The Feigenbaum's $δ$ for a high dissipative bouncing ball model

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by  Diego F. M. Oliveira

The Feigenbaum’s δ for a high dissipative bouncing ball model
Diego F. M. Oliveira∗ and Edson D. Leonel
Departamento de Estat´ıstica, Matema´tica Aplicada e Computac¸˜ao,
Universidade Estadual Paulista, CEP, 13506-900, Rio Claro, SP, Brazil
(Dated: January 25, 2011)
Wehavestudiedadissipativeversionofaone-dimensionalFermiacceleratormodel. Thedynamics
of the model is described in terms of a two-dimensional, nonlinear area-contracting map. The
dissipation is introduced via innelastic collisions of the particle with the walls and we consider the
dynamics in the regime of high dissipation. For such a regime, the model exhibitsa route to chaos
knownasperioddoublingandweobtainaconstantalongthebifurcationssocalledtheFeigenbaum’s
numberδ.
1
PACSnumbers: 05.45.Pq,05.45.Ac
1 Keywords: BouncingBallModel,Dissipation,LyapunovExponent, Feigenbaum number
0
2
n I. INTRODUCTION chaos. This route shows a sequence of doubling bifur-
a cationsconnectingregular-periodictochaoticbehaviour.
J Themaingoalofthis paperisthentoexamineandshow
The origin of cosmic radiation and a mechanism
4 that the Fermi-Ulam model obey the same convergence
trough which it could acquire enormous energies has in-
2 ratioasthatobtainedbyFeigenbaum[21]forthedynam-
trigued many physicists and mathematicians along the
ics of the logistic map. Such a constant value has been
] last decades. In particular, in the year of 1949 En-
D further called as “The Feigenbaum’s δ”.
rico Fermi [1] proposed a very simple model that qual-
The organisation of this paper is as follows: In sec-
C itatively describes a process in which charged particles
tionII wepresentallthedetails neededforthe construc-
. werebouncedviainteractionwithmovingmagneticfields.
n tion of the nonlinear mapping. Our numerical results
Thisheuristicideawasthenlatermodifiedto encompass
i are discussedin section III and our conclusions andfinal
l in a suitable model that could give quantitative results
n remarks are drawn in section IV.
[ ontheoriginalFermi’smodel. Afterthat,manydifferent
versionsofthe modelwere proposed[2–9] andstudiedin
1
different context and considering many approaches and
v
modifications [10–17]. One of them is the well known II. A DISSIPATIVE FERMI-ULAM MODEL
6
Fermi-Ulammodel(FUM).Suchmodelconsistsofaclas-
9
5 sical point-like particle moving between two walls in the Let us describe the model and obtain the equations of
4 total absence of any external field. One of the walls is themapping. Themodelwearedealingwithconsistsofa
1. consideredto be fixed while the other one moves period- classical point-like particle confined in and bouncing be-
0 ically in time. Despite this simplicity, the structure of twentworigidwalls. Oneofthemis assumedtobe fixed
1 the phase space is rathercomplex and it includes a large atthepositionx=landtheotheronemovesperiodically
1 chaoticseasurroundingKAMislandsandasetofinvari- in time according to x (t) = εcos(ωt). Thus, the mov-
w
v: ant spanning curves limiting the energy of a bouncing ing wallvelocityis givenbyvw(t)=−εωsin(ωt) whereε
i particle. However, the introduction of innelastic colli- denotes the amplitude ofoscillationandω is the angular
X
sions on this model is enough to destroy such a mixed frequency of the moving wall, respectively. Aditionally,
r structure and the system exhibits attractors. Depend- the motion of the particle does not suffer influence of
a
ing on the initial conditions and controlparameters,one anyexternalfield. Weassumethatallthe collisionswith
can observe a chaotic attractor characterised by a posi- both walls are inelastic. Thus, we introduce a restitu-
tiveLyapunovexponent. Byasuitablecontrolparameter tion coefficient for the fixed wall as α ∈ [0,1] while for
variation, the chaotic attractor might be destroyedvia a the moving wall we consider β ∈ [0,1]. The completely
crisis event [18–20]. After the destruction, the chaotic inelastic collision happens when α = β = 0 and that a
attractor is replaced by a chaotic transient [19]. single collision is enough to terminate all the particle’s
In this paper, we revisit a dissipative version of the dynamics. On the other hand, when α = β = 1, corre-
Fermi-Ulam model seeking to understand and describe spondingtoacompleteelasticcollision,alltheresultsfor
the behaviour of the dynamics for the regime of high the nondissipative case are recovered[17].
