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Resonant solutions in the presence of degeneracies for quasi-periodically perturbed systems
No full text2012
1. Abstract
1.1 Statement of the results
Let us consider the ordinary differential equation
˙
= !0(B) + "F(!t, ,B),
˙B
= "G(!t, ,B),
(1.1.1)
where ( ,B) ∈ T × B, with B an open subset of R, F,G : Td+1 × B → R and !0 : B → R are
real-analytic functions, ! ∈ Rd with d ≥ 2 and " is a (small) real parameter called the perturbation
parameter ; hence the perturbation (F,G) is quasi-periodic in t with frequency vector !. Without loss
of generality we can assume that ! has rationally independent components. Take the solution for the
unperturbed system given by ( (t),B(t)) = ( 0 + !0(B0)t,B0), with B0 such that !0(B0) is resonant
with !, i.e. such that there exists ( 0, ) ∈ Zd+1 for which !0(B0) 0 + ! · = 0. We want to study
whether for some value of 0, that is for a suitable choice of the initial phase, such a solution can be
continued under perturbation.
The resonance condition between !0(B0) and ! yields a “simple resonance” (or resonance of order
1) for the vector (!0(B0),!). The main assumptions on (1.1.1) are a Diophantine condition on the
frequency vector of the perturbation and a non-degeneracy condition on the unperturbed system. More
precisely we shall require that the vector (!0(B0),!) satisfies the condition
X
n≥0
1
2n log
inf
( 0, )∈Zd+1
( 0, )∦( 0, ),0<|( 0, )|≤2n
|!0(B0) 0 + ! · |
−1
< ∞ (1.1.2)
and that !′
0(B0) 6= 0.
Up to a linear change of coordinates, we can (and shall) assume !0(B0) = 0, so that the vector ,
such that !0(B0) 0 +! · = 0, must be the null vector. Therefore it is not restrictive to formulate the
assumptions on B0 and ! as follows.
1
Abstract
Hypothesis 1. !0(B0) = 0 and ! satisfies the Bryuno condition B(!) < ∞, where
B(!) =
X
n≥0
1
2n log
1
n(!)
, n(!) = inf
∈Zd
0<| |≤2n
|! · |. (1.1.3)
Hypothesis 2. For B0 as in Hypothesis 1 one has !′
0 (B0) 6= 0.
Note that if ! satisfies the standard Diophantine condition |! · | ≥
| |− for all ∈ Zd∗
, then it
also satisfies the Bryuno condition, since m(!) ≥
2−m in that case.
Let us write
F( , ,B) =
X
∈Zd
ei · F ( ,B), G( , ,B) =
X
∈Zd
ei · G ( ,B), (1.1.4)
and note that, since F and G are real-valued functions, one has
F− ( ,B) = F ( ,B)∗, G− ( ,B) = G ( ,B)∗. (1.1.5)
By analogy with the periodic case, the function (1)
0 ( ) := G0( ,B0) will be called the first order
Melnikov function.
We look for a quasi-periodic solution to (1.1.1) with frequency vector !, that is a solution of the
form ( (t),B(t)) = ( 0 + b(t),B0 + e B(t)), with
b(t) =
X
∈Zd
ei ·!tb , e B(t) =
X
∈Zd
ei ·!tB . (1.1.6)
Of course the existence of a quasi-periodic solution with frequency ! in the variables in which
!0(B0) = 0 implies the existence of a quasi-periodic solution with frequency resonant with ! in terms
of the original variables (that is, before performing the change of variables leading to !0(B0) = 0).
If we set (t) := !0(B(t)) + "F(!t, (t),B(t)) and (t) = "G(!t, (t),B(t)), and write
(t) =
X
∈Zd
ei ·!t , (t) =
X
∈Zd
ei ·!t , (1.1.7)
in Fourier space (1.1.1) becomes
(i! · )b = , 6= 0, (1.1.8a)
(i! · )B = , 6= 0, (1.1.8b)
0 = 0, (1.1.8c)
0 = 0. (1.1.8d)
2
1.1 Statement of the results
According to the usual terminology, we shall call (1.1.8a) and (1.1.8b) the range equations, while
(1.1.8c) and (1.1.8d) will be referred to as the bifurcation equations.
