissipation, e=1, and �� �0,4��, while Fig. 1�b� shows the
corresponding plot for the average velocity as a function of
the collision number n. In our simulations we have evolved
the maps up to 106 iterations and we observed that the aver-
age velocity saturates around �V��0.1. On the other hand, in
Fig. 2�a� we consider a great restitution coefficient e
=0.9999, characterizing a weakly dissipate system, which in

our opinion is the most typical in the nature, and we see that
the invariant tori which were above the value −V��0.25,
when e=1, now were all destroyed. The zoom in the neigh-
borhood of �� ,−V����2� ,0.18�, seen in Fig. 2�b�, shows
that the elliptic fixed point has bifurcated in a stable focus so
that the attractor is visible. We emphasize that as the dissi-
pation is weak enough, there is stickiness �see more about
stickiness in �19–22�� along the phase space and that many
initial conditions �IC� spend a long time orbiting the attract-
ing fixed point before being captured. We can see in Fig. 2�c�
that the average velocity has saturated around �V��0.17�
�1 /2��, thus leading to a gain around 	70% for the aver-
age velocity of the particle in comparison with the nondissi-
pative regime.

The calculation of the average velocities was developed in
fact through two averages. First we chose a line with k
=103 IC, in the bottom of the phase space, for −V�=0.02 and
�= �0,4��, so that for the jth IC we iterate the maps and we

(b)

(a)

FIG. 1. �Color online� �a� Phase space for the concentric pulsat-
ing annular billiard representing elastic collisions �e=1� of a par-
ticle with the boundaries. The parameters used were r0=0.4, �
=0.5, �r=�R=0.01, �0=�0=� /4, and v0=0.02. �b� Average veloc-
ity vs the number of collisions for the concentric pulsating elastic
case. The saturation occurs for �V��0.1.

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make an average along the orbit in terms of the number of
iterations: �Vj�= �1 /n�
i=1

n Vi,j up n=106 iterations. After re-
peating this first average for all IC, we construct an ensemble
of �Vj�, and from there we perform the average along the

ensemble: �V�= �1 /k�
 j=1
k �Vj�. This average velocity is plot-

ted in Figs. 1�b� and 2�c�. As this attractor occurs for a ve-
locity ��0.19 in Fig. 2�a�� above the value of the average
velocity for the nondissipative case ��V��0.1 in Fig. 1�b��,
the system experiences a gain on its average energy. So this
procedure constitutes a mechanism that may provide an in-
creasing on the average energy of any weakly dissipative
dynamical system, which should satisfy the dynamical con-
ditions described here.

We also point out that this scenario may be destructed if
the dissipation becomes stronger. In Fig. 3�a� we present four
plots for the average velocity of the particle as function of
the collision number, which show that it is possible to have
energy gain in a certain interval of the restitution coefficient.
For e=0.999 the average velocity saturates near the value of
the conservative case, �V�	0.1, for e=0.9999 and e
=0.99999, �V� attains higher values showing gain of energy
and for e=0.8 the dissipation is very strong and there is loss
of energy. In order to verify the dependence of the average
saturation velocity of the particle, �Vsat�, with respect to the
restitution coefficient e, we have performed the averages
cited above for 15 values of e, around the value of e for
which we have observed the energy gain. In Fig. 3�a� we

(b)

(a)

(c)

FIG. 2. �Color online� �a� Phase space for the concentric pulsat-
ing annular billiard representing inelastic collisions �e=0.9999� of a
particle with the boundaries. The parameters used were the same as
Fig. 1�a�. �b� Zoom in the neighborhood of the attractive fixed point
whose velocity is �0.19. �c� Average velocity vs the collision num-
ber for the pulsating inelastic case. The saturation occurs for �V�
�0.16, which is a higher value than the elastic pulsating case.

(b)

(a)

FIG. 3. �Color online� �a� Average velocity vs the collision num-
ber for four values of the restitution coefficient. �b� Average satu-
ration velocity of the particle as function of the restitution coeffi-
cient exhibiting a range of e where the particle can gain energy. A
nonlinear curve fits the data very well.

DISSIPATION AS A MECHANISM OF ENERGY GAIN PHYSICAL REVIEW E 77, 036204 �2008�

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show the behavior of �Vsat� in terms of e, where the squares
correspond to the numerical data that we have computed and
the continuous curve corresponds to a nonlinear fitting of the
type

�Vsat�e = �Vsat�e0 + a�1 − exp� e − e0

b

�P

exp� e − e0

c

 ,

where the parameters have assumed the values a=1.567, b
=2.634	10−2, c=1.6	10−4, P=0.472, e0=1 �conservative
case�, and 0.999
e
1. We observe that the system presents
a growth in its average energy until a pick around e
	0.99993, which corresponds to �V�	0.162, and from there
the system still gains energy but in a decreasing fashion up
e	0.999. For smaller values of the restitution coefficient, or
stronger dissipation, the behavior of the system does not
obey this fitting curve anymore and there is not any other
interval of e which presents energy gain.

We would like to emphasize that, based on our investiga-
tions, this mechanism of energy gain depends strongly on the
phase space architecture. We only have observed this energy
gain when �i� the resonance structures, which will generate
the attractors, have velocities higher than the saturation ve-
locity of the conservative case, �ii� the resonance structures

are immerse in the chaotic sea, and �iii� the phase space does
not present any spanning curve between this chaotic sea and
the velocities chosen as initial conditions. Based on these
observations we evaluate that our results are sufficiently ge-
neric to be observed in many other dynamical systems.

As concluding remarks, we point out that our choice of a
weakly dissipative system followed the ideas presented in
�23� where, among others, it is posed that in practice the
systems are more often neither strongly dissipative nor con-
servative. In this intermediary regime, the inelasticity on the
annular billiard boundaries may be considered as a small
perturbation which can be used to control chaos or to stabi-
lize regular dynamical behaviors �24�, besides providing a
gain in the average energy for the particle as reported in this
paper. In that sense, even though our approach had been
developed in a particular model, it constitutes a general
mechanism that can be used in different fields of science.

ACKNOWLEDGMENTS

The authors acknowledge fruitful discussions with E. D.
Leonel and also thank the Brazilian scientific agency
FAPESP for partial financial support through the process 06/
51653-0 and 06/56477-6.

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