dissipation. Moreover, we investigate a phenomenon We describe the dynamics of the particle using a map
quite common observed for a large variety of dissipative T where the dynamical variables are (vn,tn) and vn is
andnonlinear systems knownas period doubling route to the velocity of the particle at the instant tn. The in-
dex n denotes the nth collision of the particle with the
moving wall. Starting then with an initial condition
(v ,t ), with initial position of the particles given by
n n
∗Electronicaddress: [email protected] xp(t) = εcos(ωtn) with vn > 0, the dynamics is evolved
2
and the map T gives a new pair of (vn+1,tn+1) at the III. NUMERICAL RESULTS
(n+1)th collision. It is important to say that we have
threecontrolparameters,namely: ε,landωandthatthe
In this section we discuss some numerical results con-
dynamicsdoesnotdependonallofthem. Thusitiscon-
sidering the case of high dissipation. We call as high
venient to define dimensionless and more appropriated
dissipation the situation in which a particle loses more
variables. Therefore, we define ǫ = ε/l, V = v /ωl and
n n than 50% of its energy upon colision with the moving
measure the time in terms of the number of oscillations
wall. Moreover, we have considered as fixed the values
of the moving wall, consequently φ = ωt . Incorporat-
n n of the damping coefficients α = 0.85 and β = 0.50. To
ing this set of new variables into the model, the map T
illustrate that the dynamics of the system exhibits dou-
is written as
bling bifurcation cascade, it is shown in Fig. 1(a) the
T : Vn+1 =Vn∗−(1+β)ǫsin(φn+1) , (1) bcoenhtarvoiloupraroafmtheteerasǫy,mwphteorteictvheelosceiqtuyepnlcoettoefdbaigfuaricnasttiothnes
(cid:26)φn+1 =φn+∆Tn mod(2π)
is evident. A similar sequence is also observed for the
where the expressions for both V∗ and ∆T depend on asymptotic variable φ, as it is shown in Fig. 1(b). Note
n n
what kind of collision occurs. There are two different that all the bifurcations of same period in (a) and (b)
possible situations, namely: (i) multiple collisions with happen for the same value of the parameter ǫ. At the
themovingwalland,(ii)singlecollisionwiththemoving point of bifurcations, one can also observe that the posi-
wall. Consideringthefirstcase,theexpressionsareV∗ = tive Lyapunov exponent shows null value, as it is shown
n
−βV and∆T =φ ,whereφ isobtainedasthesmallest in Fig. 1(c). It is well known that the Lyapunov ex-
n n c c
solution of G(φ ) = 0 with φ ∈ (0,2π]. A solution for ponents are important tool that can be used to classify
c c
G(φ ) = 0 is equivalent to have that the position of the
c
particle is the same as that of the moving wall at the
instant of the impact. The function G(φ ) is given by
c
G(φ )=ǫcos(φ +φ )−ǫcos(φ )−V φ . (2)
c n c n n c
Ifthe functionG(φ )doesnothavearootinthe interval
c
φ ∈(0,2π], we can conclude that the particle leaves the
c
collisionzone(thecollisionzoneisdefinedastheinterval
x∈[−ǫ,ǫ]) without suffering a successive collision.
Considering now the case (ii), i.e. the case where the
particle leaves the collision zone, the corresponding ex-
pressionsareV∗ =−αβV and∆T =φ +φ +φ,with
n n n c r l
the auxiliary terms given by
1−ǫcos(φ ) 1−ǫ
n
φ = , φ = . (3)
r l
V αV
n n
Finally, the term φ is obtained as the smallest solution
c
of F(φ )=0 for φ ∈[0,2π). The expression of F(φ ) is
c c c
given by
F(φ )=ǫcos(φ +φ +φ +φ )−ǫ+αV φ . (4)
c n r l c n c
Both, Eq. (2) and Eq. (4) must be solved numerically.
After some algebra it is easy to shown that the deter-
minant of the Jacobian Matrix for the case (i) is given
by
V +ǫsin(φ )
det(J)=β2 n n , (5)
(cid:20)Vn+1+ǫsin(φn+1)(cid:21)
while for the case (ii) we obtain
V +ǫsin(φ )
det(J)=α2β2 n n . (6)
(cid:20)Vn+1+ǫsin(φn+1)(cid:21) FIG.1: Bifurcation cascadefor(a) φand(b)V bothplotted
against ǫ. In (c) it is shown the Lyapunov exponent associ-
Wethereforecanconcludebasedontheaboveresultthat ated to (a) and (b). The damping coefficients used for the
areapreservationisobtainedonlyforthecaseofα=β = constructionofthefigures(a),(b)and(c)wereα=0.85and
1. β=0.50.