We start by looking for a formal solution ( (t),B(t)), with
(t) = (t; ", 0) = 0 +
X
k≥1
"kb(k)(t; 0) = 0 +
X
k≥1
"k
X
∈Zd
ei ·!tb(k)
( 0),
B(t) = B(t; ", 0) = B0 +
X
k≥1
"kB(k)(t; 0) = B0 +
X
k≥1
"k
X
∈Zd
ei ·!tB(k)
( 0)
(1.1.9)
and set U(t) := !0(B(t)) − !′
0 (B0)(B(t) − B0) and (t) = U(t) + "F(!t, (t),B(t)). Then define
recursively for k ≥ 1
b(k)
( 0) =
1
(i! · )
(k)
( 0) +
!′
0 (B0)
(i! · )2 (k)
( 0), 6= 0
B(k)
( 0) =
1
(i! · )
(k)
( 0), 6= 0
B(k)
0 ( 0) = −
1
!′
0(B0)
(k)
0 ( 0),
(1.1.10)
where we denoted (k)
( 0) = [G(!t, (t),B(t))](k−1)
and (k)
( 0) = [U(t)](k)
+ [F(!t, (t),B(t))](k−1)
,
with U(1)
( 0) = 0, so that (1)
( 0) = G ( 0,B0) and (1)
( 0) = F ( 0,B0), while, for k ≥ 2,
[U(t)](k)
=
X
s≥2
1
s!
@sB
!0(B0)
X
1+...+ s=
i∈Zd, i=1,...,s
X
k1+...+ks=k,
ki≥1
Ys
i=1
B(ki)
i ( 0), (1.1.11)
and
[P(!t, (t),B(t))](k−1)
=
X
s≥1
X
p+q=s
X
0+...+ s=
0, j∈Zd j=p+1,...,s
i∈Zd
, i=1,...,p
1
p!q!
@p
@q
BP 0( 0,B0) ×
×
X
k1+...+ks=k−1,
ki≥1
Yp
i=1
b(ki)
i ( 0)
Ys
i=p+1
B(ki)
i ( 0), P = F,G.
(1.1.12)
The series (1.1.9), with the coefficients defined as above and arbitrary 0, turn out to be a formal
solution of (1.1.8a)-(1.1.8c): the coefficients b(k)
( 0), B(k)
0 ( 0) and B(k)
( 0) are well defined for all
k ≥ 1 and all ∈ Zd∗
, and solve (1.1.8a)-(1.1.8c) order by order; moreover the functions b(k)(t; 0) and
B(k)(t; 0) are analytic and quasi-periodic in t with frequency vector !.
Note that if there exists k0 ≥ 1 such that (k)
0 ( 0) ≡ 0 for all k < k0, then the series (1.1.9) with
the coefficients b(k)
,B(k)
defined as in (1.1.10) solve the equations of motion up to order k0 − 1 and
moreover (k0)
0 is a well-defined function of 0.
3
Abstract
Assume first that the system (1.1.1) is Hamiltonian, i.e. there exists a function
H( , ,A,B) := ! ·A + h(B) + "f( , ,B), (1.1.13)
where ( , ) ∈ Td+1 and (A,B) ∈ Rd × B, with B an open subset of R, are canonically conjugate
(action-angle) variables and the functions f : Td+1 × B → R and h : B → R are real-analytic and
such that !0(B) = @1h(B), @Bf( , ,B) = F( , ,B) and −@ f( , ,B) = G( , ,B), so that the
corresponding Hamilton equations for the variables ( ,B) are given by
˙
= !0(B) + "@Bf(!t, ,B),
˙B
= −"@ f(!t, ,B),
(1.1.14)
which are exactly of the form (1.1.1).
Hypothesis 3. One has (k)
0 ( 0) := [−@ f( ,B)](k−1)
0 ≡ 0 for all k ≥ 1.
Then we shall prove the following result.
Theorem 1.1.1. Consider the system (1.1.14) and assume Hypotheses 1, 2 and 3. Then the series
(1.1.9) are convergent for " small enough.
Next we consider the more general system (1.1.1) and we assume that there exists k0 ∈ N such that
all functions (k)
0 ( 0) are identically zero for 0 ≤ k ≤ k0 −1, while (k0)
0 ( 0) is not identically vanishing.
Again, by analogy with the periodic case, we shall call the function (k0)
0 ( 0) the k0-th order Melnikov
function.
Hypothesis 4. There exist k0 ∈ N and 0 such that (k)
0 ( 0) vanish identically for k < k0 and 0 is a
zero of order ¯n for (k0)
0 ( 0), with ¯n odd. Moreover one has "k0!′
0(B0)@¯n
0
(k0)
0 ( 0) > 0.
Then we shall prove the following result.
Theorem 1.1.2. Consider the system (1.1.1) and assume Hypotheses 1, 2 and 4 to be satisfied. Then
for " small enough there exists at least one quasi-periodic solution ( (t),B(t)) with frequency vector !
such that ( (t),B(t)) → ( 0,B0) for " → 0.
1.2 Remarks about the results and sketch of their proofs
Quasi-periodic solutions to (1.1.14) with frequency vector ! describe lower-dimensional tori (ddimensional
tori for a system with d + 1 degrees of freedom). Such tori are parabolic in the sense that
the “normal frequency” vanishes for " = 0. Theorems 1.1.1 and 1.1.2 imply the following result.
4
1.2 Remarks about the results and sketch of their proofs
Theorem 1.2.1. Consider the system (1.1.14) and assume Hypotheses 1 and 2 to be satisfied. Then
for " small enough there exists at least one quasi-periodic solution ( (t),B(t)) with frequency vector !.
Proof. If all the coefficients (k)
0 = −[@ f](k−1)
0 vanish identically for all k ≥ 1 we simply apply Theorem
1.1.1. Otherwise there exists k0 ≥ 1 such that all the coefficients (k)
0 ( 0) vanish identically for all
k < k0 while (k0)
0 ( 0) is not identically zero and hence we can solve the equations of motion up to order
k0 without fixing the parameter 0. Moreover one has (k0)
0 ( 0) = @ 0g(k0)( 0) with
g(k0)( 0) := [B ˙b](k0)
0 − [h(B0 + B + B(k0))](k0)
0 − [f(!t, 0 + b,B0 + B)](k0−1)
0 ,
because, if we denote
b =
kX0−1
k=1
b(k), B =
kX0−1
k=1
B(k).
one has
@ 0 [f(!t, 0 + b,B0 + B)](k0−1)
0 = [@ f(!t, 0 + b,B0 + B)(1 + @ 0b)](k0−1)
0
+ [@Bf(!t, 0 + b,B0 + B)@ 0B](k0−1)
0
= − (k0)
0 − [ ˙B@ 0b](k0)
0 + [˙ b@ 0B](k0)
0
− [!0(B0 + B + B(k0))@ 0(B + B(k0))](k0)
0
= − (k0)
0 + @ 0[B ˙b](k0)
0 − @ 0 [h(B0 + B + B(k0))](k0)
0 .
Since g(k0) is analytic and periodic, and then it has at least a maximum ′
0 and a minimum ′′
0 . Then
Hypothesis 4 holds. Indeed, if "k0!′
0(B0) > 0 one can choose 0 = ′′
0 , while if "k0!′
0 (B0) < 0 one
can choose 0 = ′
0 and hence in both cases Hypothesis 4 is satisfied. Therefore the existence of a
quasi-periodic solution with frequency vector ! follows from Theorem 1.1.2.
Theorem 1.2.1 can be seen as the counterpart of Cheng’s result [1] in the case in which all “proper
frequencies” are fixed (isochronous case) and the perturbation does not depend on the actions conjugated
to the “fast angles” (otherwise one should add a correction like in [2]); moreover, with respect to [1], a
weaker Diophantine condition is assumed on the proper frequencies.
The proofs of Theorems 1.1.1 and 1.1.2 are organised as follows.
We first introduce a convenient graphical representation for the coefficents b(k)
( 0),B(k)
( 0) in
(1.1.10) and we shall use it in order to prove that they are well defined.
Then we shall see that, under Hypothesis 3 and if the system is Hamiltonian, there are some suitable
“cancellations” which will yield the convergence of the series (1.1.9), so that Theorem 1.1.1 will follow.