3
orbits as chaotic. As discussed in [22], the Lyapunov happen at the same rate of convergence for the bifurca-
exponents are defined as tiondiagramchangingfromperiodictochaoticbehavior.
The procedure used to obtain the Feigenbaum value δ
1
λ = lim ln|Λ | , j =1,2 , (7) consists of: let ǫ1 represents the controlparameter value
j j
n→∞n atwhichperiod-1givesbirthtoaperiod-2orbit,ǫ2 isthe
value where period-2 changes to period-4 and so on. In
n
where Λ arethe eigenvaluesofM = J (V ,φ ) and
j i=1 i i i general the parameter ǫ corresponds to the control pa-
JiistheJacobianmatrixevaluatedoveQrtheorbit(Vi,φi). rameter value at which na period-2n orbit is born. Thus,
However,adirectimplementationofa computationalal-
we write the Feigenbaum’s δ as
gorithm to evaluate Eq. (7) has a severe limitation to
obtain M. Even in the limit of short n, the compo-
ǫn−ǫn−1
nents of M can assume very different orders of magni- δn = lim . (9)
tude for chaotic orbits and periodic attractors, yielding n≫1ǫn+1−ǫn
impracticable the implementation of the algorithm. In
The numerical value for the constant number δ ob-
ordertoavoidsuchproblem,wenotethatJ canbe writ-
tainedbyEq. (9)andconsideringasufficientlargevalues
ten as J = ΘT where Θ is an ortoghonal matrix and
for n is δ = 4.66920161.... Considering the numerical
T is a right up triangular matrix. Thus we rewrite M
apesaasrmosMydeutpco=trosoJhcfenodJwJu2nΘr−teh11ac.tda.ne.MfiJbn2Θee=s1uΘasJen−1ndJ1enJtw−o1,1Jo.w2b′..th.aeJIirnn3eΘaTT22Θ1n=e−2=x1tΘJΘs2′−2t−1T1e11JpJ.2,′1iT.atnhAides dtmusiialonaotnctdae,ieoolttnbhhsisteeaoδniFnnuele=ymidgeet4thrnh.ie6bcr4aaob(ulu4imfrg)ueh.’srsuctWahlδtteseiooLabnhrytsaeaavopivenfueercnfdyooounvhfrsoatierdhrxdeptrthoetoeondeeFbiningeetrhsmootcbhuiat-rlUaocsirulnidamlemaedr--
so on. Using this procedure, the problem is reduced to
for higher orders in the bifurcations. Our result is in a
evaluate the diagonal elements of T : Ti ,Ti . Finally,
i 11 22 good agreement with the Feigenbaum’s universal δ.
the Lyapunov exponents are given by
n
1
λ = lim ln|Ti | , j =1,2 . (8)
j n→∞n jj IV. CONCLUSION
Xi=1
If at least one of the λj is positive then the orbit is As a summary, we have studied a dissipative version
classified as chaotic. We can see in Fig. (1) (c) the be- of the Fermi-Ulam model. We introduce dissipation into
havioroftheLyapunovexponentscorrespondingtoboth the model through inelastic collisions with both walls.
Figs. (1)(a,b). It is also easy to see that when the bifur- We have shown that for regimes of high dissipation, the
cations happen, the exponent λ assumes the zero value model exhibits a sequence of doubling bifurcation cas-
at same values of the control parameter ǫ where the bi- cade. Forthis cascade,we obtained the socalled Feigen-
furcation hold. Another interesting observation is that baum’s number as δ =4.64(4).
the Lyapunovexponent, insome regions,assumesa con-
stant and negative values, such behavior occurs because
the eigenvalues of the Jacobian Matrix become complex
Acknowledgements
numbers (imaginary numbers)).
Let us now use the points where the Lyapunov expo-
nent assumes the null value to characterise a very inter- DFMO gratefully acknowledges Conselho Nacional de
estingpropertyofthemodel. AsitwasshownbyFeigen- Desenvolvimento Cient´ıfico e Tecnol´ogico – CNPq. EDL
baum [21], along the bifurcations the dynamics exhibits isgratefultoFAPESP,CNPqandFUNDUNESP,Brazil-
an universal feature. It implies that all the bifurcations ian agencies.
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