On the other hand we are not able to prove the same “cancellations” for the system (1.1.1) without
Hypothesis 3 and the assumption that the system is Hamiltonian. Hence, in order to prove Theorem
5
Abstract
1.1.2, besides the system (1.1.8) we shall consider first the system described by the range equations
(i! · )b = , 6= 0, (1.2.1a)
(i! · )B = , 6= 0, (1.2.1b)
i.e. with no condition for = 0, and we shall prove that, if some further conditions (to be specified
later on) are found to be satisfied, it is possible to find, for " small enough and arbitrary 0,B0, a
solution
( 0 + b(t),B0 + e B(t)), (1.2.2)
to the system (1.2.1), with b(t) and e B(t) as in (1.1.6) depending on the free parameters ", 0,B0; such
a solution is obtained via a ‘resummation procedure’, starting from the formal solution of the range
equations (1.2.1). The conditions mentioned above can be illustrated as follows. The resummation
procedure turns out to be well-defined if the small divisors of the resummed series can be bounded
proportionally to the square of the small divisors of the formal series. However, it is not obvious at all
that this is possible, since the latter are of the form (i! · )−1 with ∈ Zd∗
, while the small divisors
of the resummed series are of the form (det((i! · )1 −M[n](! · ; ", 0,B0)))−1, for suitable 2 × 2
matrices M[n] The bound on the small divisors of the resummed series is difficult to check without
assuming any non-degeneracy condition on the perturbation. Therefore we replace M[n](x; ", 0,B0)
with M[n](x; ", 0,B0) n(det(M[n](0; ", 0,B0))), for suitable ‘cut-off functions’ n, in such a way that
the bound automatically holds. The introduction of the cut-offs changes the series in such a way that if
on the one hand the modified series are well-defined, on the other hand in principle they no longer solve
the range equations: this turns out to be the case only if one can prove that the cut-offs can be removed.
So, the last part of the proof consists in showing that, by suitably choosing the parameters 0,B0 as
continuous functions of ", this occurs and moreover, for the same choice of 0,B0, the bifurcation
equations (1.1.8c) and (1.1.8d) hold; hence for such 0,B0, the function (1.2.2) is a solution of the whole
system (1.1.1)
Modello di scelta per la distribuzione delle residenze per la pianificazione della mobilità urbana
No full textIn this scientific research, in the land use and transport interaction
framework, the type of link (a correlation or causality) between the built
environment and the travel behavior has been investigated: the
relationship between neighborhood characteristics and travel behavior is
taken into account to understand if attitudes and neighborhood
preferences influence the residential location choices and travel behavior.
It could be useful to examine, whether neighborhood design influences
travel behavior or whether travel preferences influence the choice of
neighborhood. This could lead to better understand the effect of the
transport policies. A lot of studies (Hansen 1959, Lowry 1964, Nuzzolo
2006, Cao, Mokhtarian and Handy 2008) have verified that a suitable
transport planning leads to an urban economics development wellmatched
with the land use aims (e.g. Road Pricing, Congestion Charge
and Eco-pass). The literature in transport research underlines the tools to
catch the energetic sustainable town; these tools can be divided into 2
different objectives:
1) minimize the total travel time on the road network (in other terms
shorten home-work distance)
2) maximize the modal split on behalf of public transport.
In literature it is still not fully known which are the attitudes for the
residential location choice (R.L.C.) and if between these attitudes even
the travel preferences are included, and how this influences the R.L.C.. To
investigate if there is a casuality or an effect between the two choices
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multinomial logit model in the Rome metropolitan region has been
conducted. In the research, after a literature review of the land use
transport models, the Experimental Design Theory will be presented and
the main effect of a good survey will de discussed. It will be shown how
to investigate the problem and which are the main attitudes in travel
behavior and residential location choice in Rome. The Fig.1 marks the
sub-area of the land use and transport system and how they are in
connection. The cities in Italy are changing according to the decentralization criteria even for the household economic constraint, assuming a feature as different as possible from the old compact urban centre; new roads grow up; the buildings and the blocks are divided by streets of more than two lanes per direction and this gives an increase of the total travel time on the network for the home-work transfer. It’s easy to prove that the residential and activities replacing follows the accessibility increase or decrease. The aim of the paper is to individuate the main attributes of the utility function of the R.L.C. to evaluate the different transport policies measure to build the town to serve with an optimal transport solution. In order to obtain a result it will be necessary to make a survey on the population with an optimal experimental design - efficient designs - in order to understand the behavior of the 3 millions Roman residents with the minimum sample and with the minimum error.
Through the survey it is possible to explore the connections among residential accessibility, employment, income, and auto ownership and it will be possible to focus the attention on the potential discrete choice model which will be able to predict the residential choice function of the different transport policies. A first survey has been made and the first results are useful to calibrate a choice tasks survey through the Experimental Design Theory. Later through the “Biogeme” software, the mixed utility model parameters are calibrated and with the statistical test the validity of the results are estimated. A distribution of the parameters after the calibration with biogeme is shown. For the parameters calibration even the normal, log-normal and Johnson distribution for the latent class are taken into account in order to view the differences between the outputs and underline which distribution better represents the real R.L.C., even trying to build a new Model
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No full textKafaalo Magazine waa majallad sannadeed oo xubin rasmi ah ay ka tahay Kafaalo Academy, oo ah iskuullo Soomaaliyeed oo lagu taageero kobcinta aqoonta, waxayna majalladu ka soo baxdaa dalka Norway. Mawaadiicdeedu ugu doorka rooni waxay ka hadlayaana: saynis, tarbiyad, waxbarasho, arrimaha bulshada, taariikhda, diinta iyo dhaqaalaha oo waxtar weyn u leh bulshada Soomaaliyeed. Majaladdan waxaa loogu talaggalay dadka akhriya af Soomaaliga, wuxuuna ka hadlaa mawduucyo wada taabanaya qofka noloshiisa, haddana kobcinaya qaab fekerkiisa iyo aqoontiisa.Kafaalo Magazine is primarily designed to serve as an overview on research for Somalis worldwide, to keep them informed on their economical and social issues. It is well known in different countries with analytical results and recommendations. The KM is also an educational, scientific and innovation magazine aiming to enhance professional and educational aspirations amongst Somalis. It promotes ideas and experiences from the work of educational professionals, as well as from academic journals and business innovations.Kafaalo Magazine è destinato principalmente a fornire una visione d'insieme sulla ricerca per i somali in tutto il mondo, al fine di tenere aggiornati gli utenti sugli aspetti economici e sociali. E' conosciuta in diversi paesi come testimoniano risultati analitici e raccomandazioni. Il KM è anche una rivista educativa, scientifica e sull'innovazione che ha lo scopo di stimolare aspirazioni formative e professionali tra i somali. Promuove idee ed esperienze dal lavoro dei professionisti, così come da riviste accademiche e innovazioni aziendali
Effectiveness Review: Somalia Drought Response 2011/12
No full textThis report presents the findings of an effectiveness review carried out on Oxfam's response to the 2010/11 drought-related food crisis in Somalia.Warbixintani waxay muujinaysa natiijadii falanqaynta hawlihii ay qabatay Oxfam, oo ka jawaabaysay nafaqadarradi ka timid abaarihii ku habsaday Soomaaliya 2010/11.Questo rapporto presenta i risultati di un'analisi dell'efficacia dell'azione intrapresa da Oxfam come risposta alla crisi alimentare avvenuta in Somalia in seguito alla siccità del 2010/11
Introduction / Introduzione
No full textMartin Bertram, Introduction / Introduzione al volume "Decretales pictae. Le miniature nei manoscritti delle Decretali di Gregorio IX (Liber Extra)". Atti del colloquio internazionale tenuto all'Istituto Storico Germanico, Roma 3-4 marzo 2010. Indici compilati da Marta Pavón Ramírez. Università degli Studi Roma Tre, Roma 201
Indici dei manoscritti: II. Altri manoscritti citati e/o riprodotti
No full textMarta Pavón Ramírez, Indici dei manoscritti: II. Altri manoscritti citati e/o riprodotti, in "Decretales pictae. Le miniature nei manoscritti delle Decretali di Gregorio IX (Liber Extra)". Atti del colloquio internazionale tenuto all'Istituto Storico Germanico, Roma 3-4 marzo 2010. Indici compilati da Marta Pavón Ramírez. Università degli Studi Roma Tre, Roma 201
Immagini: II. Miniature sistemate secondo i libri I-V delle Decretali, le Novellae e le Novissimae (Fig. 151-289)
No full textSilvia Di Paolo, Immagini: II. Miniature sistemate secondo i libri I-V delle Decretali, le Novellae e le Novissimae (Fig. 151-289), in "Decretales pictae. Le miniature nei manoscritti delle Decretali di Gregorio IX (Liber Extra)". Atti del colloquio internazionale tenuto all'Istituto Storico Germanico, Roma 3-4 marzo 2010. Indici compilati da Marta Pavón Ramírez. Università degli Studi Roma Tre, Roma